VINCENT J. VANNETELBOSCH

ALTERNATING-OFFER BARGAINING AND COMMON KNOWLEDGE OF RATIONALITY

ABSTRACT. This paper reconsiders Rubinstein’s alternating-offer bargaining game with complete information. We define rationalizability and trembling- hand rationalizability (THR) for multi-stage games with observed actions. We show that rationalizability does not exclude perpetual disagreement or delay, but that THR implies a unique solution. Moreover, this unique solution is the unique subgame perfect equilibrium (SPE). Also, we reconsider an extension of Rubinstein’s game where a smallest money unit is introduced: THR rules out the non-uniqueness of SPE in some particular case. Finally, we investigate the assumption of boundedly rational players. Perpetual disagreement is excluded, but not delay. Furthermore, we cannot use the asymmetric Nash bargaining solution as an approximation of the alternating-offer bargaining model once the players are boundedly rational ones. KEY WORDS: Bargaining, Alternating-offer, Rationalizability, Bounded rationality, Money unit

1. INTRODUCTION

‘Is uniqueness in Rubinstein bargaining model robust to weaker solution concepts, like the rationalizability concepts?’ This paper reconsiders Rubinstein (1982) two-person alternating-offer bargaining game with complete information. He has shown that there is a unique partition of the cake, which can be supported as a subgame perfect equilibrium (SPE). In this SPE, agreement is reached without delay and the less impatient player obtains a larger share of the cake. But the notions of Nash equilibrium and SPE assume that each player holds a correct conjecture about her opponent’s strategy. Once we admit the possibility that a player may have several strategies that she could reasonably use, conjectures and strategies actually played may be mismatched. This is what distinguishes Theory and Decision 47: 111–137, 1999. © 1999 Kluwer Academic Publishers. Printed in the Netherlands.

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rationalizability (Bernheim 1984; Pearce 1984) from equilibrium concepts. Indeed, rationalizability is a weaker solution concept in which the conjectures of the players about the strategy of their opponent are not assumed to be correct, but are constrained by considerations of rationality. Each player knows that the strategy chosen by her opponent is a best response to some conjecture, and, further, each player knows that her opponent knows this and hence knows that her opponent knows that her strategy is a best response to some conjecture, and so on. In other words, the rationality of the players is common knowledge. To offer an answer to our question, we solve Rubinstein’s alternating-offer bargaining game using rationalizability for multistage games with observed actions. A weaker solution concept in the same flavor as rationalizability is iterated conditional dominance. Fudenberg and Tirole (1991, pp. 129–130) have shown that iterated conditional dominance followed by an equilibrium reasoning gives a unique solution which is the SPE of Rubinstein’s bargaining game. In this paper we show that rationalizability for multi-stage games does not exclude agreements reached with delay or even perpetual disagreement. Nevertheless, a weak refinement, trembling-hand rationalizability for multi-stage games with observed actions (THR), singles out a unique solution in Rubinstein’s game. This unique solution is the SPE of the game. Adopting the rationalizability concepts, we also reconsider an extension of Rubinstein’s game, developed by Van Damme et al. (1990) and Muthoo (1991), wherein there is a smallest money unit (i.e., the number of feasible agreements is finite). They have shown that any Pareto-efficient division (belonging to the finite set of feasible divisions) can be supported as a SPE of Rubinstein’s game, provided that the period between successive offers is sufficiently small. Also, as the smallest money unit becomes very small, the SPE payoffs of the discrete bargaining game converge to the unique SPE payoff of the original game. Similar results are obtained using our rationalizability concepts. Furthermore, THR rules out the non-uniqueness of SPE in some particular case. What happens if the players are boundedly rational players? We model bounded rationality by a breakdown in the commonality of the knowledge that the players are rational: both players are playing

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k-step THR strategies. A fact is common knowledge if both players know it, both know that both know it, and so on ad finitum. A fact is mutual knowledge of order 1 if both players know it; mutual knowledge of order 2 if both players know that both players know it, and so on ad infinitum. Incorporating the forward induction principle into the definition of k-step THR, we find outcomes which exclude perpetual disagreement, but not delay. Furthermore, we cannot use the asymmetric Nash bargaining solution as an approximation of Rubinstein’s alternating-offer bargaining model once the players are boundedly rational ones. Other papers have used weaker solution concepts for solving bargaining games. All these papers deal with bargaining games with incomplete information. Watson (1998) has studied Rubinstein’s bargaining game with two-sided incomplete information about the players’ discount factors. He also stated that his solution concept, iterated conditional dominance, doesn’t lead to a unique outcome and doesn’t exclude perpetual disagreement in the complete information case. For solving a one-sided offer bargaining model under one-sided incomplete information (where a seller makes an offer at each period and a buyer has private information about his reservation value), Cho (1994) has used subgame rationalizability (Bernheim 1984) but with the restriction that the buyer must use weakly stationary strategies and that this restriction is common knowledge. The paper is organized as follows. In Section 2, the bargaining model is presented. Section 3 is devoted to rationalizability and trembling-hand rationalizability for multi-stage games. In Section 4 we give our main results on rationalizability and its refinement. In Section 5 we reconsider an extension of Rubinstein’s model where only finitely many divisions of the cake are available. Section 6 investigates the assumption of boundedly rational players. 2. THE BARGAINING MODEL

Two players i(i = 1, 2) are bargaining over the division of a cake of size one. These two players must agree on an allocation from the set X. X is called the set of possible agreements. X ≡ {(x1 , x2 ) ∈ R2 | x1 , x2 > 0, x1 + x2 6 1}

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We denote by xi player i’s share, for i = 1, 2. Consider an alternating-call bargaining procedure. Player 1 calls (offers/accepts) in even-numbered periods and player 2 calls in odd-numbered periods. Let n ∈ N be the period at which an offer is made. The game starts at n = 0 and ends when one of the players accepts the opponent’s previous offer. Note that an agreement may be reached as early as in period n = 1. In each period n, the player on the move chooses an action a(n) ∈ A ≡ X ∪ {accept}; except at the very beginning of the game where player 1 cannot accept. Let h0 = ∅ be the history at the start of play. Define an history of the game at the end of period k k − 1 by hk = (a(0), . . . , a(k − 1)) ∈ H k ≡ 5k−1 n=0 A. Given h and l < k(l, k ∈ N), we call hl a sub-history of hk if hl is the first l elements of hk , and we write hl < hk . Histories after which player ∞ H 2n . Histories after 1 has the move are contained in H1 ≡ Un=0 2n+1 . Let which player 2 has the move are contained in H2 ≡ ∪∞ n=0 H H ≡ H1 ∪ H2 . A pure strategy of player i is a function si : Hi → A which maps each possible history after which player i has the move into an action. Let Si be the set of strategies for player i, i = 1, 2. −i denotes player i’s opponent. S ≡ S1 × S2 is the set of strategy profiles. Payoffs in the bargaining game are given as functions of the player’s strategy profile according to the vN-M utility functions Ui : S → R, i = 1, 2; where Ui (s) ≡ ui (θ (s)). For any outcome θ (s) ≡ [(x1 (s), x2 (s)), n(s)] ∈ X × N that specifies an agreement on allocation [x1 (s), x2 (s)] at period n(s), without too much loss of n(s)−1 generality,1 let ui [θ (s)] ≡ δi xi (s), where δi ∈ (0, 1) is player i’s discount factor, for i = 1, 2. If under s the players fail too reach an agreement, let ui [θ (s)] = 0, for i = 1, 2. Denote by G(δ1 , δ2 ) the alternating-offer bargaining game of complete information. We denote by 1(A) the set of probability distributions on A. A behavior strategy for player i, denoted ci = [ci (h)]h∈Hi , specifies a probability distribution over actions after each h ∈ Hi , ci (h) ∈ 1(A); and the probability distributions after different histories are independent. We interpret behavior strategies of player i as a way of describing player −i’s beliefs about player i’s actions for each history after which she has to move; ci (h) ∈ 1(A) represents a belief of player −i concerning player i’s action after history h. Let Ci be the set of behavior strategies for player i, i = 1, 2; and let C ≡ C1 × C2 . Given any set Sˆi ⊆ Si , i = 1, 2, let Ci (Sˆi ) denote the set of behavior

