Imitation in Cournot Oligopolies with Multiple Markets Jonas Hedlund March 2015

Abstract This paper analyzes imitation dynamics in Cournot oligopolies when …rms imitate both rivaling …rms and …rms in other markets. The resulting tension between relative and absolute performance leads to a unique prediction strictly between the Nash equilibrium and perfectly competitive outcomes, which is fully characterized by a simple formula. The outcome becomes less competitive as the number of markets increases, i.e., as …rms receive more information about …rms in other markets. A link with relative payo¤ maximization is provided. An extension of the benchmark model reveals that sophisticated …rms imitating across asymmetric markets converge to a related but somewhat less competitive outcome. Keywords: Imitation, Experimentation, Oligopoly, Evolution. JEL Classi…cation: C73, D43

1

Introduction

The seminal contribution of Vega-Redondo (1997) initiated a literature exploring imitation dynamics in Cournot oligopolies. Instead of, for example, computing best responses, …rms are assumed to imitate actions observed to result in high payo¤s. The approach can be motivated from several standpoints. From a bounded rationality perspective, oligopolies are complex environments, and "fast and frugal" decision rules, such as imitation, may reduce decision costs (see, e.g., Pingle and Day 1996; Gigerenzer, Todd and the ABC Research Group 1999). Department of Economics, University of Heidelberg. Address: Bergheimer str. 58, 69115, Heidelberg, Germany. Email: [email protected]. Phone: +49 6221 54-3171. Fax: +49 6221 54-2997

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The empirical relevance of imitative behavior is documented both in experimental oligopoly games (see, e.g., Huck, Normann and Oechssler 1999; O¤erman, Potters and Sonnemans 2002; Apesteguia and Selten 2005; Apesteguia, Huck and Oechssler 2007) and by recent empirical work in the management literature (see, e.g., Rhee, Kim and Han 2006 and the references therein). From the point of view of evolutionary game theory, imitation conveniently models the idea (dating back to Alchian in 1950) that a population should adjust toward actions performing well relative to other current actions. If agents imitate successful behavior, the best performing current actions accordingly spread. Vega-Redondo (1997) demonstrated that this emphasis on relative performance can lead to surprising predictions in oligopolies. A market of …rms repeatedly imitating the quantity generating the highest pro…t eventually converges to the perfectly competitive outcome (i.e., to the Walrasian quantity). The result follows since …rms producing quantities close to the competitive equilibrium tend to do better than their rivals and are therefore often imitated. This argument, however, depends on the implicit assumption that …rms imitate only their rivals. Apesteguia, Huck and Oechssler (2007) …nd that less competitive outcomes occur when there is more than one market for the same product (e.g., markets located in di¤erent cities) and …rms imitate across markets.1 In this case, the performance of …rms within their respective markets and the performance across markets jointly determine which quantities are imitated. The result is a natural tension between relative and absolute performance and a corresponding tension between the perfectly competitive and Nash equilibrium outcomes. While the link between imitation and competitive behavior depends on whether …rms imitate rivals or non-rivals, the nature of this relationship is currently not well understood. Apesteguia et al.’s (2007) analysis is tailored for experiments and limited to linear triopolies with a restrictive action space and a particular assumption regarding …rms’information about other markets. The present paper shows that the impact on market outcomes of …rms’information about other markets can be tractably analyzed in a substantially more general framework. Further, this analysis leads to a novel and unique prediction, which is fully characterized by a 1

Some empirical evidence suggests that strategic decisions are sometimes guided by observing …rms in other markets. For example, Greve (1998) …nds that local US radio stations imitate the choice of market position of stations serving other regions.

2

simple formula and can be linked to optimizing behavior. While the analysis con…rms that imitation of non-rivals moderates competition under conditions well beyond those contemplated by Apesteguia et al. (2007), it also reveals that their exact results are not robust beyond their particular setup. Where Apesteguia et al. (2007) predict the Nash equilibrium outcome, the analysis here uncovers a tension between relative and absolute performance, which leads to a prediction strictly between the Nash equilibrium and perfectly competitive outcomes. The model consists of a …nite number of identical, separated markets, with a …nite number of …rms in each, re‡ecting, for example, markets for the same product located in di¤erent cities. Firms observe the quantities and pro…ts of all …rms and update behavior using an imitation rule. For clarity of exposition, and as in several papers in which di¤erent payo¤s can simultaneously be observed for the same action, I assume that imitation responds to average performance. Speci…cally, …rms use Imitate the Best Average (henceforth, IBA), i.e., they imitate the quantity generating the highest average pro…t in the previous period. By comparing averages, IBA makes comprehensive use of all available information and prescribes imitating a quantity performing well in one market only if it does not perform too poorly in another.2 Firms sometimes also randomly experiment. Following the literature, I identify the stochastically stable states of the resulting stochastic process (see, e.g., Young 1993). The analysis reveals that there is a unique stochastically stable quantity strictly between the Cournot and Walrasian quantities (Proposition 1), which is both evolutionary stable (Lemma 2) and a global invader (Lemma 3).3 The stable quantity balances two main forces of the model. While …rms producing quantities close to the Walrasian quantity tend to outperform their rivals, …rms producing quantities close to the Cournot quantity often do well compared to …rms in other markets. The main results reveal that a unique quantity between the Cournot and Walrasian quantity performs well in terms of a speci…c weighted average of these e¤ects, which, since imitation is driven by average performance, endows it with stability properties. The stable quantity is decreasing in the number of markets, since the weight of non-rivals in the 2

