Bargaining with Arriving Sellers Dongkyu Changú December 15, 2014

Job Market Paper The latest version can be found here. Abstract This paper investigates the role of buyer search in sequential bargaining with incomplete information. A market is populated by homogenous sellers holding one unit of an identical non-divisible good. The buyer continues to search for new sellers, even after a seller has arrived and begun to make price offers. Contact with a new seller randomly occurs in the middle of a negotiation, and the buyer has an option to switch to the new seller, which servers as the buyer’s endogenously generated outside option. This paper focuses on equilibria in which the sellers’ offer strategies are symmetric and monotone in their beliefs over the buyer’s reservation value. The main finding is that, if the arrival rate of new sellers is high enough, the buyer reaches an immediate agreement with the first seller in the continuous-time limit. The seller’s introductory price approaches the competitive price and the outcome is efficient. If the buyer has outside options of exogenously fixed values, on the other hand, an equilibrium with a high introductory price and a delay in agreement may exist even if the arrival rate of outside options is high. The contrast between endogenous and exogenous outside options shows that the effect of outside options on bilateral negotiations hinges on how outside options are generated. Keywords: bargaining, asymmetric information, endogenous outside option, Coase conjecture

Yale University (email: [email protected]). I am indebted to my advisors Johannes Hörner and Larry Samuelson for their continuous encouragement, support, and guidance on this project. This paper is also benefited from invaluable comments by Eduardo Faingold. I would also like to thank Dirk Bergemann, Vitor Farinha Luz, Sander Heinsalu, Yuhta Ishii, Aniko Öery, Ennio Stacchetti, and the seminar audiences at Yale University for their insightful comments. ú

Table of Contents 1 Introduction 1.1 Motivation and Preview of Results . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Related Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2

2 A Model for Bargaining with Arriving Sellers 2.1 Extensive Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Monotone Anonymous Perfect Bayesian Equilibrium . . . . . . . . . . . . . .

4 4 5

3 Preliminary Observations

11

4 Coase Conjecture for Bargaining with Arriving Sellers 4.1 Coase Conjecture . . . . . . . . . . . . . . . . . . . . . . . 4.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Proof of Main Result . . . . . . . . . . . . . . . . . . . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

15 15 17 21 22

5 Contrast to Bargaining with Exogenous Outside Options

28

6 Concluding Remarks

34

References

35

Appendix A Foundational Matching Mechanism

37

Appendix B B.1 Proof B.2 Proof B.3 Proof B.4 Proof B.5 Proof B.6 Proof B.7 Proof B.8 Proof B.9 Proof

38 38 39 39 40 41 46 47 49 66

Proofs of Lemma 1 . . of Lemma 2 . . of Lemma 3 . . of Lemma 5 . . of Lemma 10 . . of Lemma 11 . . of Proposition 2 of Proposition 3 of Proposition 1

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

1 1.1

Introduction Motivation and Preview of Results

Negotiations between buyers and sellers often determine the outcome of trade. Examples include bargaining with new-car dealerships, negotiations in housing markets, and businessto-business negotiations. Buyers typically face multiple potential sellers and continue to search for new sellers while negotiating with an existing seller. Sellers are engaged sequentially and buyers sometimes walk away to a new seller in the middle of negotiations. For example, buyers in the new-vehicle market occasionally break down negotiations with an existing dealership and shift to a new dealership. An interesting interplay emerges between the walk-away option of buyers and the bargaining outcome when buyers can endogenously choose their bargaining counterparties. For instance, some new-car buyers have higher willingness-to-pay than others, and tend to purchase a car from the first dealership they contact. On the other hand, buyers with lower willingness-to-pay are ready to insist on low prices, and they even break down negotiations if necessary. Dealerships’ bargaining strategies reflect their belief over the ratio between the two buyer types and each type’s search behavior. At the same time, each type buyer’s search can be either encouraged or discouraged by dealerships’ choice of their bargaining strategies. This paper investigates the role of buyer search in the process of sequential bargaining with incomplete information. The paper introduces and analyzes a new bargaining model in which a buyer can search for sellers during negotiations. A market is populated by homogenous sellers holding one unit of an identical non-divisible good. There is a buyer whose valuation of the good is private information. The buyer continues to search for a new seller, even after a seller has arrived and begun to make price offers. Contact with a new seller randomly occurs in the middle of a negotiation, and the buyer either switches to the new seller or stays with the original one. The opportunity to switch to future sellers serves as the buyer’s endogenously generated outside option. The aim of this paper is to raise and answer the following questions. At which price is the good traded, and how is that price reached in negotiations? Does the buyer keep floating across sellers or haggling over prices for an extended period of time? These questions amount to asking whether the Coase conjecture holds. The Coase conjecture, which has been developed in the context of exclusive bargaining situations, states that the seller’s introductory price level converges to the lowest possible buyer valuation as the time interval between successive offers vanishes. The good is immediately traded at the competitive price level, and hence the outcome is efficient. The natural question whether the Coase conjecture continues to hold in the richer environment is considered in this paper. The model generally admits multiple equilibria, and the paper focuses on equilibria in which the sellers’ offer strategies are symmetric and monotone in their beliefs over the buyer’s reservation value, with additional stationary conditions also being imposed. The main finding is that the buyer reaches an immediate agreement with the first seller in the continuous-time 1

limit, if the arrival rate of new sellers is high enough. The buyer does not haggle on a price and also does not wait for another seller. The introductory price offer of each seller converges to the competitive price, and hence the Coase conjecture holds. One may not find this efficiency result surprising. A buyer’s outside option seemingly increases competitive pressure. This pressure would push sellers to trade earlier at a lower price, whether the value of outside options is determined endogenously or exogenously. Nevertheless, Board and Pycia (2014) and Hwang (2013) show that the Coase conjecture may fail with outside options of exogenously fixed values. With exogenous outside options of the buyer, the seller can insist on a high price that only targets a high-valuation buyer, under the expectation that a low-valuation buyer already has exercised an outside option. The low-valuation buyer indeed finds it is the best response to exercise an outside option as soon as possible, and hence the seller’s expectation is fulfilled in equilibrium. The introductory price only targets the high-valuation buyer, and hence it is higher than the competitive price level. Consequently, the Coase conjecture fails in such an equilibrium. The Coase conjecture survives in this paper’s richer model where the buyer’s outside options are endogenously generated. Suppose that, as in the inefficient equilibrium for bargaining games with exogenous outside options, sellers target a buyer with a high valuation while a buyer with a low valuation keeps switching across sellers. The high-valuation buyer continues to look for new sellers only if no seller has arrived to the buyer. The likelihood of such event is small if the arrival rate of new sellers is high, and in this case sellers would believe that the buyer in active search is likely to have the low valuation. Hence, it is actually sellers’ profitable deviation to offer a low price immediately. The paper’s result demonstrates that the effectiveness of outside options in bilateral negotiations hinges on how those outside options are generated. The Coase conjecture may fail if outside options are given exogenously, so that their values are independent of decisions by sellers and buyers. On the other hand, if outside options reflect a buyer’s opportunity to switch sellers, and hence their values are endogenously determined in equilibrium, the Coase conjecture survives. This paper also explains how some decentralized markets can function well even when asymmetric information is present. Neither a sophisticated mechanism nor intensive monitoring is required to achieve the efficient outcome. It also suggests that the simple static model of perfect competition is a good approximation for a market where the actual bargaining procedure is complicated as in this paper’s model.

1.2

Related Literature

This paper contributes to the literature on sequential bargaining with incomplete information. The primary question in this literature is when and how a delay of agreement arises. Gul, Sonnenschein, and Wilson (1986) show that in the seller-offer bargaining model with incomplete information on the buyer’s valuation, there is no delay in equilibrium when the gain from trading is common knowledge. Cho (1990) obtains a similar result with two-sided asymmetric information where both seller’s cost and buyer’s valuation are each player’s private 2

information. However, Ausubel and Deneckere (1989) show that inefficient delay can arise in a nonstationary equilibrium if the gain from trading is not common knowledge. Evans (1989), Vincent (1989), and Deneckere and Liang (2006) consider a model with interdependent valuations and show a delay or even a bargaining impasse may occur when a lemons problem is present. On the other hand, Abreu and Gul (2000) explore the reputational bargaining model in which each player is possibly an irrationally obstinate type who demands the same amount indefinitely, and show that delay may occur in this case. Most papers on the incomplete information bargaining problem do not model outside options explicitly. A few exceptions include Board and Pycia (2014) and Hwang (2013). The common message from these two papers is that the Coase conjecture may fail in incomplete information bargaining problems with exogenously given outside options. In the context of a reputational bargaining problem, Compte and Jehiel (2002) show that exogenously given outside options may cancel out the effect of obstinacy in bargaining. There are several other papers that consider incomplete information bargaining games where an opportunity to interact with outside players generates an endogenous outside option. Fuchs and Skrzypacz (2010) study asymmetric information bargaining models with the random arrival of new traders. In their paper, the buyer and the seller are forced to play a new game (for example, English auction or Bertrand competition) with new traders once they arrive. The final payoff of the seller conditional upon the arrival of new traders depends on the buyer’s valuation of the good. This makes the buyer’s valuation and the seller’s value of waiting interdependent, which results in a delay of agreement. In the current paper’s model, on the other hand, the seller’s payoff conditional on the buyer’s switching to a new seller is assumed to be constant, and hence such an interdependency is not present Fudenberg, Levine, and Tirole (1987) analyze a bargaining model between one seller and multiple buyers. Their model is different from the model considered in the current paper in that it is the seller who can choose whom to negotiate with in their model. Moreover, they make an assumption that the arrival process of a new buyer is deterministic. Under these assumptions they show that the Coase conjecture may fail. Inderst (2008) studies a similar model, except that he assumes a new buyer arrives according to a Poisson process. His main finding is that the Coase conjecture holds as long as we focus on stationary equilibrium. Conlisk, Gerstner, and Sobel (1984) and Sobel (1991) consider the durable good monopoly problem, which is closely related to the incomplete information bargaining problem, where a new cohort of consumers enter in each period. In these papers’ models, the seller can serve both existing consumers and new consumers at the same time, without switching from one to another. Some papers in the bargaining literature also have studied the problem of designing efficient bargaining mechanisms. A non-exclusive list includes Myerson and Satterthwaite (1983) and Cramton, Gibbons, and Klemperer (1987). Ex ante efficient bargaining mechanisms often require the bargainers’ ability to commit to leave the bargaining table, even when it is 3

common knowledge that the gain from trading is positive. On the other hand, most sequential bargaining games presented in the literature assume that bargainers must continue to negotiate so long as they expect positive gains from continuing. The current paper considers an intermediate case where a bargaining breakdown occurs endogenously, when the buyer switches to a new seller. Finally, this paper is also related to recent works on dynamic trading with asymmetric information such as Hörner and Vieille (2009), Daley and Green (2012), Lauermann and Wolinsky (2013), and Kim (2014) among many others. These papers also consider models where an agent faces a sequence of potential trade counterparties. However, agents are not allowed to revise their offers in these papers. Once the first offer is declined by the opposite party, negotiations break down immediately, and both parties search for new trade counterparties. On the other hand, the current paper allows a seller to keep revising offers as long as the buyer does not breakdown negotiation. The paper is organized in the following order. Section 2 introduces a formal model for bargaining with multiple arriving sellers and the equilibrium refinement, monotone anonymous perfect Bayesian equilibria. Section 3 provides some preliminary results, and then the paper’s main result is presented and discussed in Section 4. In Section 5, this main result is contrasted with the outcome in bargaining with exogenous outside options. Section 6 concludes the paper by discussing implications of the main result and potential directions of the future research. Most of the proofs can be found in the Appendix.

2

A Model for Bargaining with Arriving Sellers

2.1

Extensive Form

This section introduces a discrete-time infinite-horizon bargaining game between a buyer and arriving potential sellers. The buyer’s valuation of the good, v, is randomly determined by nature at the beginning of the game. Suppose v = vH (high type buyer) with probability qˆ > 0 and v = vL (low type buyer) with the complementary probability, where vH > vL > 0. Nature’s choice of v is the buyer’s private information while qˆ is common knowledge. There are a continuum of sellers indexed by j œ [0, 1] and each seller holds one-unit of the good that the buyer demands.1 Suppose each seller’s valuation of the good is zero. There is a gap between the sellers’ cost (zero) and the lowest possible buyer valuations (vL > 0), and hence it is common knowledge that the gain from trading is always strictly positive. The efficient outcome is such that the good is traded immediately.2 Each period is indexed by n = 1, 2, . . . The time interval between two consecutive periods is denoted > 0. There are two substages in each period, the arrival stage and then the 1

The assumption of a continuum of sellers is required only to guarantee that a uniform distribution is well-defined over the population of sellers. 2 With such a gap, the standard bargaining model with a single buyer and a single seller admits a unique equilibrium for which the Coase conjecture holds (Gul, Sonnenschein, and Wilson 1986).

4

bargaining stage. In the arrival stage, a new seller (he), who is randomly drawn by nature from the pool of sellers, turns up with probability 1 ≠ e≠⁄ œ (0, 1) where ⁄ > 0. No seller is available at the beginning of the game, hence the buyer (she) has to wait for the first seller at the beginning of the game.3 Once the first seller arrives, the buyer begins to negotiate with the seller. If a new seller arrives in the middle of haggling with another seller, the buyer must choose with whom she wishes to continue negotiations. The chosen seller stays with the buyer, while the other seller leaves the market immediately. After the buyer chooses her bargaining counterparty, the game moves on to the bargaining stage. The chosen seller names a price for the good, and then the buyer accepts or rejects it. If the seller’s offer is accepted, the game immediately ends; otherwise, the game moves on to the next period. Neither previous offers nor previous sellers can be recalled. All players are equally impatient with the common discounting rate r > 0. If the game ends in period n Ø 0 by buyer’s accepting price p, the buyer’s payoff is e≠rn (v ≠ p). The payoff of seller who made this offer is e≠rn p, while all other sellers’ payoffs are zero. Payoffs of all players are zero if the buyer keeps rejecting all offers indefinitely. Each seller observes the history of declined prices made by himself but cannot observe the interaction between the buyer and other sellers. Most importantly, each seller cannot observe whether a new seller arrives or not in the midst of bargaining with the buyer. Also, sellers are not informed of the period in which they enter the market. Equivalently speaking, sellers do not know how long the buyer has been on the market. All sellers share the same prior over calendar time conditional upon arrival.4 Notice that the buyer does not pay a direct search cost. Instead, the buyer must delay her consumption of the good whenever she decides to search one more period. A delay is generally more costly for the high-valuation type than the low-valuation type. The actual cost depends on the arrival rate of new sellers (⁄) and the discounting rate (r). If the buyer’s search fails so that no new seller turns up in the next period, the buyer has to keep negotiating with the existing seller. Hence the cost of delay also depends on the existing seller’s offer strategy in the future. In this sense, the cost of search is determined endogenously.

2.2

Monotone Anonymous Perfect Bayesian Equilibrium

Let H = finØ1 Rn+ fi {?} be the set of possible histories of declined offers made by one seller and let h denote its generic element. ? œ H represents the null history, where a new seller just arrived and is chosen as the bargaining counterparty. For any h œ H and j œ [0, 1] let 3

The assumption that there is no seller at the beginning of the game can be replaced by the assumption that the buyer begins negotiations with a seller from the very beginning of the game, without letting the seller know he is the first counterparty. Combining either assumption with the assumption that sellers cannot observe calendar time, the environment of the game becomes stationary in the sense that there is no discrimination between sellers according to their arrival time. 4 Appendix A shows that if nature first draws a number of potential sellers from a geometric distribution and then these sellers arrive in random order according to a binomial process, all sellers share the same belief over calendar time conditional on their arrival. From the buyer’s perspective, an additional new seller arrives in each period according to the binomial process.

5

period n

1

period n

a new seller randomly arrives

period n + 1

the chosen seller makes an o↵er

the buyer chooses with whom to continue negotiations

the buyer accepts or declines seller’s o↵er

Figure 1: Timing of the Bargaining Game ‡j (h) denote a behavioral offer strategy (a probability distribution over non-negative prices) of seller j conditional on the event that his offers in h have been rejected and the buyer still continues to negotiate with him. Let qj (h) œ [0, 1] be seller j’s belief (subjective probability) that the buyer’s valuation is vH conditional on h œ H. The buyer also observes a history of rejected offers made by an existing seller, h œ H. In addition, she can privately observe calendar time, arrival/non-arrival of other sellers in each ‚ n be the set of all possible histories previous period, past sellers’ offers, and so on. Let H B privately observed by the buyer, which ends with the information about whether a new seller ‚ B © finØ1 H ‚ n . For any h œ H and h ‚ B , let ˆB œ H turns up or not in period n Ø 1. Also, let H B ˆ B ) œ [0, 1] be the probability that the high type switches to new seller (if one arrives) ›H (h, h ‚ B , let ·H (p; h, h ˆB œ H ˆB) ˆ B . Also, for any offer p Ø 0, h œ H, and h conditional on h and h denote the probability that the high type accepts the current seller’s offer p if made after h ˆ B . The low type buyer’s behavioral strategies ›L and ·L are defined similarly. and h The solution concept is perfect Bayesian equilibrium (PBE) as1defined in Fudenberg and 2 Tirole (1991, Definition 8.1).5 A generic PBE is denoted by – = (‡j , qj )jœ[0,1] (›i , ·i )i=H,L ˆ B ; –, , ⁄), or simply by – = (‡, q, ›, · ) without indices. For a given equilibrium –, let V H (h, h L S ˆ B ; –, , ⁄), and V (h; –, , ⁄) be the continuation payoffs of the high type buyer, low V (h, h

type buyer, and seller conditional on the buyer not switching to a new seller after h œ H and ‚ B . Hence V H (?, h ˆB œ H ˆ B ; –, , ⁄) and V L (?, h ˆ B ; –, , ⁄) are each buyer type’s value of h ‚ B . When there is no potential confusion, ˆB œ H switching to a new seller with private history h – and are often dropped, and each player’s continuation payoffs are denoted simply by H ˆ ˆ B ), and V S (h) respectively. V (h, hB ), V L (h, h This papers imposes the following refinement assumptions which will be defined more precisely in the following paragraphs: • the sellers adopt the same equilibrium offer strategy (anonymous offer strategy) and it is monotone in the seller’s belief over the buyer’s valuation (monotone offer strategy); 5 Fudenberg and Tirole defined perfect Bayesian equilibria for finite games of incomplete information only. Its generalization to infinite games is straightforward and is omitted.

6

qˆ r ⁄ h ? ˆB h

probability of v = vH time interval between periods common discounting rate arrival rate of new sellers history of declined offers null history buyer’s private history

VH VL VS

continuation payoff of high type conditional on not switching to new seller continuation payoff of low type conditional on not switching to new seller seller’s continuation payoff conditional on the buyer not switching

‡j qj ›H , ›L ·H , ·L ‡, q q –

seller j’s offer strategy seller j’s belief that v = vH buyer’s switching decision rule buyer’s acceptance rule anonymous offer strategy anonymous belief system PBE assessment

Table 1: Table of Notation • the buyer uses the same bargaining strategy against all sellers (anonymity) which also satisfies some stationary conditions (stationarity). The monotonicity and anonymity of the equilibrium offer strategy are the most important conditions and play the key roles in analysis. To emphasize this, the paper will call a PBE that satisfies these conditions monotone anonymous PBE. Let me first discuss the anonymity assumption. Roughly speaking, this assumption requires that (i) all sellers adopt the same offer strategy and belief system, while (ii) the buyer also uses the same bargaining strategy against all sellers. A similar refinement assumption has been frequently employed in the literature. Examples range from Rubinstein and Wolinsky (1985) to Kim (2014) among many others. REFINEMENT (R1). (i) There exist an anonymous offer strategy ‡ and a belief system q of sellers such that ‡(h) = ‡j (h) and q(h) = qj (h) for all h œ H and j œ [0, 1]. (ii) Buyer’s strategies are measurable with respect H, and do not depend on the buyer’s private observations.6

From now on, the paper focuses on PBEs for which the refinement assumption R1 is ‚ B will be suppressed in the buyer’s behavioral strategies and their continˆB œ H satisfied. h uation payoffs for simplicity of notation. Similarly, a subscript index for seller’s identity can be dropped from equilibrium offer strategy and belief system if all sellers adopt the same strategy and belief system, and ‡(·) and q(·) would stand for all sellers’ common offer strategy and belief system. ˆ B ) = ›H (h, h ˆ ÕB ), ›L (h, h ˆ B ) = ›L (h, h ˆ ÕB ), ·H (p; h, h ˆ B ) = ·H (p; h, h ˆ ÕB ) and ·L (p; h, h ˆB) = Formally, ›H (h, h Õ ˆ B ) for any h œ H, h ˆB œ H ‚B , and p Ø 0. ·L (p; h, h 6

7

We now turn to monotonicity of equilibrium offer strategies. The following refinement assumption requires that the more a seller believes he faces the high type buyer, the higher offer he offers in equilibrium. REFINEMENT (R2). The following holds for any j œ [0, 1] and for any histories h and hÕ such that qj (h) > qj (hÕ ): p œ supp ‡j (h), pÕ œ supp ‡j (hÕ ) =∆ p Ø pÕ .