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strategies of player i when her set of pure strategies is Sˆi . Note that a pure strategy is a special kind of behavior strategy in which the distribution after each histories is degenerate. (si , c−i ) generates a probability distribution over outcomes in the obvious way, and hence gives rise to an expected payoff for each player. Player i’s expected payoff given (si , c−i ) is denoted by Ui (si , c−i ). Given any set B ⊆ A, let 10 (B) be the set of all non-degenerate probability distributions on B. That is, if B is not a singleton, 10 (B) is the set of all probability distributions that do not attach probability one to a single action in the set B. A non-degenerate behavior strategy for player i (or non-degenerate conjecture of player −i), denoted cˆi = [cˆi (h)]h∈Hi , specifies a probability distribution over actions after each history; cˆi (h) ∈ 10 (A) represents a non-degenerate belief of player −i concerning player i’s action after history h. Let Cˆ i be the set of non-degenerate behavior strategies for player i, i = 1, 2; and let Cˆ ≡ Cˆ 1 × Cˆ 2 . Given any set Sˆi ⊆ Si , i = 1, 2, let ˆ Sˆi ) denote the set of non-degenerate behavior strategies of player C( i when her set of pure strategies is Sˆi . Given a history h ∈ H , let G(h) be the continuation game, i.e. the game that begins after h. Given (si , c−i ) and a history h, the continuation strategy profile is the one induced by (si , c−i ) in G(h). Given h ∈ H k , we denote by Ui (si , c−i |h) [Ui (si , cˆ−i |h)] the expected payoff of player i in the game conditional on h describing the play through period k (or stage k) and (si , c−i )[(si , cˆ−i )] describing the play thereafter.

3. RATIONALIZABILITY FOR MULTI-STAGE GAMES

3.1. Definitions. Given the nature of the alternating-offer bargaining game (infinite multi-stage game with observed actions), instead of using Pearce’s (1984) complex definition of rationalizability for extensive-form games or Bernheim’s (1984) subgame perfect rationalizability concept, we extend the definition of rationalizability for normal-form games to our multi-stage game with observed actions. Rationalizability for multi-stage games with observed actions is based on the following assumptions: (B1) both players are rational, (B2) B1 is common knowledge, and (B3) the structure of the game (strategy sets, payoffs functions) is common knowledge. Formally,

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rationalizability for multi-stage games is defined by the following iterative process. DEFINITION 1. Consider the game G(δ1 , δ2 ). Let R 0 ≡ S. Then R k ≡ R1k × R2k (k > 1) is inductively defined as follows: for k−1 i = 1, 2, si ∈ Rik if: (i) si ∈ Rik−1 ; (ii) ∀h ∈ Hi ∃c−i ∈ C−i (R−i ) k−1 0 0 such that ∀si ∈ Ri , Ui (si , c−i |h) > Ui (si , c−i |h). The set of k rationalizable strategy profiles is R ∞ ≡ ∩∞ k=0 R . Assumption B1 means that both bargainers are playing strategies which are rational. In Definition 1, the set Ri1 is the set of player i’s rational strategies. A strategy si ∈ Si is rational if, after each individual history h ∈ Hi , there exists a conjecture, c−i ∈ C−i , such that it is a best response to this conjecture in G(h). That is, a rational player maximizes her expected payoff after each sub-history played, given her conjecture (i.e., given her collection of beliefs about her opponent’s action for each history after which he has to act). Assumption B2 means that it is common knowledge that both bargainers are rational. We call a fact mutual knowledge of order 1 between two players if they both know it, mutual knowledge of order 2 if they both know that both know it, and so on for any order. Let mutual knowledge up to order n be denoted by mk(n). Thus mk(1) is mutual knowledge of order 1 and mk(∞) is common knowledge. Mutual knowledge of order 1 of rationality means that player i knows that player −i is rational. Mutual knowledge up to order 2 (mk(2)) of rationality means that rationality is mk(1) and that player i knows that player −i knows that player i is rational. Mutual knowledge up to order 3 (mk(3)) of rationality means that rationality is mk(2) and that player i knows that player −i knows that player i knows that player −i is rational; and so on. In Definition 1, the set R 2 relies on the assumption of mutual knowledge of order 1 (mk(1)) of rationality. mk(1) of rationality means that player i should not have arbitrary conjectures or collections of beliefs about player −i’s actions for each history after which he has to act. She should hold 1 ). That is, she should exconjectures, c−i , which belong to C−i (R−i pect her opponent to use only strategies that are rational; she should expect her opponent to use only rational actions. Let si (h) be the action specified by the strategy si after the history h. Then, an action ai ∈ A of player i is rational (after history h ∈ Hi ) if and only if

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there exists some rational strategy, si ∈ Ri1 , of player i such that si (h) = ai . In Definition 1, the set R 3 relies on the assumption of mutual knowledge up to order 2 (mk(2)) of rationality. mk(2) of rationality means that player i should hold conjectures, c−i , which 2 ). That is, she should expect her opponent to use belong to C−i (R−i only strategies that are best responses to some conjectures (or collections of beliefs) that he might have. These opponent’s collections of beliefs shouldn’t be arbitrary, it should give weight only on player i’s actions that are rational; and so on for higher-order of mutual knowledge of rationality. Remark that in Definition 1, {R k ; k > 0} is a weakly decreasing sequence, i.e. ∅ 6 = R k+1 ⊆ R k ∀k ∈ N ∪ {∞}. The limit set is k k given by R ∞ ≡ limk→∞ R k = ∩∞ k=0 R . R denotes the set of pure strategy profiles which survive k round of iteration. Each higher step of the iteration requires a higher-order of mutual knowledge of assumption B1. Condition (ii) in Definition 1 restricts conjectures or collections of beliefs to the set of behavior strategies associated with the set of pure strategies that have not been eliminated at a previous stage. Notice that Definition 1 assumes that player i expects player −i to behave rationally in the future even if player −i behaved irrationally in the past. We denote by R k (δ1 , δ2 ) the set of k-step rationalizable strategy profiles and by R ∞ (δ1 , δ2 ) the set of rationalizable strategy profiles for the bargaining game G(δ1 , δ2). 3.2. A refinement: trembling-hand rationalizability (THR). Next we extend the definition of trembling-hand rationalizability for normal-form games2 (due to Herings and Vannetelbosch 1999) to our multi-stage game with observed actions. The starting point of trembling-hand rationalizability for multi-stage games with observed actions (THR) is that the rationality concept is strengthened by asking that each player’s strategy be optimal not only given her conjecture but also given perturbed or cautious conjectures. It can be interpreted as if the players have some doubt about the strategies played by their opponent. Formally, THR is defined by modifying the iterative procedure of Definition 1 as follows. DEFINITION 2. Consider the game G(δ1 , δ2 ). Let T 0 ≡ S. Then T k ≡ T1k ×T2k (k > 1) is inductively defined as follows: for i = 1, 2, k−1 si ∈ Tik if: (i) si ∈ Tik−1 ; (ii) ∀h ∈ Hi ∃cˆ−i ∈ Cˆ −i (T−i ) such that

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∀si0 ∈ Tik−1 , Ui (si , cˆ−i |h) > Ui (si0 , cˆ−i |h). The set of THR strategy k profiles is T ∞ ≡ ∩∞ k=0 T . In Definition 2, the set Ti1 is the set of player i’s trembling-hand rational strategies. A strategy, si ∈ Si , is trembling-hand rational if, after each individual history h ∈ Hi , there exists some nondegenerate conjecture cˆ−i ∈ Cˆ −i against which si is a best response in G(h). At step k of the iteration, a strategy si ∈ Tik−1 belongs to Tik if, after each h ∈ Hi , there exists some non-degenerate conjecture k−1 cˆ−i ∈ Cˆ −i (T−i ) against which si is a best response in G(h). k {T ; k > 0} is a weakly decreasing sequence, i.e. ∅ 6 = T k+1 ⊆ k T ∀k ∈ N ∪ {∞}. The limit set is given by T ∞ ≡ limk→∞ T k = k k ∩∞ k=0 T . Let T (δ1 , δ2 ) be the set of k-step THR strategy profiles ∞ and T (δ1 , δ2 ) be the set of THR strategy profiles for G (δ1 , δ2 ).