Ellison and Fudenberg (1993), Eshel, Samuelson and Shaked (1998), Apesteguia et al. (2007) and Bergin and Bernhardt (2009), among others, consider similar imitation rules. While focusing on IBA allows a clear exposition, one can of course think of alternative rules. For example, more optimistically biased …rms may imitate the quantity with the highest maximum pro…t. Section 3.5 discusses such behavior and argues that it leads to results closely paralleling Alós-Ferrer’s (2004) analysis of imitation among …rms with …nite memory. 3 The "Cournot quantity" is the quantity produced by each …rm in the symmetric Nash equilibrium.

3

computation of average pro…ts is increasing in the number of markets, leading to predictions closer to the Cournot quantity. Adding a market therefore reduces overall competition. This prediction, which is not present in standard models, may be appropriate for empirical testing. Finally, the stability of the Walrasian quantity when …rms imitate only their rivals is closely related to the fact that the Walrasian quantity is a Nash equilibrium in a game of relative payo¤ maximization (see, e.g., Scha¤er 1989). Here the stable quantity can be obtained as a Nash equilibrium of a game in which …rms are concerned about pro…ts in both absolute and relative terms (Proposition 2). The di¤erence between the result here and Apesteguia et al.’s (2007) corresponding result is mainly due to di¤erent assumptions regarding …rms’information about other markets. Apesteguia et al. (2007) assume that …rms only observe non-rivals in the same "role", which restricts how information is revealed across markets. In particular, successful choices are not observed by all …rms at once, limiting the extent to which they can spread. This stabilizes the Cournot quantity in their setup. The focus on ex-ante identical markets is obviously restrictive. In an extension of the benchmark model, inverse demands are allowed to di¤er across markets. A model of imitation across ex-ante asymmetric markets requires an assumption of how …rms take asymmetries into account in their choices, ranging from negligence to more sophisticated behavior. Here, imposing a particular and sophisticated understanding of the asymmetries yields a tractable analysis and clear results. Speci…cally, …rms are assumed to calculate the pro…t the observed …rm would have obtained if its market were ex-ante identical to the imitating …rm’s market. Such a calculation is possible if …rms correctly understand how inverse demands di¤er across markets, which is more likely when markets simply di¤er in size (for example). An implication is that a …rm is less impressed by the pro…ts obtained by another …rm if that …rm faces a larger demand. Observations across markets are consequently not misleading per se and the resulting model therefore parallels the benchmark model fairly closely. The analysis reveals a unique stochastically stable quantity that is related to that of the benchmark model but is less competitive (Proposition 3). Intuitively, …rms’ understanding of the asymmetries leads markets to behave fairly independently, which reveals the bene…ts of more collusive behavior.

4

There is a close analogy between the analysis in the present paper and Bergin and Bernhardt’s (2009) and Alós-Ferrer’s (2004) analyses of …rms that remember a …nite number of past quantities and pro…ts (see also Alós-Ferrer and Shi 2012). Their models can be understood as models with several markets, in which the additional "markets" exist in the memories of the …rms. The memory model, however, limits the order in which imitation across markets occurs. For example, a quantity cannot be imitated in all markets at once. This di¤erence can be crucial or inconsequential. In contrast to the result here, Bergin and Bernhardt (2009) …nd the symmetric collusive outcome to be stochastically stable if …rms imitate according to IBA. As argued in Section 3.5, however, under an alternative imitation rule the prediction here coincides with Alós-Ferrer’s (2004) corresponding result. Finally, this paper is related to the various extensions of Vega-Redondo’s (1997) framework, e.g., to asymmetric oligopolies (Tanaka 1999), markets with di¤erentiated goods (Tanaka 2000), more general technical conditions (Schenk-Hoppé 2000) more general classes of games (AlósFerrer and Ania 2005) and markets with both optimizers and imitators (Schipper 2009).