The refinement condition that seller’s offer strategy be monotone is used by Hwang (2013), where the author imposes the same condition for some results. On the other hand, Rubinstein (1985) places a related monotonicity condition when he analyze an alternating-offer bargaining game with incomplete information about one bargainer’s time preference. Roughly speaking, Rubinstein rules out sequential equilibria in which an informed party’s rejection can indicate that he or she is is more likely to be the impatient type; see Assumption B-2 in Rubinstein (1985). Condition R2 differs from Rubinstein’s in that it requires the seller’s strategy to be monotone in belief, while Rubinstein requires the uninformed party’s belief dynamics to be monotone. However, Condition R2 implies that the seller’s belief decreases whenever the buyer rejects an equilibrium offer made by sellers, just as does Rubinstein’s monotonicity condition on beliefs. See Lemma 6 in the next section. The literature often makes use of other forms of monotonicity. Bikhchandani (1992) considers an alternating-offer bargaining game where bargainers pay a fixed cost per period as long as they keep on bargaining, and one player’s fixed cost is private information. He places the following monotonicity restriction on off-the-equilibrium-path beliefs: whenever the informed party demands more than equilibrium strategy, the informed party should believe the uninformed party is more likely to be a type with lower fixed cost, relative to his equilibrium belief level. On the other hand, Gul and Sonnenschein (1988) focus on equilibria where buyer’s acceptance decisions is stationary and monotone in seller’s belief. Ausubel, Cramton, and Deneckere (2002) also point out that a notion of weak-Markov equilibrium also embodies a form of monotonicity. A few papers impose some monotonicity conditions on the seller’s (uninformed party’s) equilibrium payoff. For example, Fudenberg, Levine, and Tirole (1987) restrict attention to an equilibrium where the seller’s (expected) equilibrium payoff at the beginning of a period strictly decreases as the sellers’ beliefs about the current buyer become more pessimistic. A similar monotonicity is imposed in the recent paper by Lee and Liu (2013). I also impose an additional stationarity property on the low type’s behavioral strategy. This is motivated by the following lemma stating that sellers never offer a price strictly less than vL in a PBE that satisfies R1. Its proof follows the standard argument in the literature. Suppose that the minimum (infimum) offer that sellers possibly offer on equilibrium path, say p is strictly less than vL . If the seller’s equilibrium offer strategy ‡ prescribes to offer p (or¯ a price very close to p) after h œ H, it is in fact a profitable deviation for the seller ¯to make a slightly higher offer ¯ which would be accepted by both types for sure due to their impatience (i.e., r > 0). 8

LEMMA 1. In every PBE that satisfies R1, sellers never offer p < vL . Proof. In Appendix B.1. An immediate implication of the lemma is that, in equilibrium, the low type buyer’s payoff is always zero in any continuation game. This means the low type is always indifferent between switching to a new seller and not. The following refinement assumption rules out a multiplicity of equilibria due to such indifference by assuming that the low type uses a simple behavioral strategy in equilibrium. REFINEMENT (R3). The low type accepts price p if and only if p Æ vL . The probability of the low type’s switching to new seller is independent of histories of negotiations, and is either zero or one.7 The requirement that ·L be independent of h œ H plays no role in the following sense: if ·L is not stationary in a PBE that satisfies R1, there is always an equivalent PBE (that yields the same equilibrium payoff profile) such that the low type’s acceptance decision is stationary. To see this, recall that sellers never offer strictly lower than vL in a PBE that satisfies R1 (Lemma 1). Thus, the high type always accepts p = vL while the low type is indifferent between accepting and not. If the low type accepts p = vL with probability strictly less than 1 on the equilibrium path, a seller can increase his profit by charging slightly lower price. Hence the low type possibly rejects p = vL only off the equilibrium path. Formally: LEMMA 2. For any PBE – = (‡, q, ·, ›) that satisfies R1, there is another PBE –Õ = (‡ Õ , q Õ , · Õ , › Õ ) that also satisfies R1, yields the same equilibrium payoff profile and ·LÕ (h) = {p Æ vL } for any h œ H. Proof. In Appendix B.2. Finally the paper imposes an additional condition on the high type buyer’s equilibrium switching decision rule ›H . It requires that the high type never switches to a new seller, even if one is available, after she just declined an offer strictly less than vH ≠ e≠r V H (?; –, , ⁄) made by the original seller. This refinement condition actually does not impose any restriction to the equilibrium payoff profile set in the sense that if – = (‡, q, ·, ›) does not satisfy this condition, there is always another PBE –Õ such that the condition holds while the equilibrium payoff profile of –Õ is identical to –. LEMMA 3. For any PBE that satisfies R1, there is an equivalent PBE – = (‡, q, ·, ›) that also satisfies R1 and

7

Y ] 0 if pn < vH ≠ e≠r V H (?; –, , ⁄) ›H (h) = [ 1 if p > v ≠ e≠r V H (?; –, , ⁄) and · (p ; h) > 0 n H H n

Formally, (i) ·L (p; h) = {p Æ vL } for any h œ H, and (ii) either ›L (·) © 0 or ›L (·) © 1.

9

for any history h = (p0 , . . . , pn ) œ H that yields the same equilibrium payoff profile. Proof. In Appendix B.3. To catch the basic idea behind Lemma 3 note first that the high type rejects any offer p < vH ≠ e≠r V H (?; –, , ⁄) only if she expects the continuation payoff to be strictly higher than V H (?; –, , ⁄). Because the payoff from switching to new seller is V H (?; –, , ⁄), the high type’s rejection of p < vH ≠ e≠r V H (?; –, , ⁄) can be justified only if she strictly prefers continuing negotiations with the original seller to switching to another seller. Hence, on the equilibrium path, the high type never switches after she just rejected p < vH ≠ e≠r V H (?; –, , ⁄) in the last period. REFINEMENT (R4). For any history h = (p0 , . . . , pn ) œ H,

Y ] 0 if pn < vH ≠ e≠r V H (?; –, , ⁄) ›H (h) = [ 1 if p > v ≠ e≠r V H (?; –, , ⁄) and · (p ; h) > 0. n H H n

This paper will focus on PBEs for which R1–R4 hold, which will be termed monotone anonymous PBE. DEFINITION 1. A monotone anonymous perfect Bayesian equilibrium (monotone anonymous PBE) refers to a PBE that satisfies R1–R4. In addition, a PBE that only satisfies R1 will be referred to as anonymous PBE. Throughout the paper, let EM ( , ⁄) and EP ( , ⁄) denote the set of all monotone anonymous PBEs and the set of all anonymous PBEs respectively. Many bargaining papers focus on weak-Markov equilibrium (a Markov equilibrium with the state space consisting of seller’s belief and last rejected offer) as a refinement. Notice that the notion of monotone anonymous PBE is a priori neither stronger nor weaker than weak-Markov equilibrium because monotone anonymous PBE does not require equilibrium strategy to be Markovian. This section concludes by defining some terms which will be frequently used throughout the paper. Given a monotone anonymous PBE –, an offer strictly lower than vH ≠ e≠r V H (?; –, , ⁄) is referred to as secure offer. By making a secure offer, the seller can catch (secure) the high type buyer against other potential sellers. The offer p = vL will be called a winning offer, since it is accepted by both types for sure in any monotone anonymous PBE. As is standard, an offer is called serious if it is accepted by the buyer with a positive probability in equilibrium. An offer is referred to as losing offer if it is supposed to be declined by the buyer with probability 1.

10

3

Preliminary Observations

This section provides some preliminary results that will be frequently used throughout the paper. The first lemma, which immediately follows from Lemma 1, states that sellers never make offers strictly lower than (1 ≠ e≠r )vH + e≠r vL . Note that the high type is indifferent between accepting p = (1 ≠ e≠r )vH + e≠r vL right now and accepting the winning offer p = vL in the next period. Because all sellers never make an offer strictly lower than vL , the high type has to accept any offers strictly less than (1 ≠ e≠r )vH + e≠r vL in equilibrium. Understanding this, sellers never offer a price between (1 ≠ e≠r )vH + e≠r vL and vL . LEMMA 4. In any monotone anonymous PBE, a seller never offers p < (1 ≠ e≠r )vH + e≠r vL .

Proof. Pick any equilibrium assessment – = (‡, q, ·, ›) œ EM (⁄, ) such that ›L (·) © x œ {0, 1}. Suppose there are h œ H and p œ (vL , (1≠e≠r )vH +e≠r vL ) such that p œ supp ‡(h). Because a seller never offers strictly less than vL in equilibrium, the high type has to accept

p for sure (·H (p; h) = 1). Therefore, conditional upon the buyer rejecting p, q(h, p) = 0 and supp ‡(h, p) = {vL }. The sellers’ conditional profit V S (h, p; –, , ⁄) after h is then V S (h, p; –, , ⁄) = pq(h) + e≠r

Ë

È

e≠⁄ + (1 ≠ e≠⁄ )(1 ≠ x) (1 ≠ q(h))vL .

On the other hand, if a seller deviates to a slightly higher offer pÕ œ (p, (1≠e≠r )vH +e≠r vL )

after h, again ·H (pÕ ; h) = 1, q(h, pÕ ) = 0, and supp ‡(h, pÕ ) = {vL } by the same argument. The seller’s profit from such the deviation is pÕ q(h) + e≠r

Ë

È

e≠⁄ + (1 ≠ e≠⁄ )(1 ≠ x) (1 ≠ q(h))vL > V S (h, p; –, , ⁄)

contradiction. The next lemma states that sellers make the winning offer vL with positive probability in any continuation game. The proof follows the standard argument that appears in the bargaining literature (Fudenberg, Levine, and Tirole 1985) and can be found in Appendix B.4. The basic idea is as follows. As long as the high type buyer switches to new sellers whenever possible, the sellers’ belief drops toward zero rapidly, and the winning offer is eventually made soon. Hence, if the winning offer is never offered in equilibrium, the high type’s continuation payoff must be no lower than V H (?; –, , ⁄) at some point. This can be done only when a seller makes a serious offer, say p, not higher than vH ≠ V H (?; –, , ⁄). The high type must be indifferent between accepting and rejecting this serious offer p, hence the seller must offer a serious offer not higher than vH ≠ V H (?; –, , ⁄)/e≠r in the future, conditional on the high type’s rejection of p. The iteration of this argument shows that the seller eventually offers zero after a certain history which is reached with positive probability 11

in equilibrium. But this is impossible because any seller can guarantee the minimum profit vL > 0 in any continuation games. LEMMA 5. Let – = (‡, q, ·, ›) be a monotone anonymous PBE. For any h œ H, there is a history hÕ such that

• hÕ is reached in equilibrium with a positive probability in the continuation game beginning after h;

• supp ‡(hÕ ) contains the winning offer vL . Proof. In Appendix B.4 The following two lemmas will play key roles in analysis. The first lemma shows that the sellers’ belief only decreases along the equilibrium path of monotone anonymous PBE. A similar belief dynamics is also observed in other seller-offer bargaining models without outside options, and it has been called skimming property. See Fudenberg, Levine, and Tirole (1985), Gul, Sonnenschein, and Wilson (1986), and Fuchs and Skrzypacz (2010) among many others. In alternating-offer bargaining model, researchers often exclude an equilibrium without the skimming property (Rubinstein 1985). LEMMA 6 (skimming property). In any monotone anonymous PBE – = (‡, ·, ›, q), q(h) Ø q(h, p) for any history h œ H and p œ supp ‡(h). Proof. Suppose not and consider a history h œ H and p œ supp ‡(h) such that q(h) < q(h, p). If ›H (h, p) = 1, obviously q(h) Ø q(h, p), hence let ›H (h, p) < 1 and V H (h, p) Ø V H (?)

without loss. By monotonicity, any offer in supp ‡(h, p) has to be no lower than p. If all offers in supp ‡(h, p) are serious, the high type would have accepted p for sure which results in q(h, p) = 0. Thus, there is a losing offer pˆ1 œ supp ‡(h, p) and e≠r V H (h, p, pˆ1 ) Ø V H (h, p) Ø V H (?).

Because V H (h, p, pˆ1 ) > V H (?), ›H (h, p, pˆ1 ) = 0 and q(h, p, pˆ1 ) Ø q(h, p) > q(h). Again,

the monotonicity of ‡ implies any offer in supp ‡(h, p, pˆ1 ) is no lower than p, and hence there

is another losing offer pˆ2 œ supp ‡(h, p, pˆ1 , pˆ2 ) such that e≠r V H (h, p, pˆ1 , pˆ2 ) Ø V H (h, p) Ø

V H (?). The iteration of this argument identifies a sequence of losing offers pˆ1 , pˆ2 , . . . , pˆk , . . . that a seller possibly makes in sequence on equilibrium path after p is declined. This means that the seller’s equilibrium payoff is zero. But this is impossible because the seller can always earn vL > 0 as a payoff by charging the winning offer vL . 12

The skimming property determines the direction of the seller’s belief update on an monotone anonymous PBE’s equilibrium path. But it does not tell us to which level the seller’s belief should be updated conditional on his offer being rejected. The next lemma answers this question. To state the lemma, define p+ (q Õ ) œ [0, Œ) by the infimum offer that a seller possibly offers in equilibrium with belief higher than q Õ œ [0, 1]: p+ (q Õ ) = inf{p Ø 0 : ÷h œ H such that q(h) > q Õ , p œ supp ‡(h)} Similarly, let p≠ (q Õ ) = sup{p Ø 0 : ÷h œ H such that q(h) < q Õ , p œ supp ‡(h)} Note that both p+ (·) and p≠ (·) are nondecreasing. In addition, p+ (q Õ ) Ø p≠ (q Õ ) for any q Õ œ [0, 1] if seller’s offer strategy ‡ is monotone. LEMMA 7. Let – = (‡, ·, ›, q) be a monotone anonymous PBE. Suppose the seller offers p Æ vH ≠ e≠r V H (?) such that (1 ≠ e≠r )vH + e≠r p≠ (q Õ ) < p < (1 ≠ e≠r )vH + e≠r p+ (q Õ ) after a history h with q(h) Ø q Õ for some q Õ œ [0, 1].8 Then, the seller’s updated belief followed by the buyer’s rejection of p is q(h, p) = q Õ .

Proof. For a contradiction, suppose q(h, p) > q Õ . Assume the high type is willing to reject p without loss; otherwise, q(h, p) = 0 hence q(h, p) cannot be larger than q Õ . Then, e≠⁄ V H (h, p) + (1 ≠ e≠⁄ ) max{V H (h, p), V H (?)} Ø

vH ≠ p , e≠r

and the right-hand side is strictly larger than vH ≠ p+ (q Õ ) by assumption. Because p Æ

vH ≠ e≠r V H (?), the high type would be willing to forgo outside option in the following period after she turns down p, and hence V H (h, p) Ø V H (?). Combining these inequalities, V H (h, p) Ø V H (?) and V H (h, p) > vH ≠ p+ (q Õ ). By definition of p+ (q Õ ), any offer in supp ‡(h, p) is no lower than p+ (q Õ ), and hence vH ≠ pÕ Æ vH ≠ p+ (q Õ ) < V H (h, p) for any pÕ œ supp ‡(h, p). If all offers in supp ‡(h, p) are serious, therefore, the high type is 8

p may not be in supp ‡(h).

13

actually not willing to decline p which contradicts the hypothesis that the high type is wiling to reject p. Hence there must be a losing offer, say pˆ1 , in supp ‡(h, p) such that e≠⁄ V H (h, p, pˆ1 ) + (1 ≠ e≠⁄ ) max{V H (h, p, pˆ1 ), V H (?)} Ø The right-hand side is in turn lower than neither V H (?)/e≠r

V H (h, p) e≠r

nor (vH ≠ p+ (q Õ ))/e≠r

hence

V H (h, p, pˆ1 ) > max{V H (?), vH ≠ p+ (q Õ )}. Hence ›H (h, p, pˆ1 ) = 0 and q(h, p, pˆ1 ) Ø q(h, p) > q Õ . By definition of p+ (q Õ ) again, any offer in supp ‡(h, p, pˆ1 ) is no lower than p+ (q Õ ). Applying the argument above again, there is another losing offer pˆ2 œ supp ‡(h, p, pˆ1 ) such that V H (h, p, pˆ1 , pˆ2 ) > max{vH ≠ p+ (q Õ ), V H (?)}. The iteration of this argument identifies a sequence of losing offers pˆ1 , pˆ2 , pˆ3 , . . . that the seller is willing to offer in a sequence after history (h, p). This means seller’s expected payoff conditional on p being turned down is zero, which is impossible. In conclusion, q(h, p) Æ q Õ .

Now suppose q(h, p) < q Õ . In this case, the high type is willing to accept p. Otherwise,

because the high type’s payoff from accepting p is no lower than e≠r V H (?), ›H (h, p) = 0, q(h, p) Ø q(h) Ø q Õ hence q(h, p) Ø q Õ . But if q(h, p) < q Õ , vH ≠ p < e≠r (vH ≠ p≠ (q Õ )) Æ e≠r (vH ≠ pÕ ) for any pÕ œ supp ‡(h, p) hence p is actually losing, contradiction. Notice that Lemma 7 holds even if p in the statement of the lemma is not an equilibrium offer. This feature would be exceptionally useful to figure out what happens after a seller deviates from the equilibrium play. It is also worth to note that Lemma 7 makes clear the sellers’ tradeoff in monotone anonymous PBEs. Both p+ (·) and p≠ (·) are nondecreasing, thus the lemma guarantees that a seller can increase the probability of agreement by making a lower offer than equilibrium offers. To state this in a different manner, a seller has a tradeoff between speed of negotiations and price in any monotone anonymous PBEs.

14

4

Coase Conjecture for Bargaining with Arriving Sellers

4.1

Coase Conjecture

The Coase conjecture originally has been developed in the context of durable good monopoly (Coase 1972). The conjecture states that a durable goods monopolist would lose its monopoly power if it could make frequent price offers. In the bargaining literature, the Coase conjecture refers to a similar outcome: According to the Coase conjecture, if the gains from trade are bounded away from zero · · · · · · , the seller’s introductory offer converges to the lowest possible buyer valuation as the length of the time period between successive seller offers vanishes (Deneckere and Liang 2006). The main result of the current paper is that a qualitatively similar result holds for the bargaining model with arriving sellers when the interval between periods vanishes ( æ 0) and the arrival rate of new sellers goes to infinity (⁄ æ Œ). The following definition makes clear what the Coase conjecture means in the context of bargaining with arriving sellers. Here, the Coase conjecture is defined in terms of the high type buyer’s equilibrium payoff. However, we will shortly see that the Coase conjecture in this definition also implies that the trade occurs almost immediately at the price equal to the winning offer vL , and consequently the equilibrium outcome is efficient. Also note that the Coase property and the Coase conjecutre are defined not only for monotone anonymous PBEs but also for any anonymous PBEs.

DEFINITION 2. (i) (Coase Property) The Coase property holds along a sequence of anonymous PBEs (–k,m )k,mØ0 œ

r

k,mØ0 EP (

k , ⁄m )

such that limkæŒ

lim inf lim inf V H (?; –k,m , mæŒ

kæŒ

k , ⁄m )

k

= 0 and limmæŒ ⁄m = Œ if

= vH ≠ vL .

(ii) (Coase Conjecture) The Coase conjecture holds if the Coase property holds for any sequence of anonymous PBEs (–k,m )k,mØ0 œ 0 and limmæŒ ⁄m = Œ.

r

k,mØ0 EP (

k , ⁄m )

such that limkæŒ

k

=

Notice the order of limits in Definition 2. Roughly speaking, it is said that the Coase property or Coase conjecture holds if the high-valuation buyer’s equilibrium payoff in the continuous type limit converges to vH ≠ vL as the arrival rate of new sellers goes to infinity. What is crucial is that the probability that a new seller arrives in each period, 1 ≠ e≠⁄ becomes negligible in the limit. 15

Even though the Coase conjecture is defined in terms of the high type’s equilibrium payoff only, it also has the clear implication on the introductory price and the equilibrium delay of agreement. For any anonymous PBE – œ EP ( , ⁄), let d(–; , ⁄) œ N fi {Œ} denote the number of periods before a seller finally makes the winning offer in –, counting from the period in which the first seller arrives to the buyer. Then · d(–; , ⁄) represents the delay of agreement in equilibrium. · d( , ⁄) is a random variable, particularly when sellers use a mixed offer strategy. The following lemma shows that it converges to zero in probability along any sequence of anonymous PBEs with the Coase property. LEMMA 8. Fix ÷ > 0, ‘ > 0, a sequence of time intervals ( arrival rates (⁄m )mØ0 such that limkæŒ anonymous PBEs

(–k,m )kØ0,mØ0

œ

r

k

k )kØ0 ,

and a sequence of

= 0 and limmæŒ ⁄m = Œ. For any sequence of

kØ0,mØ0 EP (

k , ⁄m )

with the Coase property, there are

a positive integer M and a sequence (km )mØM such that P

Ó

k

· d(–k,m ;

k , ⁄m )

Ô

<÷ >1≠‘

if m Ø M and k Ø km . Proof. Let ÷ ú = ÷/2 > 0. Fix an arbitrary positive real number ‘ > 0, and let ‘ú = ‘(1 ≠ e≠r÷ )(vH ≠ vL ). By the Coase property, there are an integer M > 0 and sequence of ú

integers (km )mØM such that

V H (?; –k,m ,

k , ⁄m )

œ (vH ≠ vL ≠ ‘ú , vH ≠ vL ]

if m Ø M and k Ø km . Let Âk,m be the probability that a seller makes the winning offer within

Ï

÷ú

k

Ì

periods in anonymous PBE –k,m .9 Then,

V (?; – H

k,m

,

k , ⁄m )

≠r

Æ Âk,m (vH ≠ vL ) + (1 ≠ Âk,m )e

Ï

÷ú k

Ì

k

(vH ≠ vL )

Æ Âk,m (vH ≠ vL ) + (1 ≠ Âk,m )e≠r÷ (vH ≠ vL ) ú

and ‘ú > (vH ≠ vL ) ≠ V H (?; –k,m ,

k , ⁄m )

9

1

Ø (1 ≠ Âk,m ) 1 ≠ e≠r÷

ú

2

(vH ≠ vL )

For any real number z œ R, ÁzË represents the smallest integer following z. More precisely, ÁzË = min{n œ Z : n Ø x}.

16

thus Âk,m > 1 ≠ ‘, and hence ;

1 ≠ ‘ < P d(– =P

;

k

k,m

;

k , ⁄m )

· d(–k,m ;

<

9 ú :< ÷ k

k , ⁄m )

<

;

Æ P d(–

÷ + 2

k

<

k,m

;

k , ⁄m )

Ó

Æ P d(–k,m ;

<

÷ú k

k , ⁄m )

+1

<

< ÷/

k

Ô

if m Ø M and k Ø km . The Coase conjecture also implies that the efficient outcome is approached as ⁄ æ Œ and æ 0. Formally, let l(–; , ⁄) be the efficiency loss in anonymous PBE – œ EP ( , ⁄) relative to the efficient outcome: l(–; , ⁄) = (ˆ q vH + (1 ≠ qˆ)vL ) ≠

È e≠r (1 ≠ e≠⁄ ) Ë H L S q ˆ V (?; –, , ⁄) + (1 ≠ q ˆ )V (?; –, , ⁄) + V (?; –, , ⁄) . 1 ≠ e≠(r+⁄)

Here the first term (ˆ q vH + (1 ≠ qˆ)vL ) on the right-hand side represents the maximal potential ≠r ≠⁄ ) surplus without search friction. Term e 1≠e(1≠e is present due to the finite arrival rate of ≠(r+⁄) the first seller, and it converges to one as æ 0 and ⁄ æ Œ. An immediate implication of the Coase property is that the limit infimum of qˆV H (?; –, , ⁄) + (1 ≠ qˆ)V L (?; –, , ⁄) in the bracket is qˆ(vH ≠ vL ). On the other hand, V S (?; –, , ⁄) is bounded below by vL . Hence the efficient outcome is approached, along any sequence of anonymous PBEs that satisfies the Coase property. LEMMA 9. Fix a sequence of time intervals ( sellers (⁄m )mØ0 such that limkæŒ anonymous PBEs (–k,m )kØ0,mØ0 œ

r

k

k )kØ0

= 0 and limmæŒ ⁄m = Œ. Along any sequence of

kØ0,mØ0 EP (

lim sup lim sup l(–k,m ; mæŒ

and a sequence of arrival rates of new

kæŒ

k , ⁄m )

with the Coase property,

k , ⁄m )

= 0.

In conclusion, if the Coase conjecture holds true, both the delay of trade and the efficiency loss vanish as ⁄ æ Œ and æ 0.