4. MAIN RESULTS

Rubinstein (1982) has shown that if the two players have time preferences with constant discount factors, then the two-person alternating-offer game has a unique SPE outcome θ ∗ = [(x1∗ , x2∗ ), 1] with agreement (x1∗ , x2∗ ) = [1 − δ2 , δ2 (1 − δ1 )]/(1 − δ1 δ2 ) reached immediately. Nevertheless, we show that rationalizability for multistage games does not exclude inefficient outcomes. Agreement reached with delay or even perpetual disagreement may occur. 4.1. Non-uniqueness of the rationalizable solution. In order to show the non-uniqueness of the rationalizable solution, we first state some properties of the set of k-step rationalizable strategy profiles R k (δ1 , δ2 ) for our bargaining game G(δ1 , δ2 ) of complete information.3 LEMMA 1. Consider a bargaining game G(δ1 , δ2 ). All s ∈ R k (δ1 , δ2 ) are such that player i never offers x−i > (1 − δi δ−i )−1 (1 − δ−i )(δi δ−i )k−1 + (1 − δi δ−i )−1 (1 − δi )δ−i , accepts all xi > (1 − δi δ−i )−1 (1 − δi )(δi δ−i )k−1 + (1 − δi δ−i )−1 (1 − δ−i )δi , and rejects all xi < (1 − δi δ−i )−1 (1 − δ−i )δi (1 − (δi δ−i )k−1 ), i = 1, 2. Proof. From Definition 1, R 0 = S. All si ∈ Ri1 are such that player i accepts (after any history h) all xi > δi , for i = 1, 2. Player

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i with discount factor δi is indifferent between receiving δi xi today and xi tomorrow. Since xi = 1 is the biggest share i can obtain, she accepts all xi > δi . All si ∈ Ri2 are such that player i never offers x−i > δ−i and rejects all xi < δi (1 − δ−i ), for i = 1, 2. Indeed, player i never accepts an offer which gives her less than δi (1 − δ−i ), because she could obtain close to 1 − δ−i next period by offering x−i = δ−i + ε 1 ), are such with ε small. All player i’s conjectures, c−i ∈ C−i (R−i that, for all h ∈ H−i , c−i (h) gives probability one on player −i’s acceptance of δ−i + ε; and ∀ξ ∈ (0, 1)∃ε ∈ (0, 1) such that δi (1 − δ−i −ε) > δi (1−δ−i )−ξ . R 2 tells us that the largest share of the cake player i could obtain in a continuation game, G(h ∈ Hi ), where she calls first is 1 − δ−i (1 − δi ). All si ∈ Ri3 are such that player i never offers x−i > δ−i (1 − δi (1 − δ−i )), rejects all xi < δi (1 − δ−i (1 − δi (1−δ−i ))) and accepts all xi > δi (1−δ−i (1−δi )). Indeed, player i never accepts an offer which gives her less than δi (1−δ−i (1−δi (1− δ−i ))), because she could obtain close to 1 − δ−i (1 − δi (1 − δ−i )) next period by offering x−i = δ−i (1 − δi (1 − δ−i )) + ε with ε small. 2 ), are such that, for all All player i’s conjectures, c−i ∈ C−i (R−i h ∈ H−i , c−i (h) gives probability one on player −i’s acceptance of δ−i (1 − δi (1 − δ−i )) + ε; ∀ξ ∈ (0, 1)∃ε ∈ (0, 1) such that δi (1 − δ−i (1 − δi (1 − δ−i )) − ε) > δi (1 − δ−i (1 − δi (1 − δ−i ))) − ξ . Then, R 3 tells us that the largest share of the cake player i could obtain in a continuation game, G(h ∈ Hi ), where she calls first is 1 − δ−i (1 − δi (1 − δ−i (1 − δi ))); and so on. k Let {xik }∞ k=0 be strict monotonic decreasing sequences with xi ≡ δi (1 − δ−i (1 − xik−1 )) and starting with xi0 ≡ 1, i = 1, 2. Since δ1 , δ2 ∈ (0, 1), xik ≡ δi (1 − δ−i (1 − xik−1 )) can be written as   −k  δi (1 − δ−i ) 1 1 − δi k xi = + (1) 1 − δi δ−i δi δ−i 1 − δi δ−i speed of convergence

All si ∈ Rik are such that player i: (a) never offers (after any history k−1 , (b) accepts all xi > xik−1 , (c) rejects all xi < δi (1 − h) x−i > x−i k−1 x−i ). These assertions (a), (b), (c) can be easily checked. 2 PROPOSITION 1. Take the bargaining game G(δ1 , δ2 ). A strategy si ∈ Ri∞ (δ1 , δ2 ) if and only if player i offers x−i = (1−δi δ−i )−1 (1−

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δi )δ−i , accepts all xi > (1 − δi δ−i )−1 (1 − δ−i )δi , and rejects all xi < (1 − δi δ−i )−1 (1 − δ−i )δi , i = 1, 2. Proof. In the limit, from Lemma 1, si ∈ Ri∞ is such that player i ∞ = δ (1 − δ )[1 − δ δ ]−1 , offers (after any history h) x−i = x−i −i i i −i accepts all xi > xi∞ = δi (1 − δ−i )[1 − δi δ−i ]−1 , and rejects all ∞ ) = δ (1 − δ )[1 − δ δ ]−1 . xi < δi (1 − x−i 2 i −i i −i Proposition 1 tells us that the rationalizable strategy profiles s ∈ R ∞ (δ1 , δ2 ) are not unique. One of them is the SPE, where player 1 offers x2 = δ2 (1 − δ1 )/(1 − δ1 δ2 ) at period n = 0, offer which is accepted by player 2 at period n = 1. But R ∞ (δ1 , δ2 ) is not a singleton. The conjectures and the strategies actually played by both players may be mismatched. Indeed, player 2’s expected payoff is equal to δ2 (1−δ1 )/(1−δ1 δ2 ). Therefore, player 2 may reject an offer [1 − δ2 , δ2 (1 − δ1 )]/(1 − δ1 δ2 ), hoping that his counter-offer [δ1 (1 − δ2 ), 1 − δ1 ]/(1 − δ1 δ2 ) will be accepted by player 1 next period; player 2 may hold a collection of beliefs, c1 ∈ C1 (R1∞ (δ1 , δ2 )), such that after any history h player 1 accepts his counter-offer. A similar reasoning can be made for player 1. Therefore, players may even play a strategy profile s ∈ R ∞ (δ1 , δ2 ) leading to an agreement reached with delay or to perpetual disagreement. COROLLARY 1. In the bargaining game G(δ, δ2 ), there are strategy profiles s ∈ R ∞ (δ1 , δ2) leading to agreements reached with delay or to perpetual disagreement. As already mentioned in the introduction, Fudenberg and Tirole (1991, pp. 129–130) have argued that iterated conditional dominance gives a unique solution which is the SPE one. In fact after having applied iterated conditional dominance, they used an equilibrium reasoning to single out the unique SPE. Therefore, the relevance that player 1 (player 2) is indifferent between [δ1 (1−δ2), 1− δ1 ]/(1 − δ1δ2 ) today (tomorrow) and [1 − δ2 , δ2(1 − δ1 )]/(1 − δ1δ2 ) tomorrow (today) is turned down. But as discussed above, these indifferences matter once we apply the rationalizability or iterated conditional dominance concept alone.4 4.2. Uniqueness of the trembling-hand rationalizable (THR) solution. The next lemma is useful in proving the uniqueness of the