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The model

2.1

Market structure

Let K := f1; 2; :::; kg be a set of k

2 identical and separated markets with n

2 identical

…rms in each. Let N := f1; 2; :::; ng. A market is identi…ed by an index j 2 K and a …rm is identi…ed by a double index ij 2 N quantity qij (t) 2

K. In each period t = 0; 1; 2; :::, each …rm ij chooses a

at cost C(qij (t)) where

= f0; ; 2 ; :::; v g is a …nite grid. The cost function

C : R+ ! R+ is assumed twice continuously di¤erentiable, with C 0 ( ) > 0 and C 00 ( ) step size of

0. The

should be thought of as "small."

In each period t, …rms in each market j face an inverse demand P (Qj (t)), where Qj (t) := P

i2N

qij (t) for all j 2 K. The inverse demand function P : R+ ! R+ is assumed continuous

and twice continuously di¤erentiable. Further, P (0) > C 0 (0), P 0 ( ) < 0, P 00 ( ) [0; Q] with Q > 0 and P (Q) = 0 for all Q 4

0 on some

Q.4 The pro…t of …rm ij in t is (qij (t); Qj (t)) :=

Some, but not all, results hold under log-concave inverse demands. For example, Lemma 2 below can be

5

C(qij (t)).5

P (Qj (t))qij (t)

The Walrasian quantity is the quantity produced by each …rm in equilibrium under pricetaking behavior. De…nition 1 The Walrasian quantity q w is such that P (nq w )q w

C(q w )

P (nq w )q

C(q),

for all q 2 R+ . The Cournot quantity is the quantity produced by each …rm in symmetric Nash equilibrium. De…nition 2 The Cournot quantity q c is such that P (nq c )q c C(q c )

P ((n 1)q c +q)q C(q),

for all q 2 R+ . Given the assumptions on the cost and demand functions q w and q c uniquely exist and are de…ned by P (nq w )

C 0 (q w ) = 0 and P 0 (nq c )q c + P (nq c )

C 0 (q c ) = 0. For simplicity, I assume

that q w ; q c 2 .

2.2

Decision and dynamics

In each period t, each …rm is picked with independent and identical probability revise its choice. When

2 (0; 1] to

2 (0; 1), the choices are revised with inertia. Inertia re‡ects both

an unwillingness to adjust the quantity too often and the possibility that …rms do not adjust their quantities in a perfectly coordinated way. Firms know neither the demand nor the cost function and must base their choices on observed behavior and associated performance. In particular, a …rm revising its choice in t observes the quantities and pro…ts in t

1 of all …rms and applies IBA, i.e., assigns positive

probability to all quantities generating the highest average payo¤ in t t

(q) := fij 2 N

K : qij (t

1.6 Formally, let

1) = qg. The revising …rm assigns positive probability to all

quantities in

arg max qij (t 1):ij2N

1 t K j (qij (t

1))j

X

i0 j 0 2 t (q

ij (t

(qi0 j 0 (t

1); Qj 0 (t

1)):

(1)

1))

shown to hold under this assumption, but counterexamples reveal that Lemma 3 does not. 5 Henceforth, time indexes are omitted whenever possible. 6 Notice that the informational environment bears some resemblance to the multiple locations model in Anwar (2002) and Ely (2002), see also Alós-Ferrer and Weidenholzer (2008).

6

IBA aggregates all observed information in a balanced way by computing simple averages and implies that a quantity performing well in one market is imitated only if it does not perform too poorly in another market. The model outlined so far de…nes a …nite Markov chain, where a state is a tuple of quantities (q11 ; :::; qn1 ; q12 ; :::; qn2 ; :::; q1k ; :::; qnk ), the state space is determined by

(m)

kn

and the transition probabilities are

and IBA. Following convention, this process is referred to as the unperturbed

process. Let Px;y be the probability of reaching y 2 set A

kn

kn

in m steps, starting from x 2

kn

. A

(1)

is an absorbing set if (i) for all x 2 A and y 62 A, Px;y = 0 and (ii) for all x; y 2 A (m)

there is some m 2 N such that Px;y > 0. A singleton absorbing set is an absorbing state, i.e., a state that once entered is never left. Below it is argued that any absorbing set is an absorbing state. Let

kn

denote the set of absorbing states. Let !(q) denote a monomorphic state

such that all …rms produce q 2 . Let states. Obviously,

M

M

:= f!(q) : q 2 g denote the set of monomorphic

.