4.2

Example

The current paper’s main result, which will be stated and proven in Sections 4.3 and 4.4 respectively, is that the Coase property holds for any sequence of monotone anonymous PBEs r (–k,m )kØ0,mØ0 œ kØ0,mØ0 EM ( k , ⁄m ) such that limmæŒ ⁄m = Œ and limkæŒ k = 0. The proof will be by contradiction. That is to say, the proof is done by showing that a sequence of monotone anonymous PBEs that fails the Coase property cannot exist. This strategy necessitates us to establish the existence of a sequence of monotone anonymous PBEs for which the Coase property holds true. 17

We can actually construct a sequence of monotone anonymous PBEs that satisfies the Coase property by applying the standard technique of backward induction in the sequential bargaining literature (Gul, Sonnenschein, and Wilson 1986). Suppose that all players expect that the buyer would reach an agreement with the first seller almost immediately. Under this expectation, the probability that a new seller turns up during negotiations is negligible, and hence the buyer’s option to switch her bargaining counterparty makes no difference relative to the bargaining game between a single buyer and a single seller without outside sellers. Then the argument of the Coase conjecture would kick in, and hence this expectation is actually fulfilled in equilibrium. LEMMA 10. For any (⁄m )mØ0 and (

k )kØ0

such that limmæŒ ⁄m = Œ and limkæŒ

there exists a sequence of monotone anonymous PBEs (–k,m )kØ0,mØ0 œ along which the Coase property holds.

r

kØ0,mØ0 EM (

k

= 0,

k , ⁄m )

Sketch of Proof. The proof is done by construction and here a sketch of the construction is provided. The proof can be found in Appendix B.5. Fix any sequence ( such that limkæŒ

k

k )kØ0

and (⁄m )mØ0

= 0 and limmæŒ ⁄m = Œ. For each k and m, let us denote the

candidate of monotone anonymous PBE by –k,m = (‡ k,m , q k,m , › k,m , · k,m ) œ EM ( In each

–k,m ,

k , ⁄m ).

suppose the high type never switches to new seller even if one arrives. On the

other hand, suppose that the low type switches to a new seller with probability x œ {0, 1}

k,m whenever possible (see Refinement R3). In the current paper’s notation, ›H (·) = 0 and

›Lk,m (·) = x œ {0, 1}. For each k and m, let — k,m (q, q Õ ) be the solution of the following equation:

qÕ =

q(1 ≠

— k,m (q, q Õ ))

q(1 ≠ — k,m (q, q Õ )) ! + (1 ≠ q) e≠⁄m k + (1 ≠ e≠⁄m

k

",

)(1 ≠ x)

for any belief levels q and q Õ such that q > q Õ . If a seller with belief q expects that the high type accepts seller’s next offer with probability — k,m (q, q Õ ), and then observes it is actually rejected, the seller’s updated belief in the following period would be q Õ . Now an equilibrium such that any sellers charges the winning offer almost immediately obtains by applying the standard method for solving an equilibrium for the bargaining model without outside option (Gul, Sonnenschein, and Wilson 1986). In the constructed equilibrium, any seller charges p˜j © (1 ≠ e≠rj

k

)vH + e≠rj

k

vL on the equilibrium path whenever

x ), where (˜ his belief is q œ [˜ qjx , q˜j+1 qjx )jØ0 will be identified soon. The high type accepts p˜j

x ) so that seller’s updated belief after buyer’s rejection to p with probability — k,m (q, q˜j≠1 ˜j is x . q˜j≠1

On the equilibrium path, with an integer N , the seller charges p˜N , p˜N ≠1 , . . . , p˜0 in se-

x x x quence while seller’s belief evolves from q(?) to q˜N ˜N ˜N ≠1 , then from q ≠1 to q ≠2 , and so on

18

S

h:q k,m (h)=q

supp

k,m

(h)

p˜3

p˜2

p˜1 p˜0 q˜0x

q˜1x

q˜2x

q˜3x



q

Figure 2: The support of equilibrium offer strategy (for a case with NC = 3). The horizontal axis represents seller’s belief levels. The vertical axis represents the set of offers that a seller possibly makes with each belief level. When a seller believes that the buyer is the highvaluation type with probability q˜jx (j Ø 1), he offers p˜j with probability 1 on the equilibrium path, while he possibly randomizes over p˜j and p˜j≠1 off the equilibrium path. until it finally reaches q˜0x where x x qˆ Ø q(?) = q˜N > q˜N ˜0x = 0.10 ≠1 > . . . > q

For each j, the high type is indifferent between accepting p˜j in current period and accepting p˜j≠1 in next period. Note that high type’s payoff from starting bargaining with a new seller is V H (?; –,

k , ⁄m )

= e≠rN

k

(vH ≠ p˜0 ) = e≠rN

k

(vH ≠ vL )

(1)

and it is straightforward to check vH ≠ e≠r

k

V H (?; –,

k , ⁄m )

> p˜N .

As a result, all equilibrium offers are secure offers and the high type buyer indeed has no k,m incentive to deviate from the equilibrium switching decision ›H (·) © 0.

(˜ qjx )jØ0 has to be chosen so that a seller is indifferent between p˜j and p˜j≠1 with belief

10

Note that the high type also accepts an offer whenever the low type accepts one. This means the fraction of high type buyers in market only decreases over time. Hence, a new seller’s belief over buyer’s type at the null history is no higher than qˆ in any equilibrium with ›H (·) © 0.

19

q˜jx . For example, with belief q˜1x , the seller is indifferent between charging the winning offer p˜0 = vL immediately and charging p˜1 in current period and then p˜0 in the next period. The latter path yields p˜1 q˜1x + e≠(r+⁄m )

k

paths if and only if q˜1x =

(1 ≠ q˜1x )vL and hence the seller is indifferent between two (1 ≠ xe≠(r+⁄m ) k )vL ≠r k (1 ≠ xe≠⁄m k )v H +e

(1 ≠ e≠r

For any choice of ›L (·) = x œ {0, 1} and identifies all belief cutoffs

q˜0x

<

q˜1x

<

(as long as

k

q˜2x

k

k

)vL

(2)

.

is small enough), Appendix B.5

< . . . and shows that no player has incentive to

deviate from the equilibrium play with these cutoffs.11 Moreover, NC = min{j œ N : q˜jx > qˆ} is uniformly bounded across all small

k ’s

and any x œ {0, 1} and any ⁄k > 0, and thus

seller’s initial offer is bounded above by p˜NC = (1 ≠ e≠rNC of sellers and the high type buyer, V

S

V S (?; –k,m ,

and V

H,

k , ⁄m )

k

)vH + e≠rNC

k

vL . The payoffs

are respectively

Æ p˜NC = vL + O(

k)

and V H (?; –k,m ,

k , ⁄m )

Ø vH ≠ p˜NC = vH ≠ vL + O(

k)

hence the Coase property holds along any sequence of equilibria constructed here. Regarding off-equilibrium path, if the seller offers a price higher than equilibrium offer, then both type reject it for sure; the seller’s belief moves to q Õ Ø q in the next period and he immediately comes back to equilibrium play. On the other hand, suppose the seller offers a

lower price, say p œ [˜ pj Õ ≠1 , p˜j Õ ) deviating from equilibrium offer p˜j (j Õ Æ j). In this case only

x ) where q is seller’s belief when he is the high type accepts p with probability — k,m (q, q˜j≠1 x about to deviate. In the following period, the seller’s updated belief is q˜j≠1 and he randomizes

over p˜j≠1 and p˜j≠2 with probabilities justifying the high type’s randomization over accepting and rejecting p in the previous period. Note that the offer strategy is monotone (see Figure 2). x = (‡ x , q x , › x , · x ) denote the assessment constructed above For future reference, let –C C C C C where c stands for Coasean and x œ {0, 1} represents the probability that low type switches x (?) = q to new seller (›cx L (·) © x). If qC ˜jx for an integer j the seller is actually indifferent between p˜j and p˜j≠1 at the null history ? œ H. In the above equilibrium construction, all 11

See Section B.5, in particular equation (14) and the ensuing discussion

20

sellers are supposed to offer p˜j at the null history, even though they are actually indifferent between two offers. However, it is easy to check that there are actually a continuum of monotone anonymous PBEs which are different only in probability assigned to p˜j .12 All these equilibria are also referred to as Coasean equilibria. Let EC ( , ⁄) µ EM ( , ⁄) be the set of all Coasean equilibria. Finally note that Lemma 10 does not rule out the possibility that there is a sequence of PBEs (–k,m )kØ0,mØ0 such that none of them is the Coasean equilibria constructed above, but (–k,m )kØ0,mØ0 still satisfies the Coase property.

4.3

Main Result

The current paper’s main result is that the Coase conjecture holds for monotone anonymous PBE. That is to say, the Coase property holds for any sequence of monotone anonymous PBEs (–k,m ) œ EM ( k , ⁄m ) with limkæŒ k = 0 and limmæŒ ⁄m = Œ. Roughly speaking, the efficient outcome is approached along any sequence of monotone anonymous PBEs as the fictions in search and bargaining process vanish. In bilateral negotiations between a single seller and a single buyer, without the arrival of new sellers, it is difficult for the seller to insist on a serious offer that only targets the high-valuation type. Once such an offer were made and then rejected, the seller would infer that the buyer is likely to be the low-valuation type, and hence the seller is quickly tempted to make the winning offer. Understanding this, the high-valuation type is willing to reject the offer targeting to her and delay. In equilibrium, the seller’s offer quickly decreases to the low type buyer’ s valuation and hence the Coase conjecture holds. With the buyer’s opportunity to search for new sellers, a seller seemingly can escape from the curse of the Coase conjecture. It seems that a seller can insist on a high price that targets the high type, under the expectation that the low-valuation type has already switched to a new seller and hence the seller faces the high valuation type. If the low-valuation type indeed keeps switching to other sellers, the seller’s belief stays high enough to prevent him from making the winning offer. However, if sellers understand that only the low-valuation type actively searches for new sellers, sellers will believe that it is more likely to encounter the low valuation type than the high-valuation type in the market. Hence, sellers would find that it is a profitable deviation to make the winning offer immediately in order to avoid the cost from delay of agreement. On the other hand, suppose both types actively search and switch to new sellers while negotiating with an existing seller. Then the negotiation between the buyer and the existing seller is equivalent to bilateral negotiations between a single buyer and a single seller, with small chance of a breakdown. The argument of the Coase conjecture kicks in, and hence the existing seller quickly makes the winning offer. But if this is the case in an anonymous 12

For example consider an assessment – = (‡, q, ·, ›) in which if ›L (·) © x = 0, ›H (·) © 0, ·H (·, ·) = ·Cx (·, ·), x and ‡(h) = ‡C (h)|x=0 for all histories h other than the null history. Both types of the buyer never leave the first seller hence q(?) = qˆ. If qˆ = q˜jx |x=0 for some integer j Ø 1 all sellers are indifferent between offering p˜j and p˜j≠1 at the null history and hence there are a continuum of Coasean equilibria each of them only differs in seller’s initial randomization over p˜j and p˜j≠1 .

21

monotone PBE, then the high type buyer actually has no reason to switch. Because all sellers adopt the same equilibrium offer strategy, it is always better to continue to negotiate with the existing seller rather than going through all haggling process with a new seller that the buyer already has came through with the existing seller. The reasoning in the last paragraphs suggests that the Coase conjecture continues to hold even with the buyer’s option to search for new sellers, which is formalized in the following proposition. PROPOSITION 1. The Coase conjecture holds for monotone anonymous PBE. Formally, for any sequence of monotone anonymous PBEs (–k,m )kØ0,mØ0 œ that limkæŒ

k

= 0 and limmæŒ ⁄m = Œ,

lim inf lim inf V H (?; –k,m , mæŒ

kæŒ

k , ⁄m )

r

kØ0,mØ0 EM (

k , ⁄m )

such

= vH ≠ vL .

Combining Proposition 1, Lemma 8 and Lemma 9, the following corollary also obtains. COROLLARY 1. The continuous-time limit of monotone anonymous PBE outcomes converges to the static competitive equilibrium outcome as the arrival rate of new sellers goes to infinity. That is to say, the buyer reaches an immediate agreement with the first seller at the static competitive equilibrium price. As long as the arrival rate of a new seller is large enough, the static competitive market that appears in standard microeconomic theory textbooks is probably a good approximation to a market with both incomplete information and an option to haggle in prices possibly over a long period. The corollary also suggests why some decentralized markets function well with minimal monitoring and intervening, especially when a cost of searching for a new trading counterparty is not significant.

4.4

Proof of Main Result

The proof of the the main result, Proposition 1, consists of two parts. The first part of the proof characterizes all offer strategies that sellers possibly use in a monotone anonymous PBE. The second part then establishes that the only consistent belief of sellers is then such that the buyer is very likely to be the low valuation. With this belief, all sellers make the winning offer in a twinkling. Here I prove the proposition for the special case: the Coase property holds for any r sequence of monotone anonymous PBEs (–k,m )kØ0,mØ0 œ kØ0,mØ0 EM ( k , ⁄m ) such that ›Lk,m (·) © 1 for all k and m, and limmæŒ ⁄m = Œ while limkæŒ k = 0. It is proved in the appendix that any monotone anonymous PBEs – with ›L (·) = 0 has to be a Coaean equilibrium constructed for Lemma 10, and hence the Coase property holds for any sequence of monotone anonymous PBEs in which the low type never switches to a new seller. 22

For a contradiction, suppose there are a positive number µ œ (0, 1), a sequence of intervals ( k )kØ0 converging to zero, a sequence of arrival rates (⁄m )mØ0 growing to infinity, and a sequence of monotone anonymous PBE –k,m = (‡ k,m , q k,m , · k,m , › k,m ) œ EM ( k , ⁄m ) such that ›Lk,m (·) © 1 and µ(vH ≠ vL ) > V H (?; –k,m , k , ⁄m ) (3) for any k and m. The following observation is useful in analyzing the behavior of (–k,m )kØ0,mØ0 in the limit. For all large k and m, a seller has to make the winning offer immediately if –k,m happens to be a Coasean equilibrium in EC ( k , ⁄m ). Notice that q k,m (?) Æ qˆ < q˜x1 -x=1 for all large k and m where q˜1x -x=1 is as defined in (2), because -

q˜1x -x=1 =

and limm,kæŒ

e≠r

(1 ≠

e≠r

1≠e≠r k k (1≠e≠⁄m

(1 ≠ e≠(r+⁄m ) k )vL ≠r k (1 ≠ e≠⁄m k )v H +e k)

k

)vL

> e≠r

vL 1≠e≠r k k (1≠e≠⁄m

k)

+ vL

= 0. This means that, for all large k and m, sellers never try -

xthe following two-period screening in –C œ EM ( x=1

k , ⁄m ).

• The seller offers p œ (vL , (1 ≠ e≠r k )vH + e≠r k vL ]. Because the seller never offers strictly less than vL , the high type’s best response is to accept p and q(h, p) = 0. • In the following period, the seller makes the winning offer with the zero profit. -

xHence the outcome of any Coasean equilibrium –C œ EM ( k , ⁄m ) is such that sellers x=1 make the winning offer in the first round of bargaining. This observation also means that –k,m are not Coasean equilibria in EC ( k , ⁄m ) for all large k and m. If –k,m œ EC ( k , ⁄m ),

V H (?; –k,m ,

k , ⁄m )

= vH ≠ vL > µ(vH ≠ vL )

which contradicts to the assumption (3). Hence, without loss, suppose that –k,m is not Coasean. Equivalently speaking, –k,m is not in EC ( k , ⁄m ) for every k and m. Now fix k and m. Even though –k,m is not Coasean, there is q¯k,m œ (0, q k,m (?)) such that a seller makes the winning offer for sure whenever his belief is strictly lower than q¯k,m . This belief level is well-defined because the winning offer yields a profit strictly higher than all other possibilities if seller’s belief is close to 0. The following lemma shows that (i) there is indeed a history h œ H such that q k,m (h) = q¯k,m and supp ‡ k,m (h) contains an offer other than vL , and (ii) for any history h such that q k,m (h) = q¯k,m , supp ‡ k,m (h) actually does not include a losing offer. Its proof is not difficult but is involved with some lengthy and complicated arguments which are not particularly useful to catch the idea behind Proposition 1. To prevent an unnecessary distraction, the proof of the lemma is put off to Appendix B.6. LEMMA 11.

(i) There is h œ H such that q k,m (h) = q¯k,m and supp ‡ k,m (h) contains an

offer other than vL .

23

(ii) For any h œ H such that q k,m (h) = q¯k,m , supp ‡ k,m (h) includes no losing offer. Proof. In Appendix B.6. The proof of the proposition is done in two steps. First, the offer strategy –k,m is characterized for all k and m. And then, using this characterization, the sellers’ belief q k,m (?) will be computed, and it will turn out that the competed q k,m (?) converges to zero in the limit. Finally we will see that the equilibrium condition requires that q k,m (?) has to remain larger than vL /vH even in the limit which contradicts to the observation that q k,m (?). Characterization of Offer Strategy ‡ k,m Fix h œ H such q k,m (h) = q¯k,m . Then q k,m (h, p) = q¯k,m for any p œ supp ‡ k,m (h) which implies that the belief of a seller is stuck at q¯k,m once it reaches q¯k,m . Remember that Lemma 6 guarantees q k,m (h, p) Æ q¯k,m . For a contradiction suppose this inequality holds strictly. Then, after the buyer rejects p œ supp ‡ k,m (h), the seller would offer vL in the following period by construction of q¯k,m . But, the high type strictly prefers rejecting p and waiting for seller’s winning offer in the first place, which actually leads to q k,m (h, p) Ø q¯k,m . Applying Lemma 11, it is easy to see that supp ‡(h) never includes an offer strictly larger than vH ≠ e≠r k V H (?; –k,m , k , ⁄m ) if q(h) = q¯k,m . If supp ‡ k,m (h) contains p > vH ≠ e≠r k V H (?; –k,m , k , ⁄m ), which is necessarily serious by Lemma 11 (that is to say, k,m high type accepts p with positive probability ·H (p; h) > 0), seller’s belief in the next period is k,m q¯k,m (1 ≠ ·H (p; h))e≠⁄m k < q¯k,m , k,m k,m ≠⁄ k,m ≠⁄ m m k k q¯ (1 ≠ ·H (p; h))e + (1 ≠ q¯ )e k,m hence ‡ k,m (h, p) = vL for sure. But ·H (p; h) cannot be positive because the high type always prefers delaying one more period for the seller’s winning offer in the next period. The following lemma establishes that in the non-Coasean equilibrium –k,m , once a history h œ H such that q k,m (h) = q¯k,m is reached, the high type stops searching for other sellers and never switches to a new seller. At the same time, all sellers randomize between vH ≠ e≠r k V H (?, k , ⁄m ) and vL indefinitely.

LEMMA 12. For any h œ H such that q k,m (h) = q¯k,m Ó

(i) supp ‡ k,m (h) = vL , vH ≠ e≠r

k

V (?; –k,m ,

Ô

k , ⁄m )

(ii) the high-valuation buyer never switches to a new seller after she rejects the offer vH ≠ e≠r

k

V (?; –k,m ,

k , ⁄m )

after h.

Proof. The idea is that whenever a seller decides to catch the high type buyer by offering one of secure offers, a higher secure offer always yields a higher profit than a lower one. Hence, in equilibrium, the seller does not offer a secure offer other than the highest one, 24

vH ≠ e≠r

k

V H (?; –k,m ,

k , ⁄m ).

For a contradiction, let the set of secure offers (other than

the winning offer) that sellers possibly make with belief q¯k,m in equilibrium –k,m , S©

Y ]



[



hœH:q k,m (h)=¯ q k,m

supp ‡ k,m (h) : vL < p < vH ≠ e≠r V H (?; –k,m ,

Z ^

k , ⁄m )

\

be nonempty and denote the minimum (infimum) offer in this set by p† . Since sellers never ¯ charge a price strictly lower than (1 ≠ e≠r )vH + e≠r vL (Lemma 4), p† Ø (1 ≠ e≠r )vH + ¯ e≠r vL . Fix a history h œ H such that q k,m (h) = q¯k,m and p† œ supp ‡ k,m (h). If p† ”œ S, consider ¯ ¯ an offer slightly higher than p† is in supp ‡ k,m (h) instead of p† . First note that p† has to be ¯ ¯ ¯ k,m serious; otherwise, because ›Lk,m (h, p† ) = 1 and ›H (h, p† ) = 0, q k,m (h, p† ) > q k,m (h) = q¯k,m ¯ ¯ ¯ which contradicts to the skimming property (Lemma 6). On the other hand, because p† is not ¯ the winning offer, the high type rejects it with a positive probability. Actually q k,m (h, p† ) = ¯ q k,m (h) = q¯k,m by the skimming property, and hence the high type actually accepts p† with ¯ probability 1 ≠ e≠⁄m k . In the following period, to make the high type willing to reject p† , ¯ the seller charges the winning offer with a positive probability. Hence seller’s expected profit conditional on h is V S (h) = p† q¯k,m (1 ≠ e≠⁄m ¯

k

) + e≠(r+⁄m )

k

vL .

Now suppose the seller deviates to a slightly higher secure offer, say pÕ , instead of p† . The ¯ monotonicity of supp ‡ k,m implies p† Æ p+ (¯ q k,m ) < (1 ≠ e≠r ¯

k

)vH + e≠r

k

p+ (¯ q k,m ).

On the other hand, p† Ø (1 ≠ e≠r ¯

k

)vH + e≠r

k

p≠ (¯ q k,m ) = (1 ≠ e≠r

k

)vH + e≠r

k

vL ,

thus there exists pÕ such that p† < pÕ < vH ≠ e≠r k V H (?; –k,m , k , ⁄m ).13 If the seller ¯ deviates to pÕ after h such that q k,m (h) = q¯k,m , by Lemma 7, the seller’s belief in case the buyer rejects pÕ is again q¯k,m . This also requires pÕ is accepted by the high type with probability 1 ≠ e≠⁄m

. As long as the gap between pÕ and p† is small enough, the high type ¯ is willing to accept pÕ only if seller’s next offer is vL with positive probability. So the seller’s 13

k

See page 13 for the definitions of p+ (·) and p≠ (·).

25

profit from this deviation is pÕ q¯k,m (1 ≠ e≠⁄m which is strictly higher than V S (h; –k,m ,

k

) + e≠(r+⁄m )

k , ⁄m ).

k

vL

However, this cannot be true in equi-

librium. This contradiction is resolved only if the set S is empty. The only possibility is that



hœH:q(h)=¯ q k,m

supp ‡ k,m (h) = {vL , vH ≠ e≠r V H (?; –k,m ,

k , ⁄m )}.