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THR solution. Moreover, it is also useful when we analyze the case of boundedly rational players in Section 6. LEMMA 2. Consider a bargaining game G(δ1 , δ2 ). All s ∈ T k (δ1 , δ2 ) are such that player i never offers x−i > (1 − δi δ−i )−1 (1 − δ−i )(δi δ−i )k−1 + (1 − δi δ−i )−1 (1 − δi )δ−i , accepts all xi > (1 − δi δ−i )−1 (1 − δi )(δi δ−i )k−1 + (1 − δi δ−i )−1 (1 − δ−i )δi , and rejects all xi < (1 − δi δ−i )−1 (1 − δ−i )δi (1 − (δi δ−i )k−1 ), i = 1, 2. Proof. From Definition 2, T 0 = S. si ∈ Ti1 are such that player i accepts (after any history h) all xi > δi , for i = 1, 2. Player i accepts a share of δi . Given that she holds a non-degenerate conjecture, she cannot believe with probability one that her opponent accepts x−i = 0. Therefore, if she refuses a share of δi then her highest expected payoff is less than δi . Take the non-degenerate conjecture where she believes that her opponent (with probability 1 − p) accepts x−i = 0 and (with probability p) proposes xi = 1 when on the move. As p ∈ (0, 1) approaches zero, her expected payoff approaches δi . Therefore, for all cˆ−i ∈ Cˆ −i , it is a best response for player i to accept (after any history h) all xi > δi . Moreover, it is immediate that player i’s strategies which reject an offer δi xi today and propose xi tomorrow are excluded from the set Ti1 . From the proof of Lemma 1 and the argumentation here above, we have that si ∈ Tik are such that player i never offers (after any k−1 , accepts all xi > xik−1 , and rejects all xi < history h) x−i > x−i k−1 δi (1−x−i ), where xik−1 = (1−δi )(1−δi δ−i )−1 (δi δ−i )k−1 +δi (1− δ−i )/(1 − δi δ−i ). 2 Comparing Lemma 1 with Lemma 2 yields the following conclusion. The only difference between the sets Tik (δ1 , δ2 ) and Rik (δ1 , δ2 ) is that player i’s strategies in Rik (δ1 , δ2 ) which reject an offer δi xi today and propose xi tomorrow are excluded from the set Tik . The following two-player finite game in extensive-form (depicted in Figure 1) illustrates this difference between rationalizability and THR in Rubinstein’s bargaining game. Player 1 has two decision nodes. At each decision node she has two actions, ‘down’ and ‘right’. Player 2 has one decision node where he has two actions, ‘down’ and ‘right’. Every terminal node yields a payoff for both players. The first number is the payoff for player 1 and the second number the

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Figure 1. A two-player game.

payoff for player 2. Player 1 has four pure strategies; S1 = {(down, down), (down, right), (right, down), (right, right)}. Player 2 has two pure strategies; S2 = {down, right}. It is quite obvious that T11 = T1∞ = {(down, down)} and T21 = T2∞ = {down}, while R11 = R1∞ = {(down, down), (right, down)} and R21 = R2∞ = {down}. The difference between T11 and R11 is due to the THR concept which requires that player 1, when making her choice at her first node, has some doubt about her opponent’s next move (in other words, her beliefs have to put weight on both actions of player 2), while the rationalizability concept allows player 1 to believe that player 2 will play ‘down’ with probability one, a belief which rationalizes the action ‘right’ of player 1 at her first node. The next proposition shows the uniqueness of the THR solution for Rubinstein’s alternating-offer bargaining game G(δ1 , δ2 ). PROPOSITION 2. Consider the bargaining game G(δ1 , δ2 ). Then, strategy profile that is THR is unique and is the SPE: s ∈ T ∞ (δ1 , δ2 ) 1−δ2 δ2 (1−δ1 ) leads to an offer ( 1−δ , 1−δ1 δ2 ) by player 1 at n = 0, which is 1 δ2 accepted by player 2 at n = 1. Proof. In the limit, from Lemma 2, si ∈ Ti∞ is such that player ∞ = δ (1 − δ )/(1 − δ δ ), i offers (after any history h) x−i = x−i −i i i −i accepts all xi > xi∞ = δi (1 − δ−i )/(1 − δi δ−i ), and rejects all ∞ ) = δ (1 − δ )/(1 − δ δ ). This unique strategy xi < δi (1 − x−i i −i i −i profile s ∈ T ∞ is such that 1 offers x2 = δ2 (1 − δ1 )/(1 − δ1 δ2 ) at n = 0, offer which is accepted by 2 at n = 1. Therefore, s ∈ T ∞ is the SPE. The outcome (efficient) of the game is the agreement [1 − δ2 , δ2 (1 − δ1 )]/(1 − δ1 δ2 ) which is also players’ payoffs of the 2 game. For some games the commonality of the knowledge that players are trembling-hand rational runs into problems. Our previous ex-

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ample (whose extensive-form is depicted in Figure 1) illustrates this inconsistency. Remember that T11 = T1∞ = {(down, down)} and T21 = T2∞ = {down}. Mutual knowledge of order 1 of tremblinghand rationality means that player 1 knows that player 2 will play ‘down’. Therefore, player 1 is indifferent in opening the game by playing ‘down’ or ‘right’. Nevertheless the opening action ‘right’ is not THR. The problem5 is: why should player 1 plays a THR strategy if player 1 knows that player 2 will play a THR strategy? Let T¯ 0 ≡ S. Then T¯ k ≡ T¯1k × T¯2k (k > 1) is inductively defined as follows: for i = 1, 2, si ∈ T¯ik if: (i) si ∈ Si ; (ii) ∀h ∈ Hi ∃cˆ−i ∈ k−1 Cˆ −i (T¯−i ) such that ∀si0 ∈ Si , Ui (si , cˆ−i |h) > Ui (si0 , cˆ−i |h). Thus each set T¯ik consists of unconstrained best responses, while each set Tik consists of constrained best responses. Remark that in our simple two-person game: T¯12 = {(down, down), (right, down)} ⊃ T1∞ = T¯11 and T¯22 = T2∞ = {down}. Nevertheless, for the alternating-offer bargaining game G(δ1 , δ2 ) we have that, for all k ∈ N, T¯ik = Tik (for i = 1, 2).