While the unperturbed process always converges to an absorbing state,

is too large to

represent an interesting prediction.7 Following the literature (e.g., Vega-Redondo 1997), I incorporate noise, which o¤ers a tool to select between the di¤erent absorbing states. In each period, each …rm experiments with common and independent probability " to some quantity picked according to some probability distribution over

with full support.

The standard assumption that the experimentations are independent is important for the results. Its plausibility depends on the assumption that no …rm is active on di¤erent markets. A generalization involving multi-market …rms yields similar results, if the assumption of independent experimentations is maintained (i.e., if each multi-market …rm experiments with each of its sub-…rms with independent probability "). The more reasonable assumption in this case, however, seems to be that the experimentations of the sub-…rms of a multi-market …rm are correlated. This would yield a substantially di¤erent analysis, in which the predictions are likely to depend on the particular distribution of sub-…rms across markets and the correlation between the experimentations of the sub-…rms. While interesting, a careful examination of this generalization is left for future research. 7

For some values of convergence to an absorbing state is fast. For example, if enters an absorbing state from any state at which there are no ties.

7

= 1 the process immediately

The noise generates a perturbed process, which is irreducible and aperiodic, and therefore, by standard results, has a unique stationary distribution, which describes average long-run behavior. Denote this stationary distribution by u" and let u := lim"!0 u" . That is, u is the limit of the stationary distribution of the perturbed process as the experimentation probability approaches zero. The support of u is the set of stochastically stable states and is denoted by

ss

. As the experimentation probability approaches zero, the fraction of time spent in the

stochastically stable states approaches one in the long run. Hence, the set of stochastically stable states is a long-run prediction of the model. By a standard result, absorbing sets (see, e.g., Young 1993); hence, here

ss

ss

is a union of

. Intuitively, when " is small,

imitation leads to an absorbing state relatively quickly. Therefore, the process spends most of its time in absorbing states. Some absorbing states require relatively many experimentations to leave and are easily reached from other absorbing states. Such states tend to be stochastically stable. The set of stochastically stable states can be identi…ed using techniques developed by Freidlin and Wentzell (1988), Young (1993), Kandori, Mailath and Rob (1993) and Ellison (2000).

3 3.1

The stable quantity Preliminaries

The …rst step of the analysis is to identify the candidates for stochastic stability, i.e., the set of absorbing states. As in several related models (e.g., Vega-Redondo 1997, Alós-Ferrer 2004, and Bergin and Bernhardt 2009), this step is immediate. First, the fact that …rms can only imitate quantities presently produced implies that any state at which all …rms produce the same quantity is absorbing. Second, the assumption that all …rms have identical information and use identical imitation rules implies that no other state can be absorbing. In particular, in any state at which more than one quantity is produced, IBA requires all …rms to assign positive probability to the same set of quantities. All …rms consequently imitate the same quantity with positive probability. The process therefore leaves the original state for an absorbing state with positive probability, and the original state cannot be absorbing. Hence, the set of absorbing 8

states is equal to the set of states such that all …rms produce the same quantity. Lemma 1

=

M.

The next step is to pin down which of the absorbing states are stochastically stable. Whether an absorbing state is stochastically stable depends roughly on whether entering this state from other absorbing states by means of experimentation is easier than leaving it. Given an absorbing state !(q), an experimentation to q 0 by some …rm is imitated with positive probability if and only if the pro…t of the experimenting …rm is at least as high as the average pro…t of the remaining …rms. The pro…t of the experimenting …rm equals (q 0 ; (n

1)q + q 0 ), while the n

1

other …rms in the experimenting …rm’s market obtain (q; (n 1)q+q 0 ) and the remaining n(k 1) …rms obtain (q; nq). The experimentation is then imitated if (q 0 ; (n 1)q+q 0 ) is larger than [(n

1)=(nk

1)q + q 0 ) + [n(k

1)] (q; (n

1)] (q; nq). That is, the experimenting

1)=(nk

…rm must outperform a weighted average of rival and non-rival …rms. If this is the case, all …rms imitate q 0 with positive probability in the period following the experimentation; therefore, the process enters !(q 0 ) with positive probability. Let F : R2+ ! R be de…ned by F (q; q 0 ) := (q 0 ; (n

1)q + q 0 )

n 1 (q; (n nk 1

1)q + q 0 ) +

n(k nk

1) (q; nq) : 1

(2)

An experimentation to q 0 at !(q) leads to !(q 0 ) with positive probability if and only if F (q; q 0 ) 0. Whether F (q; q 0 ) is positive depends on how (q 0 ; (n 1)q +q 0 ) compares with (q; (n 1)q + q 0 ) and (q; nq). If (q 0 ; (n