The seller randomizes over the two prices with belief q¯k,m . The high type accepts vH ≠ e≠r V H (?) with probability 1 ≠ e≠⁄m

k

so that the belief stays at q¯k,m even after the

buyer rejects seller’s offer. At the same time, the high type never takes outside option while the seller randomizes over the two prices. Otherwise, making a secure offer very close to vH ≠ e≠r

k

V H (?; –k,m ,

k , ⁄m )

to catch the high type would be a profitable deviation for

the seller. From the sellers’ indifference condition, vH ≠ e≠r =∆ q¯k,m =

k

V H (?; –k,m ,

1 ≠ e≠(r+⁄m ) 1 ≠ e≠⁄m k

q k,m (1 k , ⁄m )¯

k

vH ≠ e≠r

k

≠ e≠⁄m

vL V H (?; –k,m ,

k

) + e≠(r+⁄m )

k

vL = vL (4)

k , ⁄m )

for all large k and m. On the other hand, the indifference condition of the high type 1

vH ≠ vH ≠ e≠r

k

V H (?; –k,m ,

2

≠r k , ⁄m ) = e

requires that sellers charge vL with probability y k,m © and vH ≠ e≠r tion yields

k

V H (?; –k,m ,

(vH ≠ vL )

(1 ≠ e≠r k )V H (?; –k,m , k , ⁄m ) vH ≠ vL ≠ e≠r k V H (?; –k,m , k , ⁄m ) k , ⁄m )

k , ⁄m )

= vL

V H (h; –k,m ,

k , ⁄m )

= e≠r

k,m

,

(5)

with the complementary probability. A simple calcula-

V S (h; –k,m , VL (h; –

k

k , ⁄m )

k

V H (?; –k,m ,

k , ⁄m )

=0

whenever q k,m (h) = q¯k . Notice that y k,m , V S (h; –k,m , not directly depend on the arrival rate of new sellers. 26

k , ⁄m )

and V H (h; –k,m ,

k , ⁄m )

do

Finally, the monotonicity of ‡ k,m implies that q k,m (?) = q¯k,m . LEMMA 13.

(i) q k,m (?) = q¯k,m ;

(ii) all sellers keep randomizing over two price offers vH ≠ e≠r

k

V H (?; –k,m ,

k , ⁄m )

and

vL ;

(iii) on equilibrium path, a seller’s belief keeps staying at q¯k,m indefinitely. Proof. It suffices to show (i). First notice that if q k,m (?) < q¯k,m , then sellers charge the winning offer immediately by construction. Hence, V H (?; –k,m ,

k , ⁄m )

= vH ≠ vL > µ(vH ≠ vL )

which contradicts the hypothesis V H (?; –k,m , other hand, suppose

q k,m (?)

>

q¯k,m .

k , ⁄m )

Æ µ(vH ≠ vL ) for all k and m. On the

By monotonicity of ‡ k,m , any offers from sellers at the

null history ? œ H has to be no lower than vH ≠e≠r

k

V H (?; –k,m ,

offer upon the arrival to the buyer is therefore no lower than and Ë

V H (?; –k,m ,

1

≠r k , ⁄m ) Æ (vH ≠ vH ≠ e

k

V H (?; –k,m ,

k , ⁄m ). Any seller’s first vH ≠ e≠r k V H (?; –k,m , k , ⁄m )



k , ⁄m )

< V H (?; –k,m ,

k , ⁄m )

which is a contradiction. Computation of q k,m (?) By Lemma 13, q k,m (?) = q¯k,m for all k and m. At the null history, by Lemma 12, all sellers randomize between vL and vH ≠ e≠r k V H (?; –k,m , k , ⁄m ) indefinitely in every –k,m , with the probability y k,m and 1 ≠ y k,m respectively. Recall that the high type never switches to another seller, once she begins to bargain with the first arrived seller. On the other hand, after the first seller arrives, the low type buyer purchases the good with probability y k,m in every period. Therefore, conditional upon being matched with the buyer, each seller’s belief q k,m (?) is q k,m (?) = q

≠(n≠1)⁄m nØ1 e



k

+ (1 ≠

q

≠(n≠1)⁄m k nØ1 e q q ≠(u≠1)⁄m qˆ) nØ2 n≠1 u=1 e

k

(1 ≠ e≠⁄m

k

)(1 ≠ y k,m )n≠u

Here, for each n Ø 2, the term e≠(n≠1)⁄m k represents the probability that no seller has q ≠(u≠1)⁄m k (1≠e≠⁄m k )(1≠y k,m )n≠u represents arrived up until period n≠1. The term n≠1 u=1 e the probability that one or more sellers have arrived up until period n ≠ 1, but none of them has not offered the winning offer, and hence the low type remains in market. A simple 27

calculation yields, q k,m (?) =

qˆ 1 + (1 ≠

≠⁄m qˆ) 1≠e 1≠e≠r

k k

vH ≠vL ≠V H (?;–k,m , k ,⁄m ) V H (?;–k,m , k ,⁄m )

.

By the hypothesis (3), q k,m (?) is bounded from above by qˆ 1 + (1 ≠

≠⁄m qˆ) 1≠e 1≠e≠r

k k

1≠µ µ

which converges to zero in the limit. On the other hand, from (4), q k,m (?) = q¯k,m =

1 ≠ e≠(r+⁄m ) 1 ≠ e≠⁄m k

k

vH ≠

e≠r

k

V

vL H (?; –k,m ,

k , ⁄m )

which is bounded from1 below by 2 vL /vH > 0 for any k and m. This contradicts to the k,m last observations that q (?) converges to zero, and it completes the proof of kØ0,mØ0

Proposition 1 for the special case where ›Lk,m (·) = 1 for all k and m.

5

Contrast to Bargaining with Exogenous Outside Options

Proposition 1 establishes that the Coase conjecture holds when the buyer’s opportunity to switch to new sellers serves as an endogenous outside option, under the refinement assumptions R1–R4. A natural question is whether this result is driven by the endogeneity of buyer’s outside option or by the refinement assumptions. Board and Pycia (2014) and Hwang (2013) study bargaining models in which the buyer can exercise an outside option of exogenously fixed value and show that the Coase conjecture may fails in some equilibria.14 Moreover, equilibrium strategy profiles of these non-Coasean equilibria are seemingly consistent with the refinement conditions R1–R4. This observation suggests that the endogeneity of buyer’s outside options is the key for the Coase conjecture established in Proposition 1. Of course, the bargaining models studied by Board and Pycia (2014) and Hwang (2013) are not exactly equivalent to the current paper’s, and hence this observation is only suggestive. For a more conclusive answer, I will introduce another bargaining game between a single seller and a single buyer where the buyer has outside options of exogenously fixed values. The buyer’s outside options arrive in the same way that new sellers arrive in the original model studied in previous sections. There is no difference between two models other than the values of buyer’s outside options are determined endogenously in one model, while they are exogenously fixed in another. However, the model with exogenous outside options admits an equilibrium for which the refinement conditions R1–R4 hold but the Coase conjecture fails. 14

Actually the model studied by Board and Pycia (2014) admits a unique equilibrium.

28

Consider a bargaining game where a single buyer and a single seller negotiate over the price. As in the original model presented in Section 2, the buyer’s valuation of the seller’s good is either vH or vL with probabilities qˆexo œ (0, 1) and 1 ≠ qˆexo respectively. Suppose seller’s valuation of the good is again normalized to be zero, and vH > vL > 0. At the beginning of each period n Ø 2,15 an outside option of the buyer is available with probability 1 ≠ e≠⁄ . If the buyer takes an outside option, then the game immediately ends by yielding wH to the high type buyer, wL to the low type buyer, and zero to the seller as their final payoffs. Otherwise, the buyer and seller keeps on bargaining until the buyer finally accepts a seller’s offer or takes an outside option. All other features and notations are also inherited from the original model, including the discount rates r > 0, the time interval between two consecutive offers > 0, the information structure, and so on. In particular, the seller cannot observe the arrival/non-arrival of outside options, and neither past outside options nor past offers from the seller can be recalled. h œ H generically stands for a history of seller’s all declined offers. ‡ and q represent the seller’s offer strategy and belief system respectively while ›H (›L ) and ·H (·L ) are high type (low type) buyer’s behavioral strategies, the probability of exercising an outside option (if available) and the probability of accepting seller’s offer, respectively. This game is referred to as the bargaining game with exogenous outside options throughout the paper.16 The solution concept for this game is a perfect Bayesian equilibrium (PBE) that satisfies modified versions of refinement assumptions R1–R4. Indeed, the assumption R1-(i) that all sellers adopt the same offer strategy and belief system is irrelevant in the current context because only one seller appears in this game. Also, the assumption R4 can be applied only after replacing V H (?; –, , ⁄) by wH . The assumptions R1-(ii), R2, and R3 can be imposed as they are stated in Section 2.2. A PBE that satisfies these modified versions of R1–R4 will be referred to as monotone PBE. EM,exo ( , ⁄) stands for the set of all monotone PBEs, and its generic element is denoted by – = (‡, q, ·, ›). Throughout this section, the assumption vH ≠ vL Ø wH > wL = 0

(6)

will be maintained. There are two motivations for this assumption. First, this assumption guarantees that an immediate agreement is the efficient outcome. Second, note that a similar condition always holds for the original bargaining game with endogenous outside options, 15

There is no available outside option with probability 1 at the initial period. This assumption is introduced in order to simplify notations and analysis. All results remain valid even without it. Also notice that the assumption is actually consistent with the timing structure of the original bargaining model with arriving sellers. In that model, if a seller just begins to bargain with the buyer in period n, he knows he is the only seller who arrived in period n. 16 The bargaining game with exogenous outside options is mathematically equivalent to the model studied in Hwang (2013). As the arrival rate of ⁄ goes to infinity, the physical environment of this game converges to the model studied in Board and Pycia (2014) where the buyer’s outside option is available in every period.

29

replacing wH and wL by V H (?; –, , ⁄) and V L (?; –, , ⁄ respectively. Actually, vH ≠ vL Ø V H (?; –, , ⁄) > V L (?; –, , ⁄) = 0 for any monotone anonymous PBE – œ EM ( , ⁄) for the original bargaining game with endogenous outside options. One example of a non-Coasean monotone equilibrium for the barging game with exogenous outside options is deadlock equilibrium characterized by Hwang (2013) and re-presented here. For now suppose vH ≠vL > wH > wL = 0 which is slightly stronger than the assumption (6). Fix > 0 and suppose qD ©

1 ≠ e≠(r+⁄) vL < qˆexo 1 ≠ e≠⁄ vH ≠ e≠r wH

(7)

qD <

(1 ≠ e≠(r+⁄) )vL . (1 ≠ e≠r )vH + e≠r (1 ≠ e≠⁄ )vL

(8)

and

Notice that (7) and (8) hold for all small and all large ⁄, as long as vH ≠ vL > wH > 0 as assumed and qˆexo is sufficiently close to 1. Under these parametric restrictions, I will construct a monotone PBE – for which the Coase conjecture fails. The equilibrium strategies of the players are as follows. First of all, at any history, the low type buyer always exercises the outside option whenever one arrives while she accepts seller’s offer if and only if it is no higher than vL . Formally, ›L (h) © 1 and ·L (p; h) = {p Æ vL } for any h œ H and p Ø 0. Consider a history h œ H such that q(h) < qD . At such a history, the high type does not take outside option in the arrival stage even if an outside option arrives (›H (h) = 0). In the bargaining stage, the seller immediately offers p = vL and the high type accepts it for sure. By the assumption vH ≠ vL > wH , the high type indeed has no incentive to exercise her outside option in the arrival stage. Next consider a history h œ H such that q(h) = qD < q(?) = qˆexo . At such a history, the behavioral strategies of the high type and the seller depend on the seller’s last rejected offer, say p≠1 . • Seller’s strategy at h such that q(h) = qD : The seller randomizes over vH ≠ e≠r wH and vL with probability „D (p≠1 ) and 1 ≠ „D (p≠1 ) respectively where „D (p≠1 ) =

Y _ _ ]

0 1

_ _ [ p≠1 ≠(1≠e≠r )vH ≠e≠r vL e≠r (vH ≠e≠r wH ≠vL )

if p≠1 < (1 ≠ e≠r )vH + e≠r vL if p≠1 > vH ≠ e≠r wH otherwise.

• High type buyer’s strategy at h such that q(h) = qD : If vL Æ p≠1 Æ vH ≠ e≠r wH , the 30

high type buyer never takes an outside option even if one arrives. Otherwise, the high type buyer takes an outside option for sure. That is to say, Y Ë È ] 0 if p œ v , v ≠ e≠r w ≠1 L H H ›H (h) = [ 1 otherwise.

Regarding the acceptance strategy ·H , the high type buyer accepts any offer p such that (1 ≠ e≠r )vH + e≠r vL < p Æ vH ≠ e≠r wH with probability 1 ≠ e≠⁄ . The high type buyer accepts any offers not higher than (1 ≠ e≠r )vH + e≠r vL with probability 1 while rejects any offers higher than vH ≠ e≠r wH . Y _ _ ]

0 ·H (p; h) = 1 _ _ [ 1 ≠ e≠⁄

if p > vH ≠ e≠r wH if p Æ (1 ≠ e≠r )vH + e≠r vL otherwise.

It is straightforward to show that all players have no incentive to deviate from the above behavioral strategies, once a history h œ H such that q(h) = qD is reached. Also, one can easily check that the seller’s belief would stay at qD indefinitely. On the equilibrium path, the seller keeps randomizing over two prices, vH ≠ e≠r wH and vL indefinitely. Along the equilibrium path on which the seller keeps charging vH ≠ e≠r wH , the high type accepts the seller’s offer with probability 1 ≠ e≠⁄ in each period while the low type never accepts it. Meanwhile, the high type always forgoes her outside option even if one arrives, but the low type immediately exercises her outside option whenever available. An outside option arrives with probability 1 ≠ e≠⁄ in each period, which is equal to the probability that the high type accepts the seller’s offer. Hence the seller’s belief stays at qD indefinitely. The equilibrium construction is completed by specifying each player’s equilibrium behavioral strategy for any history h such that q(h) > qD . Actually, following an argument similar to the proof for Lemma 10, we can construct an equilibrium where the seller’s offer strategy is such that † • the seller charges p†j with probability 1 whenever his belief is in (qj† , qj+1 );

• the seller randomizes p†j and p†j≠1 when his belief is exactly qj† where S

and

p†j = vH ≠ e≠r U1 ≠ e≠⁄ (1 ≠ e≠r )

j≠1 ÿ

k=0

e≠rk e≠⁄k V wH

† qD = q0† < q1† < . . . < qN = qˆexo

31

T

for any j Ø 1.

S

h:q(h)=q

supp (h)

p†3 p†2 p†1 vH

e

r

wH

vL 0

qD

q1†

q2†

q3† = qˆexo

q

Figure 3: The support of equilibrium offer strategy (for a case with ND = 3). The horizontal axis represents seller’s belief levels. The vertical axis represents the set of offers that the seller possibly makes with each belief level. On the equilibrium path, the seller sequentially offers p†3 , p†2 , and p†1 . Conditional on p†1 being rejected, the seller’s belief reaches qD , and the seller randomizes vH ≠ e≠r wH and vL indefinitely until the buyer accepts one of them. are the belief levels that rationalize the seller’s randomization over p†j≠1 and p†j with the belief equal to qj† . On equilibrium path, the seller makes serious offers p†N , p†N ≠1 , . . . , p†0 in sequence. The high type accepts each offer with a certain probability so that the seller’s † belief is updated from qj† to qj≠1 whenever the buyer rejects p†j on equilibrium path. Hence the seller’s belief reaches qD in N periods. Actually, as in the proof of Lemma 10, we can bound N by ND where ND can be chosen independent of . Hence, the seller’s belief reach qD almost immediately in the continuous-time limit. Note that the deadlock equilibrium satisfies all refinement conditions R1–R4 imposed for Proposition 1. The buyer’s behavioral strategies only depend on history of declined offers of the seller, and does not depend on the buyer’s private observation of outside options (R1) The seller’s equilibrium offer strategy is monotone in belief (R2). See Figure 3. The low type keeps exercising an outside option whenever one arrives (R3). Finally, the assumption R4 also holds because the seller never offers p œ (vL , vH ≠e≠r wH ) in this equilibrium. Also note that the seller’s first serious offer p†N is strictly higher than vH ≠ e≠r wH . Hence the high type’s equilibrium payoff is no higher than e≠r wH which is strictly smaller than vH ≠ vL by assumption. Hence, the Coase conjecture fails for a sequence of deadlock equilibria. It is worthwhile to note that a delay of agreement is also present in the deadlock equilibrium. Actually it is straightforward to show that the expected length of time until the buyer

32

finally takes either an outside option or a seller’s offer is (1 ≠ qˆexo )

vH ≠ vL ≠ wH + O( ). ⁄(vH ≠ vL ≠ wH ) + rwH

On the other hand, conditional on the event that no outside option ever arrives to the buyer, the expected length of time until the an agreement between the buyer and the seller is eventually reached is vH ≠ vL ≠ wH (1 ≠ qˆexo ) + O( ). (9) rwH Once the seller’s equilibrium belief reaches to qD then the high type would never take an outside option, and hence the seller’s belief can stay high so that the seller is not tempted to make the winning offer immediately. In the bargaining model with endogenous outside options (with arriving sellers) studied in Section 4, such a high belief cannot be sustained because sellers would immediately understand that only the low type would actively search in equilibrium. This section concludes by discussing general properties of monotone PBEs for the bargaining game with exogenous outside options. The first result (Proposition 2) identifies the condition under which a monotone PBE with a positive delay exists. With (7), (8), we have seen that there is a sequence of monotone PBEs along which the Coase property fails. The condition (8) ensures that the seller never tries the following two-period screening in equilibrium, as long as his belief is no higher than qD . • The seller offers p œ (vL , (1≠e≠r )vH +e≠r vL ]. Because the seller never offers strictly less than vL , the high type’s best response is to accept p and q(h, p) = 0. • In the following period, the seller makes the winning offer. The expected profit from the two-period screening is at most (where q is seller’s belief and q Æ qD ) Ë È q (1 ≠ e≠r vH + e≠r vL + e≠r e≠⁄ (1 ≠ q)vL which is not higher than vL if and only if (8) holds. Without (8), the seller would try the two-period screening with belief qD instead of following the equilibrium path. Appendix B.7 shows that, however, the condition (8) is dispensable for the construction of non-Coasean equilibrium by letting the seller randomizing vH ≠ e≠r wH , and (1 ≠ e≠r Ÿ )vH + e≠r Ÿ vL (instead of vL ) for an integer Ÿ Æ NC . As long as the seller keeps charing vH ≠ e≠r wH , the trade keeps being delayed. Once the seller offers (1 ≠ e≠r Ÿ )vH + e≠r Ÿ vL and then it is rejected, the seller offers (1≠e≠r (Ÿ≠1) )vH +e≠r (Ÿ≠1) vL , (1≠e≠r (Ÿ≠2) )vH +e≠r (Ÿ≠2) vL , . . ., and finally the winning offer vL within Ÿ periods. PROPOSITION 2. Suppose vH ≠ vL > wH > wL = 0. The bargaining game with exogenous

outside options admits a monotone PBE – œ EM,exo ( , ⁄) in which the high type’s expected 33

equilibrium payoff is bounded from above by wH if and only if 0<

⁄+r vL Æ qˆexo . ⁄ vH ≠ wH

Proof. In Appendix B.7. Finally, the next proposition shows that any monotone PBE for the bargaining game with exogenous outside options is qualitatively equivalent to either the Coasean equilibrium I constructed for Lemma 10, or the deadlock equilibrium. PROPOSITION 3. Any monotone PBE for the bargaining game with exogenous outside options is qualitatively equivalent to either Coasean equilibrium or deadlock equilibrium. The precise statement of Proposition 3 and its proof can be found in Appendix (see Section B.8).

6

Concluding Remarks

This paper has discussed the problem of bargaining under asymmetric information with arriving new sellers. The buyer’s has an option to switch to future sellers which serves as an endogenously generated outside option. The main result is that the Coase conjecture holds if the arrival rate of new sellers is high enough. The buyer trades with the first seller immediately at the competitive price level. This result is in contrast with the outcome in bilateral bargaining with buyer’s exogenous outside options in which the Coase conjecture may fail. The failure of the Coase conjecture with exogenous outside options, therefore, comes with caveats. The main result of the paper also has interesting implications for markets with incomplete information. It demonstrates that no sophisticated market mechanism is required to implement the efficient outcome in such markets. As long as buyers have an option to choose own bargaining counterparty freely, the efficient outcome is achieved through a conventional bargaining protocol. The result also explains why some decentralized markets function well with minimal monitoring and intervening in despite of incomplete information. There are a number of directions to extend this paper. The paper deals with the twotype model where the support of the buyer’s valuation consists of only two points. It would be interesting to study a model with more than two types of buyer. The main obstacle is that the sellers’ belief is a multi-dimensional object. The standard bargaining model with one seller does not suffer this difficulty because the seller’s belief is always one-dimensional object due to what is called the skimming property. In the model of the current paper, there is only one buyer while there are multiple sellers. The buyer has an option to switch own bargaining counterparty, but sellers do not. A natural question is what would happen if there are multiple sellers and buyers, all of who search for their counterparty. 34

References Abreu, D., and F. Gul (2000): “Bargaining and Reputation,” Econometrica, 68(1), 85– 117. Ausubel, L. M., P. Cramton, and R. J. Deneckere (2002): “Bargaining with incomplete information,” in Handbook of Game Theory with Economic Applications, ed. by R. J. Aumann, and S. Hart, vol. 3, pp. 1897–1945. Elsevier. Ausubel, L. M., and R. J. Deneckere (1989): “Reputation in Bargaining and Durable Goods Monopoly,” Econometrica, 57(3), 511–531. Bikhchandani, S. (1992): “A Bargaining Model with Incomplete Information,” The Review of Economic Studies, 59(1), 187–203. Board, S., and M. Pycia (2014): “Outside Options and the Failure of the Coase Conjecture,” American Economic Review, 104(2), 656–671. Cho, I.-K. (1990): “Uncertainty and Delay in Bargaining,” Review of Economic Studies, 57(4), 575–595. Coase, R. H. (1972): “Durability and Monopoly,” The Journal of Law and Economics, 15(1), 143–149. Compte, O., and P. Jehiel (2002): “On The Role of Outside Options in Bargaining with Obstinate Parties,” Econometrica, 70(4), 1477–1517. Conlisk, J., E. Gerstner, and J. Sobel (1984): “Cyclic Pricing by a Durable Goods Monopolist,” Quarterly Journal of Economics, 99(3), 489–505. Cramton, P., R. Gibbons, and P. Klemperer (1987): “Dissolving a Partnership Efficiently,” Econometrica, 55(3), 615–632. Daley, B., and B. Green (2012): “Waiting for News in the Market for Lemons,” Econometrica, 80(4), 1433–1504. Deneckere, R. J., and M.-Y. Liang (2006): “Bargaining with Interdependent Values,” Econometrica, 74(5), 1309–1364. Evans, R. (1989): “Sequential Bargaining with Correlated Values,” Review of Economic Studies, 56, 499–510. Fuchs, W., and A. Skrzypacz (2010): “Bargaining with Arrival of New Traders,” American Economic Review, 100(3), 802–836. Fudenberg, D., D. Levine, and J. Tirole (1985): “Infinite Horizon Models of Bargaining with One-sided Incomplete Information,” in Bargaining with Incomplete Information, ed. by A. E. Roth, pp. 73–98. Cambridge University Press, London/New York. 35

(1987): “Incomplete Information Bargaining with Outside Opportunities,” The Quarterly Journal of Economics, 102(1), 37–50. Fudenberg, D., and J. Tirole (1991): Game Theory. The MIT Press. Gul, F., and H. Sonnenschein (1988): “On Delay in Bargaining with One-Sided Uncertainty,” Econometrica, 56(3), 601–611. Gul, F., H. Sonnenschein, and R. Wilson (1986): “Foundations of Dynamic Monopoly and the Coase Conjecture,” Journal of Economic Theory, 39(1), 155–190. Hörner, J., and N. Vieille (2009): “Public vs. Private Offers in the Market for Lemons,” Econometrica, 77(1), 29–69. Hwang, I. (2013): “A Theory of Bargaining Deadlock,” PIER Working Paper. Inderst, R. (2008): “Dynamic Bilateral Bargaining under Private Information with a Sequence of Potential Buyers,” Review of Economic Dynamics, 11(1), 220–236. Kim, K. (2014): “Information about Sellers’ Past Behavior in the Market for Lemons,” mimeo. Lauermann, S., and A. Wolinsky (2013): “Search with Adverse Selection,” Working Paper. Lee, J., and Q. Liu (2013): “Gambling Reputation: Repeated Bargaining With Outside Options,” Econometrica, 81(4), 1601–1672. Myerson, R. B., and M. A. Satterthwaite (1983): “Efficient Mechanisms for Bilateral Trading,” Journal of Economic Theory, 29(2), 265–281. Rubinstein, A. (1985): “A Bargaining Model with Incomplete Information About Time Preferences,” Econometrica, 53(5), 1151–1172. Rubinstein, A., and A. Wolinsky (1985): “Equilibrium in a Market with Sequential Bargaining,” Econometrica, 53(5), 1133–1150. Sobel, J. (1991): “Durable Goods Monopoly with Entry of New Consumers,” Econometrica, 59(5), 1455–1485. Vincent, D. R. (1989): “Bargaining with Common Values,” Journal of Economic Theory, 48(1), 47–62.