5. FINITELY MANY DIVISIONS OF THE CAKE

Adopting the rationalizability concepts defined in Section 3, we reconsider an extension of the alternating-offer bargaining game, developed by Van Damme et al. (1990) and Muthoo (1991), wherein they introduced a smallest money unit. That is, the bargaining problem becomes the division of a cake (or a fixed amount of money) which can be shared only in finitely many different alternatives. They have shown that any partition of the cake can be supported by a SPE if the period between successive offers is sufficiently small. For the case of risk neutral bargainers, Van Damme et al. (1990) have shown that, as the money unit vanishes, all SPE payoffs of the discrete game approximate the unique SPE payoff of the continuum game. Moreover, in equilibrium, there is no delay in reaching an agreement if the money unit is sufficiently small. Similar results are obtained using our rationalizability concepts. Also, we provide a necessary and sufficient condition such that there is a unique solution to this discrete bargaining game that is THR. Furthermore, THR rules out the non-uniqueness of SPE in some particular case.

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Assume that both players are risk neutral and are still bargaining over the division of a cake of size one or an amount of money normalized to one. Let g > 0 be the smallest money unit. Then, the set of possible agreements becomes X ≡ {(n1 g, n2 g)|ni ∈ N, (n1 + n2 )g 6 1}

(2)

Let X e denote the set of efficient agreements, i.e., (n1 + n2 )g = 1. The negotiation proceeds according to the rules defined in Section 2. The sets of actions, histories and strategies are adapted, in an obvious way, to be fitted for this alternative bargaining game. Denote by G(δ1 , δ2 , g) the alternating-offer bargaining game with a smallest money unit. Let R ∞ (δ1 δ2 , g) and T ∞ (δ1 , δ2 , g) be, respectively, the set of rationalizable strategy profiles and the set of THR strategy profiles in G(δ1 , δ2 , g). Denote by Y ≡ {ni g|ni ∈ N, 0 6 ni g 6 1} player i’s possible share of the cake in G(δ1 , δ2 , g). In order to derive our results, we define two functions f : [0, 1] → Y and F : [0, 1] → Y , where ∀y ∈ [0, 1], f (y) ≡ y − mod[y, g] and F (y) ≡ y − mod[y, g] + g. Note that mod[y, g] means y modulo g, i.e. the remainder from dividing y by g. Next we state some properties of the sets R ∞ (δ1 , δ2 , g) and T ∞ (δ1 , δ2 , g) for our bargaining game G(δ1 , δ2 , g) of complete information. PROPOSITION 3. Take the bargaining game G(δ1 , δ2 , g). If δ1 , δ2 , and g are such that (1 − g) 6 δi (for i = 1, 2), then ∀x ∈ X ∃s ∈ R ∞ (δ1 , δ2 , g) such that θ (s) = (x, n). We can also express the bargainer i’s discount factor in terms of discount rate ri ∈ (0, 1), by the formula δi = exp(−ri 1), where 1 is the length of the bargaining period. Greater patience implies a lower discount rate and a higher discount factor: r1 > r2 ⇔ δ1 6 δ2 . Proposition 3 tells us that if the length of a single bargaining round is sufficiently small, then any (feasible) partition of the cake is rationalizable. Moreover, if (1 − g) 6 δi (for i = 1, 2), then for any efficient agreement x ∈ X e there exists a rationalizable strategy profile s ∈ R ∞ (δ1 , δ2 , g) that results in the outcome θ (s) = (x, 1). In the rest of this section, we assume that δi = δ (for i = 1, 2) where δ ∈ (0, 1). A necessary condition for G(δ1 , δ2 , g) being solvable by rationalizability for multi-stage games would be: (1 − g) > δ. Van Damme et al. (1990) have shown that in the equilibrium

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approach such a condition is not to guarantee a unique h sufficient  i 1 1−δ 1 SPE for G(δ, g). Indeed, if mod 1+δ , g < g 1+δ and 1+δ is not an integer multiple of g, then the game G(δ, g) possesses a unique SPE with outcome       1 δ 1 1 − mod ,g , + mod ,g ,1 . θ= 1+δ 1+δ 1+δ 1+δ Let z¯ be the solution of     F (δ(1−F (δ(1− z¯))))             F (δ(1−f (δ(1− z¯ ))))    z¯ ≡     f (δ(1−F (δ(1− z¯ ))))              f (δ(1−f (δ(1− z¯ ))))

( if ( if ( if ( if

mod[δ(1−F (δ(1− z¯))), g] 6 = 0 mod[δ(1− z¯ ), g] 6 = 0 mod[δ(1−F (δ(1− z¯))), g] 6 = 0 mod[δ(1− z¯ ), g] = 0 mod[δ(1−F (δ(1− z¯))), g] = 0 mod[δ(1− z¯ ), g] 6 = 0 mod[δ(1−F (δ(1− z¯))), g] = 0 mod[δ(1− z¯ ), g] = 0 (3)

and z be the solution of ( δ(1 − z¯ ) − mod[δ(1 − z¯ ), g] + g z δ(1 − z¯ )

if mod[δ(1 − z¯ ), g] 6 = 0 if mod[δ(1 − z¯ ), g] = 0 (4)

The next proposition partially characterizes the THR strategy profiles or solutions. PROPOSITION 4. Take the bargaining game G(δ, g). If δ and g are such that (1 − g) > δ, then all s ∈ T ∞ (δ, g) are such that player i never offers x−i > z¯ , and rejects xi < z, for i = 1, 2. The proof of this proposition, as well as the other proofs not in the main text, may be found in the appendix. Proposition 4 gives us an interval of possible player i’s THR expected payoffs when she

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plays a THR strategy si ∈ Ti∞ (δ, g), for i = 1, 2. Precisely, player 1’s THR expected payoffs belong to [1 − z¯ , 1 − z] and player 2’s THR expected payoffs belong to [z, z]. The game G(δ, g) is solvable by THR if and only if z = z¯ . In Proposition 4, as the money unit becomes very small (g → 0+ ), the lower bound (z) and the upper δ bound (¯z) converge to 1−δ . COROLLARY 2. If g tends to zero, the players payoffs associated  1 δ , , 1+δ to any s ∈ T ∞ (δ, g) of the game G(δ, g) converge to 1+δ that is the payoffs of the unique SPE or THR of G(δ). But in some particular case, when the smallest money unit is positive, the THR concept can rule out the non-uniqueness of SPE.  1 δ Whenever Rubinstein’s SPE partition 1+δ , 1+δ is a multiple of g and δ + g 6 1, van Damme et al. (1990) have shown   that the 1 δ discrete game G(δ, g) possesses two SPE payoff vectors 1+δ , 1+δ   1 δ and 1+δ − g, 1+δ + g . That is, the SPE of G(δ) being a SPE of G(δ, g) is not a sufficient condition for uniqueness. Are these SPE payoff vectors THR? Our answer is: only Rubinstein’s SPE partition is supported by the unique THR strategy profile. EXAMPLE. Take the bargaining game G(δ = 2/3, g = 0.05). This game has two SPE with payoff vectors (0.6, 0.4) and (0.55, 0.45). But the unique s ∈ T ∞ (δ = 2/3, g = 0.05) supports only the SPE payoff vector (0.6, 0.4). Using always the same recursive technique (see the appendix for the details), we have that all THR strategies si ∈ Tik (k > 6) are such that player i accepts all offer xi > 0.40, offers x−i = 0.40, and rejects all xi 6 0.35. Therefore, s ∈ T k (k > 6) is unique and it leads to an agreement on (0.6, 0.4) reached immediately. On the contrary, rationalizability leads to the following strategies: player i accepts all xi > 0.40, never offers x−i > 0.45 and rejects all xi 6 0.35. Both concepts tell us that the larger share player i can obtain in a continuation game where she calls first is 0.60. With the rationalizability concept, it may be optimal (for player i) to offer x−i = 0.45 if player i believes that if she offers x−i = 0.40 player −i will reject it and propose xi = 0.40 next period. Such belief is ruled out by THR.