1)q + q 0 ) > (q; (n

1)q + q 0 ), i.e., if the experimenter outperforms

its rivals, I say that q 0 leads to an improvement in relative terms. Experimentations in the direction of q w (i.e., such that q < q 0

q w or q w

q 0 < q) always lead to an improvement

in relative terms. This observation drives Vega-Redondo’s (1997) result for the single market setting. If (q 0 ; (n

1)q + q 0 ) > (q; nq), i.e., if q 0 is a better response than q, I say that q 0

leads to an improvement in absolute terms. Experimentations in the direction of q c (i.e., such that q < q 0

q c or q c

q 0 < q) always lead to an improvement in absolute terms since q 0 is

then a better response.

9

Whenever q 62 [q c ; q w ], there are experimentations that are in the direction of both q c and q w , and therefore, lead to improvements in both relative and absolute terms. For example, if q < q c then q 0 = q c is in the direction of both q c and q w and (q 0 ; (n

1)q + q 0 ) > maxf (q; (n

1)q +

q 0 ); (q; nq)g, and therefore, F (q; q 0 ) > 0. If q 2 [q c ; q w ], however, then any experimentation is either not in the direction of q c or not in the direction of q w . For example, if q = q c , then any q 0 2 (q; q w ] leads to a positive relative e¤ect and a negative absolute e¤ect. Whether F (q; q 0 ) is positive then depends on the strength of these e¤ects and the weights determined by n and k. Below it is argued that there is a quantity strictly between q c and q w that balances this tension between relative and absolute e¤ects in a unique and precise way. Any positive relative e¤ect of an experimentation at an absorbing state at which this quantity is produced is outweighed by a negative absolute e¤ect, and any positive absolute e¤ect is outweighed by a negative relative e¤ect. The properties of this quantity lead it to constitute the prediction of the model.

3.2

Evolutionary stability, global invaders, and stochastic stability

If, for some q, it holds that F (q; q 0 ) < 0 for all q 0 6= q, then no single experimentation can destabilize !(q). Imitation inevitably leads back to !(q). I refer to such a quantity as an evolutionary stable quantity (in the spirit of …nite population evolutionary stability, see, e.g., Scha¤er 1988). Since F (q; q) = 0 for all q, it must hold that if q is evolutionary stable, then F2 (q; q) = P 0 (nq) [n(k

1)=(nk

1)] q + P (nq)

C 0 (q s ) = 0. The following lemma

demonstrates that the solution to F2 (q; q) = 0 indeed de…nes a unique evolutionary stable quantity. Let

:= (n

1)=(nk

1). The proofs of all formal results are provided in the

appendix. Lemma 2 There is a unique evolutionary stable quantity q s (k; n), de…ned by P 0 (nq s (k; n))(1 )q s (k; n) + P (nq s (k; n))

C 0 (q s (k; n)) = 0. Furthermore, q s (k; n) 2 (q c ; q w ).8

Lemma 2 follows from the trade-o¤ between relative and absolute e¤ects. Notice that given = (n

1)=(nk

1) and 1

F (q; q 0 ) = [ (q 0 ; (n 8

1)q + q 0 )

= n(k

(q; (n

1)=(nk

1) it is possible to write F (q; q 0 ) as

1)q + q 0 )] + (1

)[ (q 0 ; (n

1)q + q 0 )

When possible, the arguments of q s (k; n) will be omitted in order to reduce notation.

10

(q; nq)]. (3)

For q c < q < q 0 < q w the …rst term on the right hand side of (3) is positive and the second term is negative, due to positive relative e¤ects and negative absolute e¤ects. If q and q 0 are close to q c , the relative e¤ect dominates since the absolute e¤ect is then only of the second order. This follows since q c is a Nash equilibrium, i.e., q c = arg maxq0 (q 0 ; (n

1)q c + q 0 ). If q and q 0

are instead close to q w , the absolute e¤ect dominates, since the relative e¤ect is now only of the second order. This follows since q w = arg maxq0 (q 0 ; (n

1)q w + q 0 )

(q w ; (n

1)q w + q 0 ),

which can be interpreted as q w being the Nash equilibrium in a game of relative payo¤s (see, e.g., Scha¤er 1989). The evolutionary stable quantity q s precisely balances these forces. At !(q s ), the relative improvement of a small experimentation upwards is precisely outweighed by the absolute worsening, and vice-versa for experimentation downwards. Indeed, while q c is the Nash equilibrium if only absolute pro…t matters and q w is the Nash equilibrium if only relative pro…t matters, below it is shown that q s can be understood as the Nash equilibrium when the payo¤ function is a weighted average of absolute and relative pro…ts. The left panel of Figure 1 illustrates Lemma 2 by plotting F (q; q 0 ), holding the …rst argument …xed at di¤erent values.