36

Appendix A

Foundational Matching Mechanism

In the bargaining model with arriving sellers introduced in Section 2, a new seller randomly arrives in each period with probability 1≠e≠⁄ , without letting a seller know which period he is in. In this model, all sellers cannot observe the calendar time, and hence they do not know which period they are in. All sellers share the same prior over the calendar time conditional upon their arrival to the buyer. One might ask if there is any explicit matching mechanism that gives a foundation to the model. This section provides with one explicit matching mechanism. At the beginning of the game, suppose that nature chooses the number of potential sellers, Ns , according to the geometric distribution with mean Nsú > 1. That is to say, 3

1 P{1 Æ Ns Æ N } = 1 ≠ 1 ≠ ú Ns

4N

for each N Ø 1. After Ns is realized, nature randomly draws Ns sellers from the pool of sellers [0, 1]. In each period n = 1, 2, . . ., nature sends a seller (out of Ns potential sellers) to the buyer with probability 1 ≠ e≠fl in random order. If a sent seller is not chosen as buyer’s bargaining counterparty, then he leaves the market immediately and never returns. In this case, each seller’s prior over calendar time conditional upon arriving to the buyer is P(calendar time = n ) =

n ÿ N 3 ÿ

1 1≠ ú Ns

Œ ÿ

N 3 ÿ

N =1 k=1 ¸

+

4N ≠1

˚˙

=P{Ns =N }

N =n+1 k=1

A

B

1 1 n≠1 ◊ ◊ (1 ≠ e≠fl )k efl(n≠k) ú Ns N k≠1

1 1≠ ú Ns

˝

4N ≠1

1 1 ◊ ◊ (1 ≠ e≠fl )n . Nsú N

with the convention that an empty sum is zero. In the summations, the term to nature’s shuffling the order of sellers. Notice that 3

1≠

1 Nsú

4

1 N

appears due

= P{Ns Ø n + 1|Ns Ø n} ’n

Hence if fl > 0 is chosen so that 3

4

1 1 ≠ ú (1 ≠ e≠fl ) = 1 ≠ e≠⁄ Ns

≈∆ fl =

a new seller arrives with probability 1 ≠ e≠⁄ ú s Notice lim æ0 fl = NNú ≠1 ⁄. s

37

1

3

4

Nsú ≠ 1 log , Nsú e≠⁄ ≠ 1

in the buyer’s perspective, at each period.

Appendix B

Proofs

Some Definitions Throughout the appendix, fi(h; –, ) or simply fi(h) refers to the seller’s first serious offer made after history h in assessment –. For example, suppose that, after history h, the seller is supposed to offer p1 and p2 in a sequence where p1 is losing and p2 is serious. In this case fi(h) = p2 . Note that fi(h) is a random variable when the seller is supposed to randomize after history h. From Bayes rule, seller’s posterior is updated from q Õ to q ÕÕ after observing the buyer rejects the last offer if and only if the high type was supposed to accept it with probability — ©1≠

q ÕÕ (1 ≠ q Õ ) q Õ (1 ≠ —) ÕÕ y ≈∆ q = (1 ≠ q ÕÕ )q Õ q Õ (1 ≠ —) + (1 ≠ q Õ )y

where y is the ratio of the probabilities that each does not take outside option after the last offer has been rejected. Throughout the appendix let —(q Õ , q ÕÕ , y) = 1 ≠

q ÕÕ (1 ≠ q Õ ) y (1 ≠ q ÕÕ )q Õ

for any q Õ , q ÕÕ œ [0, 1] and y > 0.

B.1

Proof of Lemma 1

Let p be the infimum of offers that the possibly makes in equilibrium. Formally let ¯ p = inf supp ‡(h). ¯ hœH For a contradiction, suppose p < vL . In this case, both types have to accept any offer ¯ )v + e≠r p. First of all, it is straightforward to check p such that p Æ p < (1 ≠ e≠r L ¯ ≠r vj ≠ p > e (vj ≠ p) for both j = H, L. ¯Hence, if buyer with valuation vj is supposed ¯ to decline p with a positive probability, she has to wait for new seller indefinitely after she declines p. But the value of waiting for new seller is at most ÿ

kØ0

e≠r

(k+1) ≠⁄ k

e

(1 ≠ e≠⁄ )(v ≠ p) =

e≠r (1 ≠ e≠⁄ ) (v ≠ p) < e≠r (v ≠ p) 1 ≠ e≠(r+⁄)

thus accepting p yields a higher payoff than waiting for outside options to both types. Now pick h œ H and p œ supp ‡(h) such that p Æ p < (1 ≠ e≠r )vL + e≠r p ¯ ¯ 38

and p œ supp ‡(h).

Because both type accepts p for sure, V S (h) = p. But if the seller deviates to pÕ œ (p, (1 ≠ e≠r )vL + e≠r p), sellers’ profit is pÕ > p, contradiction. ¯

B.2

Proof of Lemma 2

Because sellers never offer a price strictly lower than vL in any continuation games,17 the high type has to accept p Æ vL for sure whenever offered. By the same argument, the low type also accepts any offer p < vL . Suppose ·L (vL ; h) < 1 for a history h œ H. ·L (vL ; h) = 1 if vL œ supp ‡(h); otherwise, any offer slightly lower than vL dominates vL at h. So ·L (vL ; h) is possibly less than 1 only if p = vL is not in supp ‡(h). Therefore it suffices to show that there is pˆ ”= vL such that the deviation to pˆ after h yields a higher profit to the seller than the deviation to vL . Note that the profit from the deviation to vL is Ë

È

© q(h) + (1 ≠ q(h))·L (vL ; h) vL + e≠r ’L (h)(1 ≠ q(h))(1 ≠ ·L (vL ; h))vL where ’L (h) = (1 ≠ e≠⁄ )(1 ≠ ›L (h)) + e≠⁄ is the probability that the low type returns to the seller in the next period. is strictly smaller than vL as long as ·L (vL ; h) < 1 so the deviation to pˆ such that < pˆ < vL yields higher profit to the seller than the deviation to vL .

B.3

Proof of Lemma 3

By Lemma 1-(i), vH ≠ vL Ø V H (?; –, , ⁄) for any PBE – that satisfies R1. Suppose vH ≠ vL = V H (?; –, , ⁄). This equation is equivalent to (1 ≠ e≠r )vH + e≠r vL = vH ≠ e≠r V H (?; –, , ⁄). Note that, because all sellers never offer strictly less than vL , the high type has to accept any offer p strictly lower than vH ≠ e≠r V H (?; –, , ⁄) which yields her e≠r (vH ≠ vL ) as final payoff. If p is rejected, because either the buyer is the low type or the high type buyer deviates from the equilibrium strategy, the seller’s updated belief in the following period would be zero and the winning offer is made. Switching to new seller is only worth strictly less than vH ≠ vL , thus the high type would not switch between two consecutive offers p and vL indeed. Now consider the case vH ≠vL > V H (?; –, , ⁄). Suppose pn < vH ≠e≠r V H (?; –, , ⁄) and ›H (h) > 0. Such a history is actually never reached when the buyer is high type. To see this, first note that ›H (h) > 0 only if V H (h; –, , ⁄) Æ V H (?; –, , ⁄), and hence the high type buyer’s expected payoff from turning down pn is at most e≠r V H (?; –, , ⁄). 17 A continuation game refers to an information set and all following nodes that follow from that information set, plus seller’s belief over buyer’s type and her private history, inherited from the equilibrium assessment for the whole game.

39

On the other hand, if she accepted pn , she could obtain vH ≠ pn > e≠r V H (?; –, , ⁄) as final payoff. In equilibrium the high type accepts pn for sure. In particular, h is not reached conditional on that the buyer is high type. Moreover q(h) = 0, supp ‡(h) = {vL }, and V H (h; –, , ⁄) = vH ≠ vL Ø V H (?; –, , ⁄), and hence V H (h; –, ) = V H (?; –, , ⁄). Hence ›H (h) can be modified freely to construct an equivalent PBE. Next suppose pn > vH ≠ e≠r V H (?; –, , ⁄) and pn is serious but ›H (h) < 1. Because the high type would turn down outside option with positive probability after history h, e≠r V H (h, p; –, , ⁄) Ø e≠r V H (?; –, , ⁄), and hence the high type’s continuation payoff from declining p is e≠r V H (h, p; –, , ⁄). On the other hand, if the high type accepts p, her payoff is at most vH ≠ pn < e≠r V H (?; –, , ⁄) Æ e≠r V H (h; –, , ⁄). But the high type can guarantee the payoff e≠r V H (h; –, , ⁄) by rejecting pn ; hence pn could not be a serious offer in the first place, contradiction.

B.4

Proof of Lemma 5

Fix an arbitrary equilibrium assessment – = (‡, q, ·, ›) and fix a history h œ H and consider ˇœH a continuation game beginning at h. First suppose that, in equilibrium –, no history h ˇ < 1 is reached in this continuation game. In this case, there are w > 0 and k > 0 with ›H (h) such that seller’s belief drops below q(h) ≠ w in first k periods of the continuation game. Otherwise, C

D

(q(h) ≠ w)(1 ≠ q(h)) yL,k V (h; –, , ⁄) Æ 1 ≠ vH + e≠rk vH . q(h)(1 ≠ q(h) + w) yH,k S

where yj,k is the probability with which a buyer with valuation vj does not take outside option in first k periods. By assumption, yH,k = e≠⁄k and yL,k Ø e≠⁄k hence 5

6

(q(h) ≠ w)(1 ≠ q(h)) V (h; –, , ⁄)) Æ 1 ≠ vH + e≠rk vH . q(h)(1 ≠ q(h) + w) S

But this inequality cannot hold for small w and large k. The iteration of this argument shows that the seller’s posterior eventually reaches zero in equilibrium, and hence the seller eventually offers vL . ˇ with ›H (h) ˇ < 1 is reached in the continuation game beginning Now suppose a history h ˇ < 1, V H (h; ˇ –, , ⁄) has to be not smaller than V H (?; –, , ⁄). at h. In order to justify ›H (h) Because a new seller arrives in each period only with a probability strictly less than 1, the seller must offer a price no higher than vH ≠ V H (?; –, , ⁄) with positive probability in the ˇ Let p(1) be first such an offer. Suppose the high type is continuation game beginning at h. willing to decline p(1) without loss; if the high type accepts p(1) for sure, then the next offer 40

would be vL . Again the seller would offer a price no higher than vH ≠ V H (?; –, , ⁄)/e≠r , say p(2) , in the continuation game followed by the high type’s rejection of p(1) . Iterating this ˜ which is reached in equilibrium with positive probability procedure, I can find a history h ˜ and vL œ supp ‡(h).

B.5

Proof of Lemma 10

Here, I will first formally construct monotone anonymous PBEs sketched in the main text, and then show that a sequence of such equilibria satisfies the Coase property. For each integer j, define p˜j © (1 ≠ e≠rj )vH + e≠rj vL . Fix > 0 and ⁄ > 0 and consider a candidate of monotone anonymous PBE – = (‡, q, ·, ›). Fix x œ {0, 1} and suppose ›H (h) © 0

and ›L (h) © x œ {0, 1} for any h œ H.

For fixed x œ {0, 1}, ≠⁄

’L © e

+ (1 ≠ e

≠⁄

)(1 ≠ x) =

I

1 e≠⁄

if x = 0 if x = 1

be the probability that the low type comes back to the original seller without switching to new seller. Finally note that q(?) Æ qˆ because ›H (·) © 0. Lemma 1 implies ·L (p; h) = {p Æ vL } for any h œ H. To construct ‡, q, ·H , first choose any sequence of belief levels (˜ qjx )jØ0 such that q˜0x = 0 < q˜1x < q˜2x < . . . For now this sequence is chosen arbitrarily but will be pinned down shortly. For any p œ [˜ pj , p˜j+1 ) for some integer j and h, let Y x x ] 1 ≠ q˜j (1≠˜q (h)) x q˜ (h)(1≠˜ qjx ) ’L ·H (p; h) = [ 0

if q(h) Ø q˜jx otherwise

and ·H (p; h) = 1 if p < p˜0 . Accordingly, for any h q(h, p) =

I

q˜jx

q(h) q(h)+(1≠q(h))’L

if q(h) Ø q˜jx and p œ [˜ pj , p˜j+1 ) otherwise

41

and q(h, p) = 0 if p Æ p˜0 . Finally set ‡(h) = p˜j and ‡(h, p) =

Y ]

p˜j

[ p˜ j≠1

if q(h) œ (˜ qj , q˜j+1 ) for some j. p≠e≠r p˜j≠1 ≠(1≠e≠r )vH e≠r (˜ pj ≠˜ pj≠1 ) e≠r p˜j ≠p+(1≠e≠r )vH probability e≠r (˜ pj ≠˜ pj≠1 )

with probability with

(10)

(11)

if p œ (˜ pj , p˜j+1 ] and q(h, p) = q˜jx .

Of course – cannot be equilibrium for any choice of (˜ qjx )jØ0 . To identify (˜ qjx )jØ0 that Õ guarantee – is equilibrium, first define (q ) be seller’s payoff from playing any equilibrium assessment – in the continuation game with initial belief q Õ . For example, (q Õ ) = p˜0

for any q Õ œ [˜ q0x , q˜1x )

because (10) and (11) imply the seller charges the winning offer with belief less than q˜1x . With belief q œ [˜ q1x , q˜2x ), the seller charges p˜x1 and buyer’s rejection to this offer leads the seller to form posterior q˜0x = 0 so (q Õ ) = p˜1 q Õ + e≠r ’L (1 ≠ q Õ ) (˜ q0x ) for any q œ [˜ q1x , q˜2x ) In general,

C

D

x (1 ≠ q Õ ) q˜j≠1 1 ≠ qÕ ≠r x (q ) = p˜j q 1 ≠ ’ + e ’ (˜ qj≠1 ) L L x (1 ≠ q˜j≠1 )q Õ 1 ≠ q˜j≠1 Õ

Õ

x ) for j = 0, 1, . . . , N . Note that (11) requires the seller sometimes for any q Õ œ [˜ qjx , q˜j+1 randomize p˜j and p˜j≠1 with belief q˜jx . This indifference of the seller requires

(˜ qjx )

C

D

x (1 ≠ q q˜j≠1 ˜jx ) 1 ≠ q˜jx ≠r x 1≠ ’ + e ’ (˜ qj≠1 ) L L x (1 ≠ q˜j≠1 )˜ qjx 1 ≠ q˜j≠1

=

p˜j q˜jx

=

p˜j≠1 q˜jx

C

D

x (1 ≠ q q˜j≠2 ˜jx ) 1 ≠ q˜jx ≠r x 1≠ ’ + e ’ (˜ qj≠2 ) L L x (1 ≠ q˜j≠2 )˜ qjx 1 ≠ q˜j≠2

(12) (13)

for any j Ø 2. Combining all such equations for each j leads to the following system of equations: e≠r(j≠1) (vH ≠ vL ) (¸j ≠ ¸j≠1 ’L ) = (¸j≠1 ≠ ¸j≠2 ), j Ø 2 (14) ’L vH with the boundary condition ¸1 =

(1 ≠ e≠r ’L )vL (1 ≠ e≠r )(vH ≠ vL ) 42

and ¸0 = 0

where ¸j = equation.

q˜jx 1≠˜ qjx

is likelihood ratio. – can be equilibrium only if (˜ qjx )jØ0 solves the difference

Now I will show that the constructed assessment – is actually a monotone anonymous with sufficiently small . LEMMA B.1. For any sufficiently small all j Ø 1:

¸j <

, the solution of (14) satisfies the following for

qˆ vL =∆ ¸j+1 ≠ ¸j Ø . 1 ≠ qˆ vH ≠ vL

Proof. The proof is done by induction. First note that claim is trivially true for m = 1. Suppose that the statement of the lemma holds true for j = 1, 2, . . . , l ≠ 1 (¸ > 1), which also implise ¸0 < ¸1 < . . . < ¸l . Suppose ¸l < qˆ/(1 ≠ qˆ). Adjusting (14),

’L vH (¸l ≠ ¸l≠1 ) ≠ (1 ≠ ’L )¸l e≠rl (vH ≠ vL ) ’L vH qˆ Ø ¸1 ≠ (1 ≠ ’L ) (vH ≠ vL ) 1 ≠ qˆ ’L vH vL ≠ (1 ≠ ’L )vH Ø ¸1 ≠ ¸1 (vH ≠ vL ) vH ≠ vL vL = ¸1 Ø vH ≠ vL

¸l+1 ≠ ¸l =

where the second inequality is due to the fact that vL ≠ (1 ≠ ’L )vH qˆ ¸1 > (1 ≠ ’L ) vH ≠ vL 1 ≠ qˆ for any small

(15)

.

qˆ Define N? be the smallest positive integer such that ¸N? > 1≠ˆ , Lemma q . With small vL B.1 implies ¸j ≠ ¸j≠1 > vH ≠vL for any integer j Ø 0, and hence N? is well-defined. The lemma also guarantees N? is bounded from above by an integer NC which is independent of both and ⁄. Hence, any seller makes the winning offer within NC rounds of the negotiation (or even earlier) when he plays according to –. It is straightforward to check that – is a monotone anonymous PBE with out choice of (˜ qj )jØ0 as long as (15) holds. The only non-trivial part is to show that the seller has no x ) for some incentive to make a losing offer. Recall that with h œ H such that q(h) œ [˜ qjx , q˜j+1 integer j Ø 0, any offer strictly higher than p˜j is losing. Once the seller charges such a losing offer, say p, ·H (p; h) = ·L (p; h) = 0, ›H (h) = 0 and ›L (h) = 1, and hence

q(h, p) =

q˜jx > q˜jx q˜jx + (1 ≠ q˜jx )’L 43

x . In that case, supp ‡(h, p) = {˜ First suppose q(h, p) < q˜j+1 pj } and

5 5 6 6 Y x ] e≠r p˜ q(h) 1 ≠ q˜j≠1 (1≠q(h)) (’ )2 + e≠r (’ )2 1≠q(h) (˜ qj≠1 ) if j Ø 1 x ) j L L 1≠˜ qj≠1 q(h)(1≠˜ qj≠1 V S (h, p) = [ ≠r

[q(h) + (1 ≠ q(h)’L ] p˜0

e

if j = 0

Such a deviation is not profitable for the seller if (q(h)) Ø V S (h, p). When j = 0 or j = 1 this condition holds trivially. With j Ø 2, the inequality is equivalent to (q(h)) Ø

e≠r (1 ≠ ’L ) p˜j q(h). 1 ≠ e≠r ’L

(16)

Combining (12) and (13), (˜ qjx )

È 1 ≠ q˜jx Ë = p ˜ (¸ ≠ ¸ ’ ) ≠ p ˜ (¸ ≠ ¸ ) j+1 j+1 j j j+1 j L 1 ≠ e≠r ’L

(17)

and (q Õ ) = p˜j q Õ +

È 1 ≠ qÕ Ë p˜j (¸j e≠r ’L ≠ ¸j≠1 ’L ) ≠ p˜j≠1 (¸j ≠ ¸j≠1 )e≠r ’L ≠r 1≠e ’L

(18)

x ), and hence for q Õ œ [˜ qjx , q˜j+1

e≠r (1 ≠ ’L ) p˜j q(h) 1 ≠ e≠r ’L È 1 ≠ e≠r 1 ≠ q(h) Ë ≠r ≠r = p˜j q(h) + p ˜ ’ (e ¸ ≠ ¸ ) ≠ p ˜ e ’ (¸ ≠ ¸ ) j j j≠1 j≠1 j j≠1 L L 1 ≠ e≠r ’L 1 ≠ e≠r ’L

(q(h)) ≠

C

1 ≠ e≠r q(h) (e≠r ’L ¸j ≠ ’L ¸j≠1 ) ≠ e≠r ’L (¸j ≠ ¸j≠1 ) Ø p˜j (1 ≠ q(h)) + ≠r 1≠e ’L 1 ≠ q(h) 1 ≠ e≠r ’L C

1 ≠ e≠r q(h) e≠⁄ (1 ≠ e≠r ) = p˜j (1 ≠ q(h)) ≠ ¸j≠1 1 ≠ e≠r ’L 1 ≠ q(h) 1 ≠ e≠r ’L Ø