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The next proposition, whose proof is given in the appendix, generalizes the previous example. PROPOSITION 5. If (1 + δ)−1 is a multiple of g and  δ + g6 1, 1 δ and , 1+δ then the game G(δ, g) has two SPE payoff vectors, 1+δ   1 δ 1+δ − g, 1+δ + g , but there is a unique strategy profile that is   1 δ ∞ THR: s ∈ T (δ, g) leads to an offer 1+δ , 1+δ by player 1 at n = 0, which is accepted by player 2 at n = 1. 6. BOUNDEDLY RATIONAL PLAYERS

Under limited capabilities, some problems cannot be managed by the players: the complexity of the set of histories H , the complexity of a strategy, the complexity of bargainers’ reasoning processes, etc. In this section, we only focus on the player’s bounded reasoning process. We model it by a breakdown in the commonality of the knowledge that the players are rational: both players are playing k-step THR strategies. 6.1. The forward induction principle. Lemma 2 has partially characterized player i’s k-step THR strategies, for i = 1, 2. Nonetheless, perpetual disagreement may still occur with k-step THR. That is, kstep THR strategy profiles s ∈ T k (δ1 , δ2 ) are such that the player’s lower bound payoff is equal to zero. In order to tighten the interval of possible players’ payoffs we propose to incorporate the forward induction principle into the definition of k-step THR. Forward induction (FI) asserts that the behavior of rational players in a subgame depends on the options that were available to them in the earlier part of the game, before the subgame. Then, in each subgame, what is k-step THR with forward induction depends on which options and what resulting payoffs the players in the subgame had renounced beforehand.6 At step k of the iterative procedure, given player i knows that k−1 player −i will play a strategy belonging to the set T−i , we define 0 0 player i’s maxmin payoff after h = ∅ or in G(h ) by # " v i (T k−1 ) = max

si ∈Tik−1

min Ui (si , s−i |h0 ) ;

k−1 s−i ∈T−i

i = 1, 2.

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Let v(T k−1 ) ≡ (v 1 (T k−1 ), v 2 (T k−1 )). The maxmin level of player i is the highest payoff that player i can guarantee herself in G(h0 ). In our bargaining game G(δ1 , δ2 ), with boundedly rational players, a player can guarantee herself a payoff. At step k of the iterative procedure, player 1 knows that player 2 will play a strategy belonging to the set T2k−1 . That is, she knows that he will accept all x2 > [(1 − δ2 )/(1 − δ1 δ2 )](δ1 δ2 )k−2 + δ2 (1 − δ1 )/(1 − δ1 δ2 ). Therefore, in the game player 1 can guarantee herself a payoff of at least   1 − δ2 k−1 v 1 (T )= (1 − (δ1 δ2 )k−2 ). 1 − δ1 δ2 This payoff v 1 (T k−1 ) is her lower bound payoff in the game G(δ1 , δ2 ). A similar argument leads to player 2’s lower bound payoff v 2 (T k−1 ) =

δ2 (1 − δ1 ) (1 − (δ1 δ2 )k−2 ). 1 − δ1 δ2

Given an history hl ∈ H l at the end of stage l − 1, we denote by U i (hl ) the maximum between the player i’s highest payoff she could have obtained earlier in the game by accepting one of her opponent’s offer and v i (T k−1 ). That is, if ∃hi ∈ Hi such that hi < hl and hi 6 = h0 , then   l k−1 U i (h ) = max v i (T ), max Ui (si (hi ) = {accept}, ·) hi 6=h0 ∈Hi ,hi
otherwise, U i (hl ) = v i (T k−1 ). Formally, the definition of trembling-hand rationalizability with forward induction (THR-FI) or kstep trembling-hand rationalizability with forward induction (k-step THR-FI) for multi-stage games becomes: DEFINITION 3. Consider the game G(δ1 , δ2 ). Let T 0 ≡ S. Then T k ≡ T1k ×T2k (k > 1) is inductively defined as follows: for i = 1, 2, k−1 si ∈ Tik if: (i) si ∈ Tik−1 ; (ii) ∀h ∈ Hi ∃cˆ−i ∈ Cˆ −i (T−i ) such that: k−1 0 0 (iii) ∀si ∈ Ti , Ui (si , cˆ−i |h) > Ui (si , cˆ−i |h); (iv) Ui (si , cˆ−i |h) > U i (h) and U−i (si , cˆ−i |h) > U −i (h) whenever possible. The set of T k THR-FI strategy profiles is T ∞ ≡ ∞ k=0 T .

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Part (iv) in Definition 3 is the FI principle. By ‘whenever possible’ it means: if there exists a (cˆ1 , cˆ2 ) ∈ Cˆ 1 (T1k−1 ) × Cˆ 2 (T2k−1 ) such that U1 (cˆ1 , cˆ2 |h) > U 1 (h) and U2 (cˆ1 , cˆ2 |h) > U 2 (h). 6.2. A bound on delay. From Lemma 2, all s2 ∈ T2k are such that player 2 will reject all x2 < [δ2 (1 − δ1 )/(1 − δ1 δ2 )] (1 − (δ1 δ2 )k−1 ). Therefore, player 1’s upper bound payoff in the game G(δ1 , δ2 ) is equal to [δ2 (1 − δ1 )/(1 − δ1 δ2 )](δ1 δ2 )k−1 + (1 − δ2 )/(1 − δ1 δ2 ). A similar argument leads to player 2’s upper bound payoff. Therefore, k-step THR-FI (for k > 2) implies strictly positive bounds on the players’ payoffs as well as it excludes perpetual disagreement, but delay may still occur. PROPOSITION 6. Consider a bargaining game G(δ1 , δ2 ). For all k-step THR-FI strategy profiles s ∈ T k (δ1 , δ2 ), (k > 2), U1 (s) ∈ 

    δ2 (1 − δ1 ) 1 − δ2 1 − δ2 k−2 k−1 + (1 − (δ1 δ2 ) ), (δ1 δ2 ) 1 − δ1 δ2 1 − δ1 δ2 1 − δ1 δ2

and U2 (s) ∈ 

   1 − δ2 δ2 (1 − δ1 ) δ2 (1 − δ1 ) k−2 k−1 (1 − (δ1 δ2 ) ), + (δ1 δ2 ) . 1 − δ1 δ2 1 − δ1 δ2 1 − δ1 δ2

Proposition 6 gives us a tight interval of possible player i’s payoff (for i = 1, 2). The lower bound and the upper bound, as k → +∞, converge to the SPE payoff. Remember that [1−δ2 , δ2 (1−δ1 )]/(1− δ1 δ2 ) is the unique SPE payoff vector in G(δ1 , δ2 ). Let 9 we the speed of convergence to the SPE (or THR outcome) of the set of k-step THR outcomes of G(δ1 , δ2 ). We define it as the speed of convergence of xik−1 to xik in the sequence {xik }∞ k=0 defined in the −1 proof of Lemma 1: 9(δ1 , δ2 ) ≡ (δ1 δ2 ) . The speed of convergence is faster the smaller the players’s discount factors. As already mentioned, inefficient outcomes are possible with k-step THR-FI: the negotiation may involve delay. For k > 2, Proposition 6 implies a bound on delay in reaching an agreement.7 To construct a bound, define 0(k, δ1 , δ2 ) ≡ {0 ∈ M|0 > m, ∀m ∈ M}

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where

M = m ∈ N ∃(x1 , x2 ) ∈ X with (  δ1m x1 > [(1 − δ2 )/(1 − δ1 δ2 )](1 − (δ1 δ2 )k−2 ) . δ2m x2 > [δ2 (1 − δ1 )/(1 − δ1 δ2 )](1 − (δ1 δ2 )k−2 ) 