F qw , q

F qc , q

F q, qc

F q, qs

F q, q w

q qC

q1

qs

F qs , q

q2

qC

qw

q qs

qw

Figure 1. The left and right panels plot F (q; q 0 ) …xing the …rst and second argument, respectively.

Lemma 2 speci…es which absorbing states are stable against single experimentations. A related question is which absorbing states can be destabilized by a …xed experimentation. For any q 0 , f!(q) : F (q; q 0 )

0g is the set of states that can be destabilized by q 0 . I refer to a

quantity q 0 such that F (q; q 0 )

0 for all q 6= q 0 as a global invader. Given Lemma 2, only q s 11

can be a global invader. It can be shown that F1 (q 0 ; q 0 ) =

F2 (q 0 ; q 0 ) for all q 0 , and Lemma

2 therefore implies F1 (q s ; q s ) = 0. The next result establishes that q s is a global invader by proving that F1 (q; q s ) < 0 for q < q s and F1 (q; q s ) > 0 for q > q s . Lemma 3 q s is the unique global invader. The intuition behind Lemma 3 is similar to that of Lemma 2. At any state !(q) such that q < q s , the relative improvement of an experimentation to q s outweighs any absolute worsening, and vice versa for q > q s . The right panel of Figure 1 illustrates Lemma 3 by plotting F (q; q 0 ), holding the second argument …xed at di¤erent values. In Figure 1 one can also observe that experimentations in the direction of q s are imitated with positive probability. This holds generally and is presented here as a corollary since it follows straightforwardly from the proofs of Lemmata 2 and 3. Corollary 1 If q < q 0 < q s or q s < q 0 < q, then F (q; q 0 ) > 0. If q 0 < q < q s or q s < q < q 0 , then F (q; q 0 ) < 0. The fact that !(q s ) is relatively di¢ cult to leave and easy to enter from other absorbing states immediately leads to the following result. Proposition 1 Suppose q s 2 . Then !(q s ) is the unique stochastically stable state. Three dynamic properties of !(q s ) are worth emphasizing. First, the fact that q s is a global invader implies that convergence to !(q s ) is relatively fast. In particular, Theorem 1 in Ellison (2000) establishes that the expected waiting time for convergence to the set of stochastically stable states is of the order of " x , where x is the largest number of experimentations required to reach the stochastically stable states from any absorbing state. Since q s is a global invader, one experimentation is always su¢ cient, and therefore, the expected waiting time is of the order of " 1 , which is a lower bound in this class of models. Second, Corollary 1 implies that the process can move in small steps toward !(q s ) and the stochastic stability of !(q s ) is therefore robust to local experimentation. For example, !(q s ) is uniquely stochastically stable, if experimentations in t are restricted to [qij (t

1)

; qij (t

1) + ], for some

> 0. Third,

the stochastic stability of !(q s ) does not depend on the presence or absence of inertia. 12

While imitation of non-rivals reduces competition, the e¤ect is weaker than the one identi…ed by Apesteguia et al. (2007), who …nd !(q c ) to be uniquely stochastically stable under IBA. The di¤erence is due to these authors’ assumption that …rms observe only non-rivals in the same role. This implies that if ij experiments to q s at !(q c ), then …rms outside of j do not simultaneously observe ij and the other …rms in j. Firms outside of j who observe the pro…t reduction of ij then do not observe the larger pro…t reduction experienced by other …rms in j. That is, the positive relative e¤ect of the experimentation to q s is invisible for …rms outside of j, who consequently do not imitate q s . A model with multiple markets is similar to a model with one market and …nite memory. Proposition 1 reveals that the analogy does not always extend to the predictions. Bergin and Bernhardt (2009) demonstrate that if …rms have su¢ ciently long memory and update behavior through IBA, the symmetric collusive quantity q m is uniquely stochastically stable. In their model, an experimentation at !(q m ) to some q 0 > q m is initially followed by all …rms. Pro…ts then decrease, and as time passes, the low pro…ts of q 0 gain weight in the average payo¤ computation. Since the high pro…ts at !(q m ) linger in memory, …rms ultimately go back to q m . The di¤erence in predictions essentially follows since in the memory model, quantities cannot change fast enough across all markets, thus revealing the pitfall of high quantities.