5

6

D

D

1 ≠ e≠r q(h) p˜j (1 ≠ q(h)) ≠ ¸j≠1 > 0. ≠r 1 ≠ q(h) 1≠e ’L

x . In this case, p Next suppose q(h, p) Ø q˜j+1 ˜j+1 œ supp ‡(h, p), and hence

V S (h,p)/e≠r

A

B

q˜jx (1 ≠ q(h)) 1 ≠ q(h) = p˜j+1 q(h) 1 ≠ (’L )2 + e≠r (’L )2 (˜ qj ) q(h)(1 ≠ q˜jx ) 1 ≠ q˜j = ’L

C

A

B

D

q˜jx (1 ≠ q(h)) 1 ≠ q(h) p˜j+1 q(h) 1 ≠ ’L + e≠r ’L (˜ qj ) + (1 ≠ ’L )˜ pj+1 q(h) x q(h)(1 ≠ q˜j ) 1 ≠ q˜j

44

which is no larger than ’L (q(h)) + (1 ≠ ’L )˜ pj+1 q(h) Hence, it suffices to have (q(h)) Ø

e≠r (1 ≠ ’L ) p˜j+1 q(h). 1 ≠ e≠r ’L

(19)

to deter seller’s deviation. If x = 0, the left-hand side is zero, hence (19) holds true. On the other hand, if x = 1, ’L = e≠⁄ and (q(h))≠

as long as

e≠r (1 ≠ e≠⁄ ) p˜j+1 q(h) 1 ≠ e≠r e≠⁄ e≠r (1 ≠ e≠⁄ ) e≠r (1 ≠ e≠⁄ ) = (q(h)) ≠ p ˜ q(h) ≠ (˜ pj+1 ≠ p˜j )q(h) j 1 ≠ e≠r e≠⁄ 1 ≠ e≠r e≠⁄ 1 ≠ e≠r vL2 e≠r (1 ≠ e≠⁄ ) Ø ≠ (˜ pj+1 ≠ p˜j )q(h) 1 ≠ e≠r e≠⁄ vH ≠ vL 1 ≠ e≠r e≠⁄ 1 ≠ e≠r vL2 e≠r (1 ≠ e≠⁄ ) = ≠ (1 ≠ e≠r )(vH ≠ vL ) 1 ≠ e≠r e≠⁄ vH ≠ vL 1 ≠ e≠r e≠⁄ A B 1 ≠ e≠r vL2 ≠r ≠r = ≠e (1 ≠ e )(vH ≠ vL ) Ø 0 vH ≠ vL 1 ≠ e≠(r+⁄)

is small.

x = (‡ x , q c , · x , › x ) denote the equilibrium just For any > 0 and x œ {0, 1}, let –C C x C C constructed here, in which the low type keeps swathing to a new seller with probability x; ›cx L (·) © x. As in the main text, all these equilibria are referred to as Coasean equilibria. Remember that if q(?) = q˜jx for an integer j the seller is actually indifferent between p˜j and p˜j≠1 at the null history. Appealing the argument in last paragraphs, therefore, I can construct multiple of asymptotically efficient equilibria, varying the probability that the seller offers p˜j at the null history. All these equilibria are also referred to as Coasean equilibria.18 ECx ( , ⁄) be the set of Coasean equlibira with ›L (h) = x for any h œ H. The proof of Lemma 10 above shows that ECx ( , ⁄) is non-empty as long as is small enough. Also let t x x Õ EC ( , ⁄) = xœ{0,1} EC ( , ⁄). For future reference, let C (q ) be seller’s payoff from playing x in the continuation game with initial belief q Õ . any equilibrium assessment –C

Now I am ready to show Lemma 10. Fix any sequences (⁄m )mØ0 and ( k )kØ0 such that 0 œ E0 ( limmæŒ ⁄m = Œ and limkæŒ k = 0. For each k and m, let –k,m be –C k , ⁄k ). C k,m Then it is straightforward that (– )kØ0,mØ0 satisfies the Coase property. 18

x For example if ›L (·) © x = 0 and ›H (·) © 0 in equilibrium and all sellers adopt offer strategy ‡C |x=0 , x both types of the buyer transact with the first seller for sure, and hence q(?) = qˆ. With qˆ = q˜j |x=0 for some integer j Ø 1 all sellers are indifferent between offering p˜j and p˜j≠1 at the null history there are a continuum of Coasean equilibria each of them differs in seller’s initial randomization over p˜j and p˜j≠1 .

45

x = (‡ x , q c , · x , › x ) is also a monotone anonymous PBE for the bargaining Note that –C C x C C game with exogenous outside options, as long as vH ≠ vL > wH , reinterpreting that ›cx H (h) and ›cx L (h) as the probabilities that each type of buyer takes an outside option if available x after history h. Let EC,exo (ˆ q ; , ⁄) and EC,exo (ˆ q exo ; , ⁄) denote the sets of these Coasean equilibria for the bargaining game with exogenous outside options and xC (·) the seller’s x œE profit from –C q ; , ⁄). C,exo (ˆ

B.6

Proof of Lemma 11

Proof of (i). Let q¯k,m = inf{q Õ œ [0, 1] : ÷h such that q(h) = q Õ and p > vL in supp ‡ k,m (h) } = inf{q Õ œ [0, 1] : ÷h such that q(h) = q Õ and p > (1 ≠ e≠r

k

)vH + e≠r vL in supp ‡ k,m (h) }.

where the second inequality comes from the fact q¯k,m Æ q k,m (?) < q˜1x |x=1 . By monotonicity of ‡ k,m , p≠ (¯ q k,m ) = vL and p+ (¯ q k,m ) Ø (1 ≠ e≠r k )vH + e≠r vL . Now choose ‘ > 0 and pick a history h œ H such that q k,m (h) œ [¯ q k,m , q¯k,m + ‘) and supp ‡(h) includes a price other than vL . Now suppose a seller offers pÕ such that pÕ Æ vH ≠ e≠r

k

V H (?; –k,m ,

k , ⁄m )

and (1 ≠ e≠r ¸

k

)vH + e≠r

=(1≠e≠r

˚˙

k

k )vH +e≠r

p≠ (¯ q k,m ) < pÕ < (1 ≠ e≠r vL

˝

¸

k

>vH ≠e≠r

)vH + e≠r ˚˙

k

k V H (?;–k,m ,

p+ (¯ q k,m ) k ,⁄m )

˝

after h. pÕ may not be in supp ‡(h). Then, Lemma 7 implies q k,m (h, pÕ ) = q¯k,m conditional on the event that the buyer rejects pÕ . Proof of (ii). Consider a history h œ H such that q k,m (h) = q¯k,m and suppose supp ‡ k,m (h) includes a losing offer, say p. By Lemma 12-(i), p Ø vH ≠ e≠r k V H (?; –k,m , k , ⁄m ).19 fi(h, p), the first serious offer of the seller in the continuation game following the buyer’s rejection of p, should be exactly vH ≠ e≠r k VH (?; –k,m , k , ⁄m ).20 To see this, first note that by the skimming property (Lemma 6), the seller’ belief stays at q¯k,m indefinitely on equilibrium path. However, any offer strictly larger than vH ≠ e≠r k V H (?, –k,m , k , ⁄m ) can not be considered seriously if made when the seller’s belief is q¯k,m . Otherwise, the 19 20

Lemma 11-(ii) is not used in the proof for Lemma 12-(i), hence there is no circular reasoning here. See page 38 for definition of fi.

46

buyer’s rejection to such offers would lead the seller’s belief to a level strictly less than q¯k,m , and hence the seller will make the winning offer p = vL in the following period; in this case, both types of the buyer strictly wait for the winning offer. Hence, after seller’s losing offer p Ø vH ≠ e≠r k V H (?; –k,m , k , ⁄m ) is offered after h and then rejected, the seller’s next serious offer fi(h, p) should be either vL or vH ≠ e≠r k V H (?; –k,m , k , ⁄m ) for sure. Specifically suppose ‡ k,m (h, p, hÕ ) contains a serious offer, say pˆ, where hÕ is a sequence of equilibrium losing offers that follows p. Again by the skimming property, q k,m (h, p, hÕ ) = q¯k,m . If pˆ = vL , the seller’s expected profit from charing p after h is strictly less than vL , contradiction. Hence, pˆ is necessarily vH ≠ e≠r k V H (?; –k,m , k , ⁄m ). But in this case, because the seller never offers a price in (vL , vH ≠ e≠r k V H (?; –k,m , k , ⁄m ) with belief q¯k,m (Lemma 12-(i)), ‡ k,m (h, p, hÕ , pˆ) has to include vL and V H (h; –k,m , k , ⁄m ) < vL which is also impossible in equilibrium (because the seller can always guarantee the profit vL by making the winning offer).

B.7

Proof of Proposition 2

The proof is by constructing a monotone PBE whose outcome (in the continuous-time limit) is equivalent to the deadlock equilibrium constructed in Section 5. Suppose there is qD œ (0, qˆexo ) such that 1 ≠ e≠r e≠⁄ x=1 qD = (qD ). (20) (1 ≠ e≠⁄ )(vH ≠ e≠r wH ) C

x=1 in the conwhere x=1 C (qD ) is seller’s payoff from playing any equilibrium assessment –C tinuation game with initial belief qD (see the end of the last subsection, page 45).21 Pick an arbitrary equilibrium from ECx=1 (qD ; , ⁄) and let pˆ be the minimum price in supp ‡(?) in that equilibrium. Now construct an equilibrium as follows:

• The low type buyer exercises outside option with probability 1 whenever one arrives: ›L (·) © 1. • Suppose either a history h œ H such that q(h) < qD is reached or the seller offers a price no higher than (1 ≠ e≠r )vH + e≠r pˆ. Then both types of buyer and the seller x=1 (q(h); , ⁄) until either an agreement is begin to play as in an equilibrium in EC,exo reached or the buyer takes outside option. • Consider a history h œ H such that q(h) = qD . Let p≠1 be seller’s last rejected offer, if any exists. The case where p≠1 Æ (1 ≠ e≠r )vH + e≠r pˆ is already covered by the last bullet point, hence suppose p≠1 > (1 ≠ e≠r )vH + e≠r pˆ.

21 Both sides are strictly increasing in qD and exists and actually is unique for all small ’s.

x=1 C (·)

47

converges pointwise to vL . Hence such qD always

– Seller’s strategy at h such that q(h) = qD : The seller randomizes over vH ≠e≠r wH and pˆ with probability „D (p≠1 ) and 1 ≠ „D (p≠1 ) respectively where „D (p≠1 ) =

Y _ _ ]

0 1

_ _ [ p≠1 ≠(1≠e≠r )vH ≠e≠r pˆ

if p≠1 < (1 ≠ e≠r )vH + e≠r pˆ if p≠1 > vH ≠ e≠r wH otherwise.

e≠r (vH ≠e≠r wH ≠ˆ p)

– High type buyer’s strategy at h such that q(h) = qD : If (1 ≠ e≠r )vH + e≠r pˆ < p≠1 Æ vH ≠ e≠r wH , the high type buyer never takes an outside option even if one arrives. Otherwise, the high type buyer takes an outside option for sure. That is to say, ›H (h) =

I

0 if (1 ≠ e≠r )vH + e≠r pˆ < p≠1 Æ vH ≠ e≠r wH 1 p≠1 > vH ≠ e≠r wH .

Regarding the acceptance strategy ·H , the high type buyer accepts any offer p such that (1 ≠ e≠r )vH + e≠r pˆ < p Æ vH ≠ e≠r wH with probability 1 ≠ e≠⁄ . The high type rejects any offers higher than vH ≠ e≠r wH . ·H (p; h) =

I

0 1 ≠ e≠⁄

if (1 ≠

e≠r

if p > vH ≠ e≠r wH )vH + e≠r pˆ < p Æ vH ≠ e≠r wH .

– Seller’s belief in the next period, conditional on buyer’s rejecting to seller’s equilibrium offer, stays at qD in all cases. The equilibrium construction is completed by specifying each player’s equilibrium behavioral strategy for any history h such that q(h) > qD . Actually, following an argument similar to the proof for Lemma 10, we can construct an equilibrium where the seller’s offer strategy is such that † • the seller charges p†j with probability 1 whenever his belief is in (qj† , qj+1 );

• the seller randomizes p†j and p†j≠1 when his belief is exactly qj†

where

S

and

p†j = vH ≠ e≠r U1 ≠ e≠⁄ (1 ≠ e≠r )

j≠1 ÿ

k=0

T

e≠rk e≠⁄k V wH

for any j Ø 1.

† qD = q0† < q1† < . . . < qN = qˆexo

are the belief levels that rationalize the seller’s randomization over p†j≠1 and p†j with the belief equal to qj† . On equilibrium path, the seller makes serious offers p†N , p†N ≠1 , . . . , p†0 in 48

sequence. The high type accepts each offer with a certain probability so that the seller’s † belief is updated from qj† to qj≠1 whenever the buyer rejects p†j on equilibrium path. Hence the seller’s belief reaches qD in N periods. Note that this equilibrium is qualitatively similar to the deadlock equilibrium constructed in Section 5. This equilibrium construction is valid if and only if 0 < qD Æ qˆexo . Note that if qD > qˆexo the equilibrium path for a deadlock equilibrium constructed above is equivalent to one in EC,exo (ˆ q exo , , ⁄). The condition 0 < qD Æ qˆexo holds true for all small > 0 if 1 ≠ e≠r e≠⁄ æ0 (1 ≠ e≠⁄ )(vH ≠ e≠r wH )

0 < lim

x=1

(qD ) =

⁄+r vL Æ qˆexo . ⁄ vH ≠ wH

Throughout the appendix, the equilibrium constructed above is referred to as deadlock equilibrium (for the bargaining game with exogenous outside options). Let ED,exo (ˆ q exo ; ⁄, ) be the set of deadlock equilibrium and Note that there is no deadlock equilibrium with x = 0; in this case, seller’s profit should be zero in order to make his belief is stuck at some 0 œ (0, 1) which is not possible in equilibrium. qD

B.8

Proof of Proposition 3

Throughout this section, define p˜k = (1 ≠ e≠r )vH + e≠r k vL for all integer k Ø 0, as in the proof for Lemma 10. Also let q˜kx for an integer k Ø 0 be as defined in the same proof. Recall x >q that there is N ú such that q˜N ˆexo for all , x, and wH . Suppose > 0 is small enough ú to guarantee p˜N ú < vH ≠ e≠r wH . (CL) Suppose ›L (·) © x for some x œ {0, 1}. For any h œ H

(S)

’L © (1 ≠ e≠⁄ )(1 ≠ x) + e≠⁄

is the probability that the low type does not take outside option (because either no outside option arrives or low type buyer abandons one) in each period. Similarly, for any h œ H ’H (h) = (1 ≠ e≠⁄ )(1 ≠ ›H (h)) + e≠⁄ is the probability that the high type does not take outside option after h. For any h, ’H (h, p) =

I

e≠⁄ 1

if p > vH ≠ e≠r wH and p is serious if p < vH ≠ e≠r wH .

by the same argument in the proof for Lemma 3 . 49

Restatement of Proposition 3 With notation developed in the last two sections, Section B.5 and Section B.7, Proposition 3 can be formally restated as follows. PROPOSITION B.1. Suppose vH ≠ vL > wH > 0. For all small

> 0, the set of monotone

PBEs for the bargaining game with exogenous outside option is EC,exo (ˆ q exo ; , ⁄)



ED,exo (ˆ q exo ; , ⁄)

Preliminary Observation Choose an arbitrary monotone PBE – = (‡, q, ·, ›) not in EC,exo (ˆ q exo ; , ⁄). Their is a cutoff for seller’s belief, say q¯, such that all players’ equilibrium play after h such that q(h) < q¯ is equivalent to a Coasean equilibrium with ›L (·) © x. Note that, with belief q Õ œ (0, 1), the upper bound of seller’s payoff from charing other than the winning offer is vH q Õ + e≠r (1 ≠ q Õ )vL . Hence, after h œ H such that q(h) is strictly lower than q<

(1 ≠ e≠r )vL , vH ≠ e≠r vL

the seller charges vL immediately. By Lemma 1 the low type would accept this offer as in any Coasean equilibrium. vH ≠ vL > wH by assumption, and hence the high type would not switch to new seller after h even if one arrives and accepts the original seller’s winning offer, again as in any Coasean equilibrium. This observation guarantees the existence of such belief level q¯ > 0. From now on, suppose equilibrium outcome according to – is identical to Coasean equilibrium with ›L (·) = x in any continuation game played after seller’s belief reaches q Õ < q¯. Note that such q¯ is possibly non-unique; in this case, take the supremum of such q¯’s. Without loss, assume x q¯ œ [˜ qŸx , q˜Ÿ+1 ) for some Ÿ.

(21)

Also, because – is not in EC,exo (ˆ q exo ; , ⁄), there must exist h œ H such that q(h) Ø q¯. LEMMA B.2. Suppose (CL) holds and let – = (‡, ·, ›, q) be a perfect Bayesian equilibrium satisfying (S). (i) p+ (¯ q ) Ø p˜Ÿ 22 (ii) For a history h œ H and price p such that either q(h) Ø q˜jx , 22

and

p œ (˜ pj , p˜j+1 )

See page 13 for the definitions of p+ (·).

50

for some 0 Æ j Æ Ÿ

(22)

or q(h) Ø q˜jx

and

p = p˜j+1 œ supp ‡(h)

(23)

for some 0 Æ j Æ Ÿ. Then, ·H (p; h) = —(q(h), q˜jx , ’L ), ›H (p; h) = 0, q(h, p) = q˜jx and seller’s payoff by charing p after h is C

D

q˜jx (1 ≠ q(h)) 1 ≠ q(h) pq(h) 1 ≠ ’L + e≠r ’L q(h)(1 ≠ q˜jx ) 1 ≠ q˜jx

x x qj ). C (˜

(iii) supp ‡(h, p) never contains one of p˜0 , . . . , p˜j+1 if p < vH ≠ e≠r wH was losing and q(h) Ø q˜jx for some 0 Æ j Æ Ÿ.

First of all, the statements (ii) and (iii) are true with an additional condition. CLAIM. (ii) and (iii) of Lemma B.2 are true with additional condition p < (1 ≠ e≠r )vH + e≠r p+ (¯ q ).

Proof of Claim. The first claim is that the statement (ii) is true when (22) and the additional condition p < (1 ≠ e≠r )vH + e≠r p+ (¯ q ) hold. The claim is trivial for such a case if q(h) < q¯ or j = 0, thus suppose q(h) Ø q¯ > q˜jx and j Ø 1 without loss. p Æ p˜Ÿ+1 < vH ≠ e≠r wH by assumption, and hence ’H (h, p) = 1. Also note that

p < (1 ≠ e≠r )vH + e≠r p+ (¯ q ) Æ (1 ≠ e≠r )vH + e≠r p+ (˜ qjx ) and p > p˜j Ø (1 ≠ e≠r )vH + e≠r p(˜ qjx ). ¯ Hence ,Lemma 7 applies and q(h, p) = q˜jx < q¯ Æ q(h). The seller’s expected payoff by charging p after h is then 1

2

pq(h)— q(h), q˜jx , ’L + e≠r

Ë

1

2

q(h)(1 ≠ — q(h), q˜jx , ’L ) + (1 ≠ q(h))’L C

D

È

x x qj ) C (˜

q˜jx (1 ≠ q(h)) 1 ≠ q(h) ’L + e≠r ’L = pq(h) 1 ≠ x q(h)(1 ≠ q˜j ) 1 ≠ q˜jx

x x qj ) C (˜

as desired. The last argument also shows that, in equilibrium, the seller never charges p œ (˜ pj , p˜j+1 ) after a history h such that q(h) Ø q˜jx , as long as p < (1 ≠ e≠r )vH + e≠r p+ (¯ q ). Also, V S (h) is bounded from below by C

D

q˜jx (1 ≠ q(h)) 1 ≠ q(h) p˜j+1 q(h) 1 ≠ ’L + e≠r ’L q(h)(1 ≠ q˜jx ) 1 ≠ q˜jx 51

x x qj ). C (˜

(24)

under the same condition; if V S (h) is strictly lower than this, it is a profitable deviation for the seller to charge a price slightly lower than p˜j+1 after h. Now we turn to the remaining parts of the statement (ii) and the statement (iii). Again, suppose q(h) Ø q¯ > q˜Ÿx without loss, since the claim is trivial if q(h) < q¯ or j = 0. The

proof is by induction. Suppose (ii) holds with condition p < (1 ≠ e≠r )vH + e≠r p+ (¯ q ) for j = 0, 1, . . . , m ≠ 1 where 1 Æ m Æ Ÿ; and also suppose (iii) is true for j = 0, 1, . . . m ≠ 1

x and with condition p < (1 ≠ e≠r )vH + e≠r p+ (¯ q ). Consider a history h such that q(h) Ø qm

p = p˜m+1 œ supp ‡(h). By Lemma 7, q(h, p) Æ q¯.

In this case, q(h, p) is actually strictly less than q¯. For contradiction, suppose q(h, p) =

x . The high type is willing to decline p = p q¯ > q˜m ˜m+1 only if fi(h, p) is one of p˜0 , . . . , p˜m with

positive probability. Because (iii) holds for j = 0, . . . , m ≠ 1, the seller has to offer one of p˜0 , . . . , p˜m immediately after p is rejected. This implies V S (h, p) Æ where

Tj

D

C

In conclusion,

max

0ÆjÆm≠1

q˜jx (1 ≠ q(h)) 1 ≠ q(h) ’L + e≠r ’L Tj © p˜j+1 q(h) 1 ≠ q(h)(1 ≠ q˜jx ) 1 ≠ q˜jx

V S (h) Æ p˜m+1 ·q(h) · — (q(h), q¯, ’L ) + e≠r

Ë

q(h) (1 ≠ — (q(h), q¯, ’L )) + (1 ≠ q(h))’L

x x qj ). C (˜

È

max

0ÆjÆm≠1

Tj .

A simple calculation shows that the right-hand side is strictly lower than (24), contradiction. q(h, p) < q¯ from the last paragraph . By the construction of Coasean equilibria, fi(h, p) = x ,q x p˜m for sure and q(h, p) œ [˜ qm ˜m+1 ]. Also in this case, seller’s expected payoff from charring

p˜m+1 œ supp ‡(h) is

p·q(h)— (q(h), q(h, p), ’L ) + e≠r 5

= p˜m+1 q(h) 1 ≠

Ë

q(h) (1 ≠ — (q(h), q(h, p), ’L )) + (1 ≠ q(h))’L 6

q(h, p)(1 ≠ q(h)) 1 ≠ q(h) ’L + e≠r ’L q(h)(1 ≠ q(h, p)) 1 ≠ q(h, p)

x C (q(h, p)).