PROPOSITION 7 (bound on delay). Consider a bargaining game G(δ1 , δ2 ). For all k-step THR-FI strategy profiles s ∈ T k (δ1 , δ2 ), (k > 2), an agreement will be reached at period 0(k, δ1 , δ2 ) + 1 at the latest. This bound on delay, 0, is tight. For a given k, the smaller the potential delay (and maximal delay 0) for reaching an agreement the more impatient players are. It is clear that, as k → +∞, this possible inefficiency converges to zero. 6.3. Short intervals between offers. We focus on the case where the interval between offers and counteroffers is short, i.e., as the period length 1 shrinks to zero. We express the bargainer’s discount factor in terms of discount rate ri , by the formula δi = exp(−ri 1). Denote by G(r1 , r2 ) the alternating-offer bargaining game in which the period length 1 shrinks to zero. Binmore et al. (1986) have shown that the unique limiting SPE of G(r1 , r2 ), which is obtained when the length of a single bargaining period approaches zero, approximates the asymmetric Nash bargaining solution to the appropriated bargaining problem. Let T k (r1 , r2) be the set of k-step THR-FI strategy profiles in G(r1 , r2). From Lemma 2 and Proposition 6, taking the limit 1 → 0+ (while keeping constant the ratio r1 /r2 and using l’Hopital’s rule) we obtain the following result. PROPOSITION 8. Consider a bargaining game G(r1 , r2 ). Then, as 1 → 0+ , for all υi ∈ [0, 1] there exists a k-step THR-FI strategy profile s ∈ T k (r1 , r2 ) such that Ui (s) = υi , i = 1, 2. Proposition 8 tells that, as 1 → 0+ , every partition of the cake can be supported by a k-step THR-FI strategy profile s ∈ T k (r1 , r2 ). As 1 → 0+ , the outcome associated to the singleton lim1→0+ (limk→∞ T k (r1 , r2)) is the asymmetric Nash bargaining solution of

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our bargaining problem (see Binmore et al. 1986, or Osborne and Rubinstein 1990). Obviously, from Lemma 2, we have lim ( lim T k (r1 , r2)) ⊂ lim ( lim T k (r1 , r2)).

1→0+ k→∞

k→∞ 1→0+

Therefore, whenever rationality is not common knowledge, as 1 converges to zero, the set of k-step THR-FI outcomes is invariant to the order of mutual knowledge of rationality and does not converge, as k goes to infinity, to the asymmetric Nash bargaining solution. In other words, we cannot use the asymmetric Nash bargaining solution as an approximation of Rubinstein’s alternatingoffer bargaining model once the players are boundedly rational ones.

APPENDIX

Proof of Proposition 3. From Definition 1, R 0 = S. All si ∈ Ri1 are such that player i accepts (after any history h) all xi > F (δi ) = δi − mod[δi , g] + g, for i = 1, 2. Player i with discount factor δi is indifferent between receiving δi xi today and xi tomorrow. Since xi = 1 is the biggest share player i can obtain, she accepts all xi > δi − mod[δi , g] + g. All si ∈ Ri2 are such that player i never offers: x−i > F (δ−i ) = δ−i − mod[δ−i , g] + g if mod[δ−i , g] 6 = 0; x−i > f (δ−i ) = δ−i if mod[δ−i , g] = 0. All si ∈ Ri2 are such that player i rejects all: xi 6 f (δi (1 − F (δ−i ))) if mod[δi (1 − F (δ−i )), g] 6 = 0 and mod[δ−i , g] 6 = 0; xi 6 f (δi (1 − f (δ−i ))) if mod[δi (1 − F (δ−i )), g] 6 = 0 and mod[δ−i , g] = 0; xi < f (δi (1 − F (δ−i ))) if mod[δi (1 − F (δ−i )), g] = 0 and mod[δ−i , g] 6 = 0; xi < f (δi (1 − f (δ−i ))) if mod[δi (1 − F (δ−i )), g] = 0 and mod[δ−i , g] = 0. R 2 tells us that the largest share of the cake player i could obtain in a continuation game where she calls first is: 1 − F (δ−i (1 − F (δi ))) if mod[δi (1 − F (δ−i )), g] 6 = 0 and mod[δ−i , g] 6 = 0; 1 − F (δ−i (1 − f (δi ))) if mod[δi (1 − F (δ−i )), g] 6 = 0 and mod[δ−i , g] = 0; 1 − f (δ−i (1−F (δi ))) if mod[δi (1−F (δ−i )), g] = 0 and mod[δ−i , g] 6 = 0; 1−f (δ−i (1−f (δi ))) if mod[δi (1−F (δ−i )), g] = 0 and mod[δ−i , g] = 0. It follows that if F (δi ) = 1 (for i = 1, 2) then all si ∈ Rik (k ∈ N) are such that player i never offers x−i > 1 and rejects all xi < 0, for i = 1, 2. It is straightforward that, if F (δi ) = 1 (for

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i = 1, 2) then ∀x ∈ X ∃s ∈ R ∞ (δ1 , δ2 , g) such that θ (s) = (x, n). Remark that F (δi ) = 1 can be rewritten as (1−g) = δi −mod[δi , g] which is equivalent to (1 − g) 6 δi ; indeed, [F (δi ) = 1] ⇔ [(1 − g) 6 δi ], (for i = 1, 2). 2 Proof of Proposition 4. Assume that δi = δ ∈ (0, 1), (for i = 1, 2), and (1 − g) > δ. From Proposition 3 and using the same recursive technique as in the proofs of Propositions 1 and 2, we can show that all si ∈ Tik (k > 1) are such that player i never offers x−i > zk−1 , accepts all xi > zk−1 , and rejects all xi < zk−1, where {zk }∞ k=0 is 0 a monotonic decreasing sequence, starting with z = 1 and defined by  (  mod[δ(1−F (δ(1−zk−1))), g] 6= 0   F (δ(1−F (δ(1−zk−1)))) if   mod[δ(1−zk−1), g] 6= 0     (    mod[δ(1−F (δ(1−zk−1))), g] 6= 0  k−1  if F (δ(1−f (δ(1−z ))))   mod[δ(1−zk−1), g] = 0 k z ≡ (   mod[δ(1−F (δ(1−zk−1))), g] = 0  k−1  if f (δ(1−F (δ(1−z ))))   mod[δ(1−zk−1), g] 6= 0     (    mod[δ(1−F (δ(1−zk−1))), g] = 0  k−1  if f (δ(1−f (δ(1−z )))) mod[δ(1−zk−1), g] = 0 This monotonic decreasing sequence converges (in a finite number of steps) to z¯ solution of equation (3). Then, it is straightforward that s ∈ T ∞ are such that player i never offers x−i > z¯ , accepts all xi > z¯ , and rejects all xi < z, where z is solution of equation (4). 2 On the Example. Take the bargaining game G(δ = 2/3, g = 0.05). All si ∈ Ti1 are such that player i accepts all xi > F (δ) = 0.7 (for i = 1, 2). All si ∈ Ti2 are such that player i never offers x−i > F (δ) = 0.7 and rejects all xi < f (δ(1 − f (0.7))) = 0.2. All si ∈ Ti3 are such that player i never offers x−i > F (δ(1 − 0.2)) = 0.55 and rejects all xi < f (δ(1 − 0.55)) = 0.3. All si ∈ Ti4 are such that player i never offers x−i > F (δ(1 − 0.3)) = 0.5 and rejects all xi < F (δ(1 − 0.5)) = 0.35. All si ∈ Ti5 are such that player i never offers x−i > F (δ(1 − 0.35)) = 0.45 and rejects all xi < F (δ(1 − 0.45)) = 0.40. All si ∈ Tik (k > 6) are such that player i never offers x−i > f (δ(1 − 0.40)) = 0.40 and rejects