3.3

Comparative statics

The number of …rms per market and the number of markets determine the weights of the pro…ts of rivals and non-rivals in the calculation of average pro…ts and, therefore, a¤ect the value of qs. Remark 1 q s (k; n) strictly decreases in k and n and lim q s (k; n) = q c . k!1

With more markets, absolute e¤ects weigh more in the calculation of average pro…ts; therefore, q s is closer to q c . With more …rms per market, there are two contradictory forces. The pro…ts of the …rms in the own market weigh more in the calculation of average pro…ts, making experimentations toward q w more likely to be successful. However, the e¤ect on the pro…t of the …rms in the own market is smaller, making these experimentations less successful. The negative sign of @q s =@n indicates that the second e¤ect dominates. 13

Remark 1 implies that more …rms can lead to less competition. If an additional local market for a good appears, keeping the number of …rms per market …xed, competition in each market will decrease, despite the fact that the total number of …rms increases. This e¤ect does not appear in standard models and may be suitable for empirical tests.

3.4

A link to relative payo¤ maximization

The fact that q w performs well relative to other quantities implies that q w can be obtained as the symmetric Nash equilibrium quantity of a game in which …rms maximize relative pro…ts (see, e.g., Scha¤er 1989). For example, in a duopoly this happens if each …rm i’s payo¤ is given by (qi ; Q)

(qj ; Q), i.e., if …rms’preferences display spite.9 Here, it is demonstrated that q s

can be given a similar interpretation. In particular, q s can be obtained as the symmetric Nash equilibrium of a game in which …rms are concerned both about their pro…ts in absolute terms and about outperforming their competitors, i.e., …rms’preferences display moderated spite. Consider a market with n …rms in which each …rm i’s payo¤ function is

U (qi ; q i ) := (1

where q

i

) (qi ; Q) + ( (qi ; Q)

K(qi ; q i )),

(4)

= (q1 ; :::; qi 1 ; qi+1 ; :::; qn ), K(qi ; q i ) is a weighted average of the pro…ts of all …rms

except i and

2 [0; 1] captures …rms’preferences for relative versus absolute payo¤s.10 The

game is a straightforward generalization of a simple model of relative payo¤ maximization, where if

parameterizes …rms’degree of spite. If

= 0, we obtain a standard Cournot game and

= 1, we obtain a standard game of relative payo¤ maximization. Intermediate values of

generate moderated spite. By rewriting (4) as U (qi ; q i ) = (qi ; Q)

K(qi ; q i ), it becomes

clear that the preferences here simply allow …rms to attach a smaller weight to the pro…ts of rivals. Let q^ be a symmetric Nash equilibrium quantity if q^ 2 arg maxq0 2R+ U (q 0 ; (^ q ; :::; q^)). 9

Apesteguia, Huck, Oechssler, and Weidenholzer (2010) …nd such preferences to be behaviorally relevant. While it is perhaps most natural to think of K(qi ; q i ) as a simple average, it could, e.g., also be that K(qi ; q i ) = maxj2N nfig (qj ; Q), i.e., each …rm wants to maximize the distance to the runner-up. 10

14

Proposition 2 If …rms compete in a single market with preferences given by (4) there is a unique symmetric Nash equilibrium quantity q^, de…ned by P 0 (n^ q )(1

)^ q + P (n^ q)

C 0 (^ q ) = 0.

The result follows directly from arguments in the proof of Lemma 2 and exploits the fact that this proof does not depend on any parameter F (q; q 0 ) + (1

being any particular function of n and k but holds for

2 [0; 1]. By setting

it is then possible to write U (q 0 ; (q; :::; q)) =

=

) (q; nq) and the proof of Lemma 2 immediately leads to the result.

Notice that given n, the parameter ranging from q c (if

= 0) to q w (if

characterizes a continuum of equilibrium outcomes = 1). A similar characterization is possible in the

imitation model, but there the value of n determines which restricts the values that

as a function of the discrete variable k,

can take. For example, if n = 2, then

depending on the value of k. Notice also that in the imitation model A consequence is that while for any n and k there is a value of

2 f1=3; 1=5; 1=7; :::g,

takes no value in [1=2; 1].

such that the equilibria of both

models coincide, the converse is not true. Finally, notice that as the proportion of non-rivals in the population increases in the imitation model, stronger preferences for absolute pro…ts are required to generate an identical prediction. For example, if n = 2, k = 2 and k 0 = 3 the corresponding preferences for absolute pro…ts equal 1

= 2=3 and 1

0

= 4=5. Conversely,

for larger values of n weaker preferences for absolute pro…ts are required to generate an identical prediction.