È

x C (q(h, p))

x and this maximal A simple calculation shows that it is maximized if and only if q(h, p) = q˜m

profit coincides to the lower bound (24) for of V S (h). So p can be in supp ‡(h) only if x in which case the claim is true. (iii) immediately follows as well. q(h, p) = q˜m

Proof of Lemma B.2. First consider the statement (i) in the lemma. Since a case with Ÿ = 0 or Ÿ = 1 is trivial, suppose Ÿ Ø 2 without loss. For a contradiction suppose p+ (¯ q ) < p˜Ÿ and 52

pick a history h œ H and p œ supp ‡(h) such that q(h) > q¯ and p+ (¯ q ) Æ p < p+ (¯ q ) + ‘ < p˜Ÿ

(25)

for a positive number ‘ such that 0 < ‘ < (1 ≠ e≠r )(vH ≠ p+ (¯ q )). Without loss, let’s say x p œ (˜ pj≠1 , p˜j ]\{˜ pŸ } for some 1 Æ j Æ Ÿ. The last claims conclude q(h, p) = q˜j≠1 and

C

D

x (1 ≠ q(h)) q˜j≠1 1 ≠ q(h) V (h) = pq(h) 1 ≠ ’L + e≠r ’L x x q(h)(1 ≠ q˜j≠1 ) 1 ≠ q˜j≠1 S

x x qj≠1 ). C (˜

(26)

Now consider seller’s deviation to p˜Ÿ after h. By the last claims again, seller’s expected profit from such deviation is C

D

q˜x (1 ≠ q(h)) 1 ≠ q(h) ≠r p˜Ÿ q(h) 1 ≠ Ÿ≠1 ’L x ) ’L + e x q(h)(1 ≠ q˜Ÿ≠1 1 ≠ q˜Ÿ≠1

x x qŸ≠1 ). C (˜

Because q(h) > q¯ > q˜Ÿx this payoff dominates V S (h), contradiction. Next consider the statement (ii) in the lemma. It suffices to show that the statement is true with (23) for j = Ÿ. (ii) for all other cases follow the statement (i) and the last two claims above. First suppose p+ (¯ q ) > p˜Ÿ . Then, actually all arguments in the proof for the second claim above also work for this case to show the statement (ii) is true. Hence, suppose p+ (¯ q ) = p˜Ÿ without loss. Again consider a history h œ H such that q(h) > q˜Ÿx and p = p˜Ÿ+1 œ supp ‡(h). If p were losing, fi(h, p) would be no higher than p˜Ÿ with a positive

probability. But this is impossible for the last claim. Hence, p has to be serious which also implies that the seller charges p˜Ÿ after p is declined. Now consider two sub-cases in order. First suppose q(h, p) = q˜Ÿx in which case seller’s equilibrium profit is 5

6

q˜x (1 ≠ q(h)) 1 ≠ q(h) V (h) = p˜Ÿ+1 q(h) 1 ≠ Ÿ ’L + e≠r ’L x q(h)(1 ≠ q˜Ÿ ) 1 ≠ q˜Ÿx S

x qŸ ). C (˜

(27)

On the other hand if q(h, p) > q˜Ÿ , 5

V S (h) = p˜Ÿ+1 q(h) 1 ≠ where

5

= p˜Ÿ q(h, p) 1 ≠

6

q(h, p)(1 ≠ q(h)) 1 ≠ q(h) ’L + e≠r ’L q(h)(1 ≠ q(h, p)) 1 ≠ q(h, p) 6

q˜Ÿ≠1 (1 ≠ q(h, p)) 1 ≠ q(h, p) ’L + e≠r ’L q(h, p)(1 ≠ q˜Ÿ≠1 ) 1 ≠ q˜Ÿ≠1

(28)

x qŸ≠1 ). C (˜

A simple calculation shows that (27) is strictly lower than (28). Note that if (27) is the case

53

the proof is completed. Hence, suppose (27) is not the case and 5

6

q˜Ÿ (1 ≠ q(h)) 1 ≠ q(h) V (h) < p˜Ÿ+1 q(h) 1 ≠ ’L + e≠r ’L q(h)(1 ≠ q˜Ÿ ) 1 ≠ q˜Ÿ S

x qŸ ). C (˜

But now if the seller deviates to a price slightly lower than p˜Ÿ+1 at h, the claim proved above shows that the seller can enjoy a profit strictly higher than V S (h), contradiction. V S (h) is therefore as in (27), and hence q(h, p) = q˜Ÿ . It completes the proof for statement (ii). Finally (iii) follows (ii) immediately. Below I will characterize all monotone PBEs (for bargaining game with exogenous outside options) satisfying (S). Fix a monotone PBE – satisfying this. From now on suppose {h œ H : supp ‡(h) contains p > p˜Ÿ } = ÿ.

(N C)

x ) by construction, and hence if the set for the candidate of monotone PBE –. q¯ œ [˜ qŸx , q˜Ÿ+1 x above is empty, equilibrium assessment – is identical to one of equilibria in Eexo,c (ˆ q exo , , ⁄). Hence, the assumption (N C) can be made without loss. By monotonicity of ‡, the following is immediate:

q¯ = inf{q Õ œ [0, 1] : there is h such that q(h) = q Õ and p > p˜Ÿ in supp ‡(h) }

= inf{q Õ œ [0, 1] : there is h such that q(h) = q Õ and p Ø p˜Ÿ+1 in supp ‡(h) }

where the second equality comes from (ii) of Lemma B.2 which implies that the seller never offers p œ (˜ pŸ , p˜Ÿ+1 ) with posterior no lower than q˜Ÿx (because pÕ œ (p, p˜Ÿ+1 ) dominates p). Before proceed, define (q; –, ) by the collection of equilibrium offers the seller possibly makes in monotone PBE –, conditional on his posterior being q Õ . Q

(q Õ ; –, ) © a



hœH:q(h)Æq Õ

R

supp ‡(h)b



Q a



hœH:q(h)Øq Õ

R

supp ‡(h)b .

(29)

If there is a history h œ H such that q(h) = q, the above definition is equivalent to (q Õ ; –, ) ©



supp ‡(h)

hœH:q(h)=q Õ

Recall that, by assumption, – coincides with a Coasean equilibrium in if seller’s posterior reaches q Õ < q¯. Hence, it suffices to pin down each player’s behavior when seller’s posterior is no lower than q¯.

54

Equilibrium Play at h œ H such that q(h) = q¯ LEMMA B.3. Suppose (CL) holds and let – = (‡, ·, ›, q) be a monotone PBE satisfying (S) and (N C). (i) There is a history h œ H such that q(h) = q¯ (ii)

(¯ q ; –, ) contains an offer p > p˜Ÿ+1 .

Proof. (i) There is hÕ œ H such that q(hÕ ) Ø q¯ because – is not Coasean by assumption. If q(hÕ ) = q¯ there is nothing to prove. Suppose q(hÕ ) > q¯. If the seller offers p œ (˜ pŸ+1 , p˜Ÿ+2 )

after hÕ , Lemma 7 implies q(hÕ , p) Æ q¯ < q(hÕ ). If q(hÕ , p) < q¯, p has to be losing because all offers in supp ‡(hÕ , p) is no higher than p˜Ÿ ; but if p is losing, q(hÕ , p) Ø q(hÕ ) > q¯ which is a contradiction. In conclusion, q(hÕ , p) = q¯. (ii) Suppose note that

(¯ q ; –, ) does not include a price higher than p˜Ÿ+1 for contradiction. First

(q Õ ; –, ) never contains a price in (˜ pŸ , p˜Ÿ+1 ) as long as q Õ < q˜Ÿ+1 because such

a price is dominated by p˜xŸ . Also, from Lemma B.2,

(q Õ ; –, ) never includes a price in

x (˜ pxŸ , p˜xŸ+1 ) if q Õ Ø q˜Ÿ≠1 because it is dominated by slightly higher price. In conclusion,

(¯ q ; –, ) µ {˜ pxŸ≠1 , p˜xŸ }. Then, for any ‘ > 0, there is a history h œ H such that q(h) œ

x }) and supp ‡(h) contains a price larger than p (¯ q , min{¯ q + ‘, q˜Ÿ+1 ˜Ÿ+1 . By monotonicity of ‡,

supp ‡(h) does not include a price in [˜ pŸ , p˜Ÿ+1 ] as long as ‘ > 0 is small enough. Therefore

p+ (¯ q ) Ø p˜Ÿ+1 .23 Now suppose the seller charges p œ (˜ pŸ+1 , p˜Ÿ+2 ) after one of such histories.

By Lemma 7, q(h, p) Æ q¯ < q(h), and hence p has to be serious. But by hypothesis, the seller never offers prices higher than p˜Ÿ+1 once his posterior becomes weakly lower than q¯. Hence, p is actually losing, contradiction. Motivated by last Lemma define p† © inf{p œ R+ : p œ (¯ q ; –, ) and p > p˜Ÿ+1 }. ¯ If q † = q˜Ÿx , the seller is indifferent between charing p˜Ÿ and p˜Ÿ≠1 after h such that q(h) = q¯. ¯ indifference results in multiple monotone PBEs that satisfies (N C) and (S), depending This on which price, either p˜Ÿ or p˜Ÿ≠1 , is included in (¯ q ; –, ). But equilibrium outcome is invariant regardless of choice between p˜Ÿ and p˜Ÿ≠1 . From now on, without much loss, I focus on case (¯ q ; –, ) includes p˜Ÿ , not p˜Ÿ≠1 .

23

The following Lemma extends Lemma B.2 to all offers no higher than vH ≠ e≠r wH .

See page 13 for the definitions of p+ (·).

55

LEMMA B.4. Suppose (CL) holds and let – = (‡, ·, ›, q) be a monotone PBE satisfying (S) and (N C). (i) p† Ø vH ≠ e≠r wH ¯ (ii) For a history h œ H and price p such that q(h) Ø q¯ and

p˜Ÿ+1 < p < vH ≠ e≠r wH

(30)

·H (p; h) = —(q(h), q¯, ’L ), ›H (p; h) = 0, q(h, p) = q¯ and seller’s payoff by charring p after h is

5

pq(h) 1 ≠

6

q¯(1 ≠ q(h)) 1 ≠ q(h) ’L + e≠r ’L (1 ≠ q¯)q(h) 1 ≠ q¯

x q ). C (¯

Hence, the seller never charges p œ (˜ pŸ , vH ≠ e≠r wH ) after a history h such that q(h) Ø q¯.

(iii) For a history h œ H q(h) Ø q¯ and

÷p = vH ≠ e≠r wH œ supp ‡(h),

(31)

·H (p; h) = —(q(h), q¯, ’L ), ›H (p; h) = 0, q(h, p) = q¯ and V (h) = (vH ≠ e S

≠r

5

6

q¯(1 ≠ q(h) 1 ≠ q(h) wH )¯ q 1≠ ’L (1 ≠ ’L ) + e≠r ’L q(h)(1 ≠ q¯) 1 ≠ q¯

x qŸ ). C (˜

The following preliminary result (Claim B.4.1) is useful in proving Lemma B.4. CLAIM B.4.1. (ii) holds if the condition (30) and additional condition p < (1 ≠ e≠r )vH + e≠r p† holds. ¯ Proof of Claim. Note p† Æ p+ (¯ q ),24 and therefore ¯

p < (1 ≠ e≠r )vH + e≠r p† Æ (1 ≠ e≠r )vH + e≠r p+ (¯ q ). ¯ On the other hand, p(¯ q ) Ø p˜Ÿ≠1 hence p > (1 ≠ e≠r )vH + e≠r p(¯ q ). From Lemma 7, ¯ ¯ q(h, p) = q¯. p is never the winning offer for p > p˜1 by assumption; hence, the high type is willing to decline p. By assumption vH ≠ p > e≠r (vH ≠ p† ) and thus fi(h, p) < p† with ¯ ¯ positive probability which in turn implies fi(h, p) Æ p˜Ÿ+1 with positive probability. (iii) of Lemma B.2 and (ii) of the last Lemma require that supp ‡(h, p) includes one of prices strictly

x ) or one in {˜ x ). In either lower than p˜Ÿ+1 which is either p˜Ÿ (if q¯ > q˜Ÿ≠1 pŸ≠1 , p˜Ÿ } (if q¯ = q˜Ÿ≠1 24

See page 13 for the definitions of p+ (·).

56

case, seller’s continuation payoff conditional on p being rejected after h is

x (¯ C q ),

so seller’s

payoff from charing p is as in the statement of Lemma. Proof of Lemma B.4-(i). For contradiction suppose p˜Ÿ+1 Æ p† < vH ≠ e≠r wH ; this assump) ¯ tion implies p˜Ÿ+1 < min (1 ≠ e≠r )vH + e≠r p† , vH ≠ e≠r wH } = (1 ≠ e≠r )vH + e≠r p† . ¯ ¯ In this case, there must be p œ (¯ q ; –, ) such that p˜Ÿ+1 < p < (1 ≠ e≠r )vH + e≠r p† . ¯ This is trivially true if p† > p˜Ÿ+1 . On the other hand if p† = p˜Ÿ+1 there is h‘ œ H such that ¯ ¯ q(h‘ ) = q¯ and p œ (˜ pŸ+1 , p˜Ÿ+1 + ‘) by definition of p† . Such a price p satisfies p˜Ÿ+1 < p < ¯ (1 ≠ e≠r )vH + e≠r p† as long as ‘ > 0 is small. ¯ By Lemma B.4, seller’s expected profit from charging p œ (˜ pŸ+1 , (1 ≠ e≠r )vH + e≠r p† ) ¯ after h such that q(h) = q¯ is V S (h) = p¯ q (1 ≠ ’L ) + e≠r ’L

x q ). C (¯

However, if the seller deviates to pÕ œ (p, (1 ≠ e≠r )vH + e≠r p† ) instead of p, his expected ¯ profit is pÕ q¯(1 ≠ ’L ) + e≠r ’L

x q) C (¯

> V S (h)

contradiction. Proof of Lemma B.4-(ii). Combining Lemma B.4 -(i) and the last Claim proves (ii) holds under the condition (30) only. Proof of Lemma B.4-(iii). Suppose there is a history h œ H such that q(h) = q¯ and p = vH ≠ e≠r wH œ supp ‡(h). Define

5

© (vH ≠ e≠r wH )q(h) 1 ≠ Lemma B.4 implies V S (h) Ø

6

q¯(1 ≠ q(h)) 1 ≠ q(h) ’L + e≠r ’L q(h)(1 ≠ q¯) 1 ≠ q¯

x q ). C (¯

(32)

. Hence, it suffices to show that the opposite inequality also

holds. First note that p > p˜1 , and hence the high type is willing to decline p in equilibrium; this requires fi(h, p) < p with positive probability. p† Ø vH ≠ e≠r wH by Lemma B.4-(i), ¯ and fi(h, p) can be strictly less than p only if the seller has to offer p˜Ÿ+1 or even lower offer in the following period with positive probability. Putting another way, supp ‡(h, p) contains

57

an offer not higher than p˜Ÿ+1 , and thus V (h, p) Æ V S (h) Æ (vH ≠e≠r wH )q(h)a+e≠r

Ë

x (¯ C q ).

This requires q(h, p) Æ q¯.

)

q(h)(1≠a) (1≠e≠⁄ )(1≠b)+e≠⁄

*

+(1≠q(h))’L

È

x q) C (¯

(33)

where a the probability that the high type accepts p and b is the probability that, conditional on that she rejects p and then an outside option arrives in the following period, she takes outside option with probability b in equilibrium. Case I: x = 1: Note that a = 0 cannot be the case in equilibrium because in that case V S (h) <

x (¯ C q)

while the seller always can guarantee expected payoff higher than

x (¯ C q)

(by offering arbitrarily close to p˜Ÿ ). Hence, suppose a > 0 without loss; that is to say, p = vH ≠ e≠r wH is serious. If q(h, p) < q(h) = q¯, seller never charges a price strictly higher than p˜Ÿ at (h, p) and afterward in which case p could not be serious. Hence, a > 0 only if q(h, p) = q¯. This requires 5

q¯(1 ≠ q(h)) ’L (1 ≠ q¯)q(h) (1 ≠ e≠⁄ )(1 ≠ b) + e≠⁄

V S (h) Æ sup (vH ≠ e≠r wH ) 1 ≠ bœ[0,1]

+ e≠r ’L Æ

1 ≠ q(h) 1 ≠ q¯

6

x q) C (¯

as desired. Case II: x = 0: In this cue, ’L = 1 hence

= e≠r

x (¯ C q ).

If p is serious, q(h, p) < q(h) = q¯

and seller’s next offer next to p is no higher than p˜Ÿ . But then, p could not be serious in the first place because vH ≠ p < e≠r (vH ≠ p˜Ÿ ). Therefore p is losing. In other words, a = 0. Lemma 6 implies q(h, p) = q(h) = q¯, and therefore b = x = 0 and V S (h) Æ e≠r

x q) C (¯

=

.

LEMMA B.5. Suppose (CL) holds and let – = (‡, ·, ›, q) be a monotone PBE satisfying (S) and (N C).

(¯ q ; –, ) = {˜ pŸ , vH ≠ e≠r wH } and both p˜Ÿ and vH ≠ e≠r wH are serious.

Proof. First of all,

(¯ q ; –, ) does not include an offer higher than p˜Ÿ+1 other than vH ≠

e≠r wH . To show this, consider an arbitrary history h œ H such that q(h) = q¯ and suppose p > vH ≠ e≠r wH is in supp ‡(h) for contradiction. Claim: p is losing and q(h, p) = q(h) = q¯. 58

If p is serious, Lemma 3 implies q(h, p) < q(h) = q¯, and hence any offers in supp ‡(h, p) are no higher than p˜Ÿ ; vH ≠ p < e≠r (vH ≠ p˜Ÿ ) so p could not be serious in this case which contradicts the hypothesis that p is serious. To see q(h, p) = q(h) = q¯, note that if q(h, p) < q¯, V S (h) is strictly less than profit

x (¯ C q)

x (¯ C q)

because p is losing. But the seller can guarantee expected

by charging p˜Ÿ instead of p.

If the seller keep charing losing offers indefinitely after h, V S (h) is necessarily zero which is impossible in equilibrium. Hence, the seller eventually offers a serious offer which is either p˜Ÿ , p˜Ÿ+1 , or vH ≠ e≠r wH . But Lemma B.2 - (iii) requires fi(h, p) = vH ≠ e≠r wH for sure. Then, (ii) implies

V S (h) Æ e≠r

Ë

(vH ≠ e≠r wH )(1 ≠ ’L )¯ q + e≠r ’L

È

x q) C (¯

.

But the seller can obtain a payoff strictly higher than V S (h) by offering p slightly lower than vH ≠ e≠r wH (Lemma B.4). This contradiction shows higher than p˜Ÿ+1 other than vH ≠

e≠r

(¯ q ; –, ) does not include an offer

wH .

Claim: p˜Ÿ œ (¯ q ; –, ). Suppose not. Consider a history h such that q(h) Ø q¯ and suppose the seller charges

p œ (˜ pŸ , vH ≠ e≠r wH ). Lemma B.2 implies q(h, p) = q¯. Moreover, the high type is willing to decline p because p > p˜1 . But this is impossible because the seller keep charing vH ≠ e≠r wH after (h, p) indefinitely, by (ii) of Lemma B.4.

The last two lemmas (Lemma B.4 and Lemma B.5) shows that, if x = 0, (¯ q ; –, ) ≠r cannot include an offer other than p˜Ÿ and p˜Ÿ+1 ; the only candidate is vH ≠ e wH but ≠r Lemma B.4 shows that equilibrium payoff from charging vH ≠ e wH is strictly less than x (¯ C q ) which is impossible in equilibrium. In conclusion, with x = 0, there is no monotone x PBE other than ones in Eexo,c (ˆ q exo , , ⁄). From now on suppose ›L (·) © x = 1 without loss: ›L (p; h) © x = 1 ’p Ø 0 and ’h œ H

(S Õ )

hence ’L = e≠⁄ . Now define q ú be a unique solution of (1 ≠ e≠r ’L ) xC (q ú ) (1 ≠ e≠r e≠⁄ ) xC (q ú ) = = qú. (1 ≠ ’L )(vH ≠ e≠r wH ) (1 ≠ e≠⁄ )(vH ≠ e≠r wH ) Because xC (·) is strictly increasing (piecewise linear) and C (q Õ ) æ vL as æ 0 for any Õ q by construction of Coasean equilibrium, a unique solution always exists. The following lemma shows q¯ = q ú and also characterizes seller’s continuation payoff with belief q ú . 59

LEMMA B.6. Suppose (CL) holds and let – = (‡, ·, ›, q) be a monotone PBE satisfying (S Õ ) and (N C). q¯ = q ú . Seller’s expected equilibrium profit with posterior q ú is always

x (q ú ). C

Proof. From the last two lemmas (Lemma B.4 and Lemma B.5), for any h œ H such that

q(h) = q¯ and p = vH ≠ e≠r wH œ supp ‡(h), q(h, p) = q¯. Moreover, if p is rejected, the seller randomizes vH ≠ e≠r wH and either of p˜Ÿ or p˜k+1 in the next period. Hence, V S (h) = (vH ≠ e≠r wH )¯ q (1 ≠ e≠⁄ ) + e≠r e≠⁄

x q) C (¯

=

x q) C (¯

for any h œ H such that q(h) = q¯ and q¯ =

x (¯ 1 ≠ e≠r e≠⁄ C q) = qú. ≠⁄ 1≠e vH ≠ e≠r wH

The last Lemma uniquely pins down equilibrium outcome in a continuation game with seller’s posterior is q¯ = q ú . First consider a history h such that q(h) = q ú = q¯. By Lemma B.5, supp ‡(h) µ {˜ pŸ , vH ≠e≠r wH }. If seller charges p˜Ÿ after such a history h, Lemma B.2 implies x q˜Ÿ≠1 (1≠q ú ) ≠⁄ x q(h, p˜Ÿ ) = q˜Ÿ≠1 which requires the high type accepts p˜Ÿ with probability 1 ≠ (1≠˜ x qŸ≠1 )q ú e x and ’H (h, p˜Ÿ ) = 1. From the next period, the buyer and the seller play as in Coasean equilibria, all of which have the equivalent outcome. In particular, the seller charges the  periods in continuation game (in a twinkling if winning offer in N is small enough). ≠r On the other hand, suppose the seller charges p = vH ≠e wH after a history h such that ú ú q(h) = q . Lemma B.4 implies q(h, p) = q again and ’H (h, p) = 1. This requires that the high type accepts p = vH ≠e≠r wH after such a history h with probability 1≠’L = 1≠e≠⁄ . In the following period, with posterior q ú again, the seller randomizes vH ≠ e≠r wH and p˜Ÿ so that the high type was indifferent between accepting and rejecting vH ≠ e≠r wH in the previous period.

Equilibrium Play at h œ H such that q(h) > q¯

This section characterizes equilibrium profile for histories after which seller’s posterior is strictly higher than q¯. The primary purpose is that, for any qˆexo Ø q¯, the seller’s posterior reaches to q¯ in N † periods where N † is uniformly bounded regardless of . If seller never offers higher than vH ≠ e≠r wH in equilibrium, seller’s posterior obviously reaches to q¯ in one period. Hence, without loss, suppose the other way around and define q1† = inf{q(h) : h is such that ÷p œ supp ‡(h) such that p > vH ≠ e≠r wH }.