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all xi < F (δ(1 − 0.40)) = 0.40. It follows that player i’s THR strategies are such that player i accepts all offer xi > 0.40, offers x−i = 0.40, and rejects all xi 6 0.35. Proof of Proposition 5. The existence of two SPE has been shown by Van Damme et al. (1990). Since (1 + δ)−1 is a multiple of g [the cake normalized to one is also a multiple of g], so is δ(1 + δ)−1 ; δ + g 6 1. Let l be such that ( (g)−1 [F (δ) − δ[1 + δ]−1 ] if mod[δ, g] 6 = 0 l= (g)−1 [f (δ) − δ[1 + δ]−1 ] if mod[δ, g] = 0. Our results are derived using Proposition 4 and the same recursive technique as in the proof of Proposition 2. Then, all si ∈ Ti1 are such that player i accepts all xi > z¯ 1 = δ[1 + δ]−1 + lg. All si ∈ Ti2 are such that player i never offers x−i > z¯ 1 = δ(1 + δ]−1 + lg, rejects all ( F (δ[1 + δ]−1 − δlg) if mod[δ, g] 6 = 0 xi < z 1 = f (δ[1 + δ]−1 − δlg) if mod[δ, g] = 0. That is, all si ∈ Ti2 are such that player i rejects all xi < z1 = δ[1 + δ]−1 − 8(lδ)g, where 8(lδ) gives the integer closest to lδ with 8(lδ) 6 lδ. Then, T 2 tells us that the largest share of the cake player i could obtain in a continuation game where she calls first is [1 + δ]−1 + 8(lδ)g. All si ∈ Ti3 are such that player i never offers  F (δ[1 + δ]−1 + δ8(lδ)g) if mod[δ[1 + δ]−1 + δ8(lδ)g, g] 6= 0 2 x−i > z¯ = −1 −1 f (δ[1 + δ]

+ δ8(lδ)g)

where z¯ 2 can be rewritten as  δ[1 + δ]−1 + [1 + 8(δ8(lδ))]g 2 z¯ = −1 δ[1 + δ]

+ δ8(lδ)g

if mod[δ[1 + δ]

+ δ8(lδ)g, g] = 0,

if mod[δ[1 + δ]−1 + δ8(lδ)g, g] 6= 0 if mod[δ[1 + δ]−1 + δ8(lδ)g, g] = 0.

All si ∈ Ti3 are such that player i rejects all xi < z2 where

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z2 =      δ + δ8(lδ)g  F δ 1 − F  1+δ                    f δ 1 − F δ + δ8(lδ)g   1+δ               δ + δ8(lδ)g   F δ 1 − f 1+δ                     δ  f δ 1 − f 1+δ + δ8(lδ)g        

h i δ + δ8(lδ)g, g 6 = 0 and if mod 1+δ h    i δ + δ8(lδ)g mod δ 1 − F 1+δ , g 6= 0 h i δ + δ8(lδ)g, g 6 = 0 and if mod 1+δ      δ mod δ 1 − F + δ8(lδ)g ,g = 0 1+δ i h δ + δ8(lδ)g, g = 0 and if mod 1+δ h    i δ + δ8(lδ)g mod δ 1 − F 1+δ , g 6= 0 h i δ + δ8(lδ)g, g = 0 and if mod 1+δ h    i δ + δ8(lδ)g mod δ 1 − F 1+δ ,g = 0

Then, z2 can be rewritten as z2 =  δ   1+δ − [1 + 8(δ(1 + 8(δ8(lδ))))]g                δ    1+δ − δ[1 + 8(δ8(lδ))]g            δ 2   1+δ − 8(δ 8(lδ))g                  δ 2   1+δ − δ 8(lδ)g          

h i δ + δ8(lδ)g, g 6 = 0 and if mod 1+δ h    i δ + δ8(lδ)g mod δ 1 − F 1+δ , g 6= 0 h i δ + δ8(lδ)g, g 6 = 0 and if mod 1+δ h    i δ + δ8(lδ)g mod δ 1 − F 1+δ ,g = 0   δ if mod + δ8(lδ)g, g = 0 and 1+δ      δ mod δ 1 − F + δ8(lδ)g , g 6= 0 1+δ   δ if mod + δ8(lδ)g, g = 0 and 1+δ      δ mod δ 1 − F + δ8(lδ)g ,g = 0 1+δ

and so on . . . . It is straightforward that the monotonic decreask ∞ ing sequences, {¯zk }∞ k=1 and {z }k=1 , converge (in a finite number of steps) to z∞ = δ[1 + δ]−1 = z¯ ∞ . Then, all s ∈ T ∞ are such that player i never offers x−i > δ[1 + δ]−1 , accepts all xi > δ[1 + δ]−1 , 2 and rejects all xi < δ[1 + δ]−1 .

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ACKNOWLEDGMENTS

I wish to thank Pierpaolo Battigalli, Eric van Damme, Claude d’Aspremont, Pierre Dehez, Martin Dufwenberg, Françoise Forges, Ana Mauleon, Hans Peters and two anonymous referees for helpful comments and discussions. This paper was presented at seminars and conferences at the University of Bonn, University of Louvain, Hebrew University of Jerusalem, University of Valencia, University of Aarhus, and WZB (Berlin).

NOTES 1. Indeed, it would be enough to assume that players’ preferences over outcomes satisfy Osborne and Rubinstein’s (1990) assumptions A1 to A6. These assumptions impose the following conditions on players’ preferences over outcomes: (A1) disagreement is the worst outcome, (A2) cake is desirable, (A3) time is valuable, (A4) player i’s preference ordering is continuous, (A5) stationarity of player i’s preference ordering, and (A6) the larger the share of the cake the larger the compensation player i requires to be indifferent to incur a delay of one period. For ease of exposition, we assume that ui (θ (s)) ≡ δin(s)−1xi (s). The preferences that this function represents satisfy these assumptions and are called time preferences with constant discount factors. 2. Herings and Vannetelbosch (1999) have defined the trembling-hand rationalizability concept for normal-form games. They have studied the relationships between this concept and other refinements (perfect, weakly perfect, proper, and cautious rationalizability). Trembling-hand rationalizability seems to have most cutting power in many examples and it is the only solution concept that does not change when adding dominated strategies to the game. 3. The proofs employ a recursive technique, which involves establishing the relationship between bounds on payoffs or actions over time, comparable to the technique used by Fudenberg and Tirole (1991). 4. Watson (1998) mentioned that iterated conditional dominance does not exclude perpetual disagreement or delay and that this result was known to Martin Osborne. 5. The inconsistency problem has been studied by Reny (1993). To resolve such an inconsistency of common knowledge of trembling-hand rationality, Asheim and Dufwenberg (1996) have changed the object for the common knowledge: instead of common knowledge of rational choice, they assume common knowledge of rational reasoning.

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6. Forward induction (FI) was first introduced by Kohlberg and Mertens (1986); see also Van Damme (1989). Our formalization of FI is based on Hammond’s (1993) informal definition of conditional rationalizability. A similar notion of FI is implicitly incorporated in Pearce’s (1984) definition of extensive-form rationalizability for finite games. Notice that, contrary to Börgers (1991), we apply the FI at each step k of our iterative procedure. 7. Note that delay is time delay from period n = 1, since agreement cannot be reached before period n = 1.

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Address for correspondence: V.J. Vannetelbosch, CORE, University of Louvain, voie du Roman Pays 34, B-1348 Louvain-la-Neuve, Belgium, and IEP, Basque Country University, avda. Lehendakari Aguirre 83, E-48015 Bilbao, Spain Phone: 34-94-4795555; Fax: 34-94-4795555; E-mail: [email protected]