3.5

An alternative imitation rule

The presence of multiple markets implies that …rms can simultaneously observe di¤erent profits for the same quantity. The notion of following the best observed performance is therefore sometimes ambiguous, and the literature accordingly considers more than one model of behavior. Here a natural alternative to IBA is discussed, which instead of comparing averages prescribes the quantity that generated the highest pro…t in the previous period. In particular, a …rm that follows Imitate the Best Max (henceforth, IBM) assigns positive probability to all quantities in arg maxqij (t

1):ij2N K

(qij (t

1); Qj (t

11

1)).11 IBM can be understood as an

Alós-Ferrer’s (2004), Apesteguia et al. (2007) and Matros (2012), among others, consider similar imitation rules.

15

optimistically biased imitation rule, since it only responds to the best observed performance of each quantity. This optimism turns out to disfavor experimentations at monomorphic states and IBM accordingly stabilizes a larger set of absorbing states than IBA. More precisely, an experimentation to q 0 at !(q) is imitated with positive probability under IBM if and only if the experimenting …rm performs better than all remaining …rms. This is equivalent to

(q 0 ; (n

1)q + q 0 )

maxf (q; (n

1)q + q 0 ); (q; nq)g

0. That is, the

experimentation must lead to an improvement in both relative and absolute terms. As argued above, such experimentations are available for q 62 [q c ; q w ] but not for q 2 [q c ; q w ]. This endows quantities in [q c ; q w ] with a stability advantage under IBM and leads to the following result. Remark 2 If …rms imitate according to IBM and

is su¢ ciently small, then the set of sto-

chastically stable states is f!(q) : q 2 [q c ; q w ] \ g. The proof of Remark 2, which here is omitted but is available in a working paper version, adds the observation that states in f!(q) : q 2 [q c ; q w ] \ g communicate by means of two experimentations and are equally stable. The result closely parallels Theorem 2 in Alós-Ferrer’s (2004) analysis of …rms with …nite memory that imitate according to IBM. Indeed, the logic underlying both results is the same, and the arguments in Alós-Ferrer’s (2004) proof can be used to prove Remark 2. Hence, the analogy between models with multiple markets and memory models seems to be more complete under IBM than under IBA. Remark 2 implies that imitation of non-rivals moderates competition under IBM as well. This contrasts with the conclusion of Apesteguia et al. (2007), who …nd !(q w ) uniquely stochastically stable under IBM. The di¤erence is again mainly due to these authors’assumption that …rms observe only non-rivals in the same role.

4

Sophisticated imitation across asymmetric markets

This section relaxes the assumption of ex-ante identical markets by allowing the inverse demand functions to di¤er to some degree across markets. An inevitable question here is to what extent …rms would take such asymmetries into account when they imitate. There are several possibilities, ranging from negligence to more sophisticated behavior. Some candidate 16

assumptions, however, do not yield a meaningful analysis. For example, if …rms ignore the asymmetries and directly apply IBA, they tend to simply follow the largest market, which in many cases stabilizes virtually all quantities. Similar behavior follows even if …rms only imitate su¢ ciently similar markets. The second largest market then follows the largest market, while the third largest market follows the second largest market, and so on. However, it is shown here that under a particular and sophisticated understanding of the asymmetries, the model closely parallels the benchmark model and yields clear results. Speci…cally, it is assumed that …rms are su¢ ciently sophisticated and possess enough information to calculate counterfactual pro…ts, i.e., the pro…t the observed …rm would have obtained if its market were ex-ante identical to the imitating …rm’s market. It is argued below that …rms can calculate counterfactual pro…ts if they understand both how inverse demands di¤er across markets and the structure of the pro…t function (without knowing its exact shape). The implication is that while …rms in di¤erent markets play di¤erent games, observations across markets provide information about the payo¤s of the game played in the own market and therefore, are not misleading per se.12 Under this assumption, the smallest quantity (across inverse demands) satisfying the equation in Lemma 2 is uniquely stochastically stable, provided that the inverse demands are not too di¤erent, that choices are made without inertia, and given a tie-breaking rule discussed in more detail below.

4.1

The extension

The extension described below is referred to as sophisticated imitation across asymmetric markets. Market structure. Firms in each market j 2 K now face inverse demand Pj ( ) and have pro…t function

j (q; Q).

All inverse demand functions and the cost function satisfy the

assumptions in Section 2.1. For each j 2 K, let qjw be the walrasian quantity given Pj ( ) and let qjs be de…ned by Pj0 (nqjs )(1 q1s

q2s

:::

)qjs + Pj (nqjs )

C 0 (qjs ) = 0. Without loss of generality, let

qks . Let qjm be de…ned by Pj (nqjm ) + nPj0 (nqjm )