60

LEMMA B.7. qÕ Ø q†

(i) q(h, p) Æ q Õ for a history h œ H, belief level q Õ , and price p such that and

p˜1 < p < (1 ≠ e≠(r+⁄) )vH + e≠(r+⁄) p+ (q Õ ) ≠ e≠r (1 ≠ e≠⁄ )wH

(ii) q(h, p) Ø q Õ for a history h œ H, belief level q Õ , and price p such that q(h) e≠⁄ 1 ≠ q(h) ’L

Ø

qÕ q¯ Ø Õ 1≠q 1 ≠ q¯

and (1 ≠ e≠(r+⁄) )vH + e≠(r+⁄) p(q Õ ) ≠ e≠r (1 ≠ e≠⁄ )wH < p ¯ Proof. (i) Suppose not; in other words, q(h, p) > q Õ . Then, all offers in supp ‡(h, p) are p+ (q Õ ) or even higher. If all of them are serious, vH ≠ p > e≠r = e≠r

1 1

e≠⁄ V S (h, p) + (1 ≠ e≠⁄ ) max{V S (h, p), wH } 2

e≠⁄ V S (h, p) + (1 ≠ e≠⁄ )wH ,

2

and hence the high type accepts p for sure; but this is impossible because in this case seller’s next offer is vL . Hence, there must be a losing offer pˆ1 œ supp ‡(h, p) such that V S (h, p) Æ e≠(r+⁄) V S (h, p, pˆ1 ) + e≠r (1 ≠ e≠⁄ ) max{V S (h, p, pˆ1 ), wH }. Also if pˆ1 is losing q(h, p, pˆ1 ) > q¯, and hence all offers in supp ‡(h, p, pˆ1 ) are not lower than vH ≠ e≠r wH . Again, there is a losing offer pˆ2 in supp ‡(h, p, pˆ1 ). Otherwise, V S (h, p, pˆ1 ) Æ e≠r wH and

vH ≠ p Æ e≠(r+⁄) V S (h, p) + e≠r (1 ≠ e≠⁄ ) max{V S (h, p), wH } Æ (e≠r ≠ e≠(r+⁄) + e≠r e≠2⁄ )wH

thus p Ø vH ≠ (e≠r

≠ e≠(r+⁄)

+ e≠(2r+⁄) )wH ; but it contradicts to the assumption

p < (1 ≠ e≠(r+⁄) )vH + e≠(r+⁄) p+ (q Õ ) ≠ e≠r (1 ≠ e≠⁄ )wH . Moreover, V S (h, p, pˆ1 ) Æ e≠(r+⁄) V S (h, p, pˆ1 , pˆ2) + e≠r (1 ≠ e≠⁄ ) max{V S (h, p, pˆ1 , pˆ2 ), wH } and q(h, p, pˆ1 , pˆ2 ) > q¯. The repetition this procedure identifies a sequence of losing offers (ˆ pk )kØ1 that the seller offers after p in a sequence with positive probability; contradiction.

61

(ii) For contradiction suppose q(h, p) < q Õ Æ q(h). Then, for any pÕ œ supp ‡(h, p), (1 ≠ e≠(r+⁄) )vH + e≠(r+⁄) pÕ ≠ e≠r (≠1e≠⁄ )wH < p ≈∆ vH ≠ p < e≠(r+⁄) (vH ≠ p) + e≠r (1 ≠ e≠⁄ )wH

and hence p has to be losing. If p is losing, q(h, p) q(h) e≠⁄ Ø 1 ≠ q(h, p) 1 ≠ q(h) ’L

Ø

qÕ 1 ≠ qÕ

which leads to q Õ Æ q(h, p), contradiction. For any j Ø 1, define

S

p†j = vH ≠ e≠r U1 ≠ e≠⁄ (1 ≠ e≠r )

j≠1 ÿ

k=0

T

e≠(r+⁄)k V wH

and let p†0 = vH ≠ e≠r wH and p†≠1 = p˜Ÿ . Also define qj† = inf{q(h) : h is such that ÷p œ supp ‡(h) such that p > p†j≠1 } † for any j Ø 1 and q≠1 = p˜Ÿ . If there is no h such that supp ‡(h) contains a price strictly higher † † than pj≠1 , define qj = 1. Without loss, let {h : ÷p œ supp ‡(h) such that p > p†j≠1 } µ H is nonempty for j = 0, . . . , N † for the candidate of monotone equilibrium –. The primary purpose is that N † is uniformly bounded across all monotone PBE such that {h : ÷p œ supp ‡(h) such that p > p†0 } is nonempty. To achieve this purpose, it suffices to show, using induction, I † {p†j≠1 } if q Õ œ (qj≠1 , qj† ) Õ (q ; –, ) = (A.j) {p†j≠1 , p†j } if q Õ = qj†

and

† supp ‡(h) only includes serious offers if q(h) œ [qj† , qj+1 )

(B.j)

for j Ø 0. The following auxiliary results for j Ø 0 would be also proved in the following paragraphs. (C.j) supp ‡(h) does not contains an offer in (p†j≠1 , p†j ) if q(h) Ø qj† .

(D.j) for h œ H such that p†j œ supp ‡(h) and q(h) Ø qj† , V S (h) = Rj (q(h)) where S

T

† qj≠1 (1 ≠ q Õ ) ’L 1 ≠ qÕ † V + e≠r ’L Rj (q Õ ) = p†j q Õ U1 ≠ Rj≠1 (qj≠1 ) † † ≠⁄ Õ e q (1 ≠ qj≠1 ) 1 ≠ qj≠1

for any q Õ with R0 (q Õ ) =

x (q Õ ). C

62

(E.j)

† qj+1

† 1≠qj+1

e≠⁄ ’L

>

qj†

1≠qj†

.

(F.j) there is a history h œ H such that q(h) = qj† . For any such history h, V S (h) = Rj (qj† ). LEMMA B.8. Suppose (CL) holds and let – = (‡, ·, ›, q) be a monotone PBE satisfying (S Õ ) and (N C). For j = 0, (A.j) ≠ (F.j) are true. Proof. First (A.j) for j = 0, or simply (A.0) is true by Lemma B.5 and the fact that † outcome of – in a continuation game with seller’s posterior q≠1 < q Õ < q0† is identical to

one of Coasean equilibria with ›L (·) = 1. To see (B.0) is also true, note that any offers in † supp ‡(h) with q(h) < q≠1 are serious for any Coasean equilibrium with ›L (·) = 1. To see

(B.0) is true, consider a history h œ H such that q(h) œ [ˆ q † , q1† ). supp ‡(h) contains p†≠1 = p˜Ÿ

or p†0 = vH ≠ e≠r wH or both. From Lemma B.2 and B.4 these offers are accepted by high type with probabilities 1 ≠

q˜Ÿ≠1 (1≠q(h)) ≠⁄ (1≠˜ qŸ≠1 )q(h) e

and 1 ≠

qˆ† (1≠q(h)) ≠⁄ e (1≠ˆ q † )q(h)

, both positive. (C.0)

clearly holds from Lemma B.5 and the monotonicity of ‡. (D.0) and (F.0) follow Lemma B.4-(ii) and Lemma B.6 respectively. Therefore it suffices to show (E.0). By construction, q1† Ø q¯. Suppose q1† = q¯ for contra-

diction. In this case, there is a history h œ H such that positive number ‘ > 0. If there is ‘ > 0 such that ;

q¯ 1≠¯ q +‘

>

q¯ q¯ q(h) hœH: +‘> > 1 ≠ q¯ 1 ≠ q(h) 1 ≠ q¯

q(h) 1≠q(h)

>

q¯ 1≠¯ q

for any small

<

is empty, there must be a history h œ H such that q(h) = q1† = q¯ and supp ‡(h) includes an offer strictly higher than vH ≠ e≠r wH . But it is impossible by Lemma B.5. From now on fix ‘ > 0 and h œ H such that

q¯ 1≠¯ q

+‘ >

q(h) 1≠q(h)

>

q¯ 1≠¯ q,

keeping the

hypothesis q1† = q¯. Without loss suppose all prices in supp ‡(h) are strictly higher than vH ≠ e≠r wH . If seller’s posterior remains above q¯ indefinitely, seller’s profit conditional on h is V S (h) = O(‘) < vL . Hence, seller’s posterior eventually reaches to q¯. V S (h) is then V S (h) Æ vH q(h)‘ + e≠r e≠⁄ which is strictly less than

x (q(h)), C

1 ≠ q(h) 1 ≠ q¯

x q) C (¯

contradiction.

LEMMA B.9. Suppose (CL) holds and let – = (‡, ·, ›, q) be a monotone PBE satisfying (S Õ ) and (N C). For 0 Æ j Æ N † , (A.j) ≠ (F.j) are true Proof. Suppose (A.j) ≠ (F.j) holds for j = 0, 1, . . . , m ≠ 1 where m Ø 1. 63

Proof of (C.m) Suppose the seller charges p œ (p†m≠1 , p†m ) after a history h such that

† . Here, p may not be included in supp ‡(h). By monotonicity of ‡ and inq(h) Ø qm † duction hypothesis, particularly (A.j) for j Æ m ≠ 1, it is immediate that p≠ (qm≠1 ) Æ p†m≠2

† † and p+ (qm≠1 ) Ø p†m≠1 . In case of m = 1, Lemma 7 applies and therefore q(h, p) = qm≠1 = q¯.

If m Ø 1,

† q(h) Ø qm > q0†

for (E.j) holds for j = m ≠ 1 by induction hypothesis, and hence Lemma B.7 applies to † conclude q(h, p) = qm≠1 . In both cases, ’H (h, p) = e≠⁄

the high type accepts p with probability 1 ≠

† qm≠1 (1≠q(h))

because p > vH ≠ e≠r wH hence

† q(h)(1≠qm≠1 )

† to guarantee q(h, p) = qm≠1 . In

conclusion, seller’s expected profit from charging p after h is C

pq(h) 1 ≠ If

q(h) 1≠q(h)

>

† qm≠1

† 1≠qm≠1

† qm≠1 (1 ≠ q(h))

q(h)(1 ≠

† qm≠1 )

D

+ e≠r e≠⁄

1 ≠ q(h)

1≠

† qm≠1

† Rm≠1 (qm≠1 ).

(34)

, the profit above is strictly increasing in p, and p is therefore dominated

by a slightly higher offer. † . Suppose p† is Proof of (D.m) for j = m Suppose p†m is made after h such that q(h) Ø qm m

losing. The skimming property requires q(h, p) Æ q(h). Recall that I keep assuming x = 1. In order for q(h, p) is not higher than q(h), therefore, the high type has to be willing to take an outside option after she declines p. Note that vH ≠ p†m = e≠(r+⁄) (vH ≠ p†m≠1 ) + e≠r (1 ≠ e≠⁄ )wH and hence p†m can be losing only if supp ‡(h, p) contains a price not higher than p†m≠1 . By † and seller offers pÕ œ {p† , . . . , p† induction hypothesis, it requires q(h, p) Æ qm 0 m≠1 } after (h, p)

with positive probability. Hence V S (h, p) < Rm≠1 (q(h, p)) Æ Rm≠1 (q(h)). On the other hand, the seller is always able to guarantee a profit no lower than Rm≠1 (q(h)) again from (D.j) for j = m ≠ 1, contradiction.

† † ] and supp ‡(h, p) = Hence p†m œ supp ‡(h) has to be serious which requires q(h, p) œ [qm≠1 , qm

{p†m≠1 }. The induction hypothesis implies V (h) Æ S

max

† † q Õ œ[qm≠1 ,qm ]

p†m q(h)

5

6

q Õ (1 ≠ q(h)) 1 ≠ q(h) 1≠ + e≠r e≠⁄ Rm≠1 (q Õ ) q(h)(1 ≠ q Õ ) 1 ≠ qÕ

A straightforward calculation would show that the right-hand side is maximized when q Õ =

64

† qm≠1 , and hence

C

† qm≠1 (1 ≠ q(h))

V S (h) Æ p†m q(h) 1 ≠

q(h)(1 ≠

† qm≠1 )

D

+ e≠r e≠⁄

1 ≠ q(h)

1≠

† qm≠1

† Rm≠1 (qm≠1 )

(35)

Finally recall that the seller can guarantee a profit in (34) by charging p œ (p†m≠1 , p†m ) which becomes arbitrarily close to the upper bound (35) of V S (h) as p approaches to p†m . This shows V (h) = S

p†m q(h)

C

† qm≠1 (1 ≠ q(h))

1≠

q(h)(1 ≠

† qm≠1 )

D

+ e≠r e≠⁄

1 ≠ q(h)

† 1 ≠ qm≠1

† Rm≠1 (qm≠1 )

† and p† œ supp ‡(h). whenever q(h) Ø qm m † † . In this case, there is a history h œ H Proof of (E.m) For contradiction, suppose qm+1 = qm

such that such that

† qm † 1≠qm

+‘ >

q(h) 1≠q(h)

>

I

† qm † 1≠qm

hœH:

for any small positive number ‘ > 0. If there is ‘ > 0

† qm

† 1 ≠ qm

† q(h) qm +‘> > † 1 ≠ q(h) 1 ≠ qm

J

† † and supp ‡(h) includes is empty, there must be a history h œ H such that q(h) = qm+1 = qm

an offer strictly higher than p†m≠1 . But it is contradicts to (E.m ≠ 1).

† Proof of (F.m), (A.m), and (B.m) Consider a history h œ H such that q(h) Ø qm+1 . By † q(h) qm > † . 1≠q(h) 1≠qm † † (pm , pm+1 ), therefore,

(E.m), pœ

† ) Ø p† and p≠ (q † ) Æ p† Also p+ (qm m m m≠1 . If the seller charges

† conditional on that the buyer p has to be serious and q(h, p) = qm

rejects p (Lemma B.7). Then, the seller has to randomize p†m and p†m≠1 after (h, p). This † )=R † S Õ † Õ Õ † shows Rm (qm m≠1 (qm ) and so is V (h ) = Rm (qm ) for any h such that q(h ) = qm .

The last lemma, combining the characterization of – for seller’s posterior being not higher than q † ,uniquely pins down monotone PBE outcome that satisfies (S) and (N C). Consider † , q† † a history h such that q(h) œ (qm m+1 ). The last lemma shows that supp ‡(h) = {pm }. If † the seller charges p†m as he should do, Lemma B.4 implies q(h, p) = qm≠1 . Because all offers made on equilibrium path are serious by (B.j), the next offer has to be p†m≠1 , and so on. † for m = 1, 2, . . . to complete the proof. By the proof of Now it remains to identify qm † . (F.m) above, the seller is indifferent between charing p†m and p†m≠1 whenever his belief is qm Payoff form each offer is identified by (C.m). Equating two payoffs,

Q

(p†j+1 ≠p†j ) a

† qj+1

1≠

† qj+1



qj† 1≠

qj†

R

Q

b = (pj ≠ e≠r e≠⁄ pj≠1 ) a

65

qj† 1 ≠ qj†



† qj≠1 † 1 ≠ qj≠1

R b

for j Ø 1. Because pj ≠ e≠r e≠⁄ pj≠1 Ø pj ≠ pj≠1 , Q

(p†j+1 ≠ p†j ) a

† qj+1 † 1 ≠ qj+1



qj† 1 ≠ qj†

R

Q

b Ø (pj ≠ pj≠1 ) a

qj† 1 ≠ qj†



† qj≠1 † 1 ≠ qj≠1

R b

Using definition of p†j , Q

which implies

a

† qj+1 † 1 ≠ qj+1



qj† 1 ≠ qj†

† qj+1 ≠ qj† Ø

R

1

bØ e≠(r+⁄)

Q a

qj† 1 ≠ qj†



† qj≠1 † 1 ≠ qj≠1

R b

1 † † (qj† ≠ qj≠1 ) Ø qj† ≠ qj≠1 e≠(r+⁄)

for j Ø 1. Finally, from R1 (q1† ) = R0 (q1† ), q1†

=

Ë

È

⁄(vH ≠ wH ) + rwH ) Ë

q¯ 1≠¯ q

È

rwH + ⁄(vH ≠ wH ) + rwH )

q¯ 1≠¯ q

+ O( )

and hence there is N † such that seller’s posterior reaches to q¯ within N † periods as long as is small.

B.9

Proof of Proposition 1

Fix any monotone anonymous PBE, say – = (‡, q, ·, ›) and consider each type’s value switching to new seller, V H (?; –, , ⁄) and V L (?; –, , ⁄). From Lemma 1 in Section 2, a new seller never charges strictly below than vL in any monotone anonymous PBE, hence vH ≠ vL Ø V H (?; –, , ⁄) and V L (?; –, , ⁄) = 0. On the other hand, Lemma 5 ensures that a new seller offers p = vL with positive probability in any monotone anonymous PBE. Hence, V H (?; –, , ⁄) > 0. Combining these observations, vH ≠ vL Ø V H (?; –, , ⁄) > V L (?; –, , ⁄) = 0. (36) Suppose the proposition does not hold. Then, there are a positive number µ œ (0, 1), a sequence of intervals ( k )kØ0 converging to zero, a sequence of arrival rates (⁄m )mØ0 such that limmæŒ ⁄m = Œ, and a sequence of monotone anonymous PBE (–k,m )k,mØ0 œ r k,mØ0 EM ( k , ⁄m ) such that µ(vH ≠ vL ) > V H (?; –k,m , 66

k , ⁄m )

(37)

for all k and m. For each k and m, let xk,m œ {0, 1} be the number such that ›Lk,m (h) = xk,m

for all h œ H.

Because all sellers make the winning offer within NC periods in every Coasean equilibria, V H (?; –k,m , k , ⁄m ) Ø e≠rNC k (vH ≠ vL ), and hence the inequalities (37) does not hold if –k,m were Coasean. Hence, suppose –k,m is non-Coasean for any k and m. Replacing wH and wL by V H (?; –k,m , k , ⁄m ) and V H (?; –k,m , k , ⁄m ) respectively, we can see that the condition for Proposition B.1 holds. Appealing the argument of Proposition B.1, the sellers’ offer strategy ‡ k,m should be equivalent to one for a deadlock equilibrium for the case with exogenous outside options, where wH = V H (?; –k,m , k , ⁄m ), and qˆexo = q k,m (?). This also means that I may assume xk,m © 1 for any k because the deadlock equilibrium with ›L (·) © 0 does not exist. Now the proof is completed by following the argument in the main text (see pages 22-28).

67

Bargaining with Arriving Sellers

Dec 15, 2014 - ъYale University (email: [email protected]). I am indebted to ... Examples ..... Notice that the buyer does not pay a direct search cost.

2MB Sizes 0 Downloads 256 Views

Recommend Documents

Recursive Bargaining with Endogenous Threats
Nov 28, 2009 - activities in order to influence their value with outside competitors. ...... workers can largely affect the acquisition of skills outside formal training arrangements, ... with potential customers, rather than establishing a more soli

Recursive Bargaining with Endogenous Threats
Nov 28, 2009 - Contracts, assigning residual rents, are subject to bargaining. .... The intuition follows the idea that an agent's income is solely derived from the ...

Gambling Reputation: Repeated Bargaining With ...
must take into account not only the amount of information that this decision will disclose ... player has an incentive to build a reputation for having a good distribution ..... sults in inefficiency of bargaining outcomes as well as discontinuity in

Bilateral Matching and Bargaining with Private Information
two$sided private information in a dynamic matching market where sellers use auctions, and ..... of degree one), and satisfies lim+$$. ; (2,?) φ lim-$$ ... matching technology is assumed to be constant returns to scale, it is easy to see that J(") .

Bargaining with Interdependent Values ... - Stanford University
Mar 30, 2010 - pacz: Stanford University, Graduate School of Business. e-mail: ..... Of course, proving it for all n would be sufficient (since it would imply that the.

Gambling Reputation: Repeated Bargaining with ...
May 28, 2012 - with the treatment of reputation with imperfect public monitoring in Fudenberg ..... has private information about technology or consumer brand.

Bargaining with incomplete information: Evolutionary ...
Jan 2, 2016 - SFB-TR-15) is gratefully acknowledged. †Corresponding author. Max Planck Institute for Tax Law and Public Finance, Marstallplatz 1,.

Equity bargaining with common value
Jan 30, 2015 - Keywords Asymmetric information bargaining · Information ... In this paper, we ask to what degree players can aggregate information in bilateral.

Bargaining with Revoking Costs
... of possible links between the non cooperative game and the Propor- tional Bargaining Solution. I thank the Center for Research in Economics and Strategy (CRES), in the. Olin Business School, Washington University in St. Louis for support on this

Gambling Reputation: Repeated Bargaining with ...
May 28, 2012 - bargaining postures and surplus division in repeated bargaining between a long-run player ..... When reputation is above the upper threshold p∗∗, both types of the long-run player accept the short-run players' low ..... forth, and

Bargaining with Interdependent Values, Experience ...
Nov 8, 2009 - to our paper, DL assume that the value of the lowest type is strictly lower ... equilibrium takes place in short, isolated bursts of activity, .... indifferent between buying immediately at р) and waiting for the lower price р)-%. ).

Women roadside sellers in Madang
clear that many women enjoy the company and the social ... store (K350–1,800). Food crops, fish, betel, ..... number of relatively high-income earners among the ...

Bargaining with Commitment between Workers and Large Firms
both vacancy-posting and entry coincide with the social marginal values of ...... the more general problem where the planner avails herself of the full set of tools.

Bargaining with Commitment between Workers and ...
firms in the Job Opening and Labor Turnover Survey for the United States. 5BC studying the ..... However, because of risk neutrality and the availability of transfers, many bargaining microfoundations other ... net payments (in excess of the value of

Delay in Bargaining with Outside Options
Oct 12, 2016 - For example, Apple reportedly delayed launching its live TV service in 2015 as price negotiations with content providers stalled.1 Such delays ...

Bargaining with Arrival of New Traders
Nov 22, 2009 - We study dynamic bargaining with asymmetric information and arrival of exogenous events, which represent ... terms of trade, maybe new information will arrive reducing the information asymmetry, etc. Traders compare ... Out of many fac

Bargaining with Commitment between Workers and ...
I thank Mark Aguiar, Mark Bils, Manolis Galenianos, Leo Kaas, Philipp Kircher,. Rafael Lopes de Melo, Giuseppe Moscarini, Eric Smith, and seminar ... for example, that large firms pay higher wages (Brown and Medoff 1989; Davis and Haltiwanger. 1991),

Evolutionary Bargaining with Intentional Idiosyncratic Play
May 7, 2010 - Santa Fe Institute and University of California-Berkeley ‡ Uni- .... transition probability matrix of U-process is irreducible and aperiodic, so the ...

Bargaining with a Property Rights Owner
potential users of this property. A specific context is an innovator of a new technology which is superior to that used by firms in an oligopolistic industry. The IPRO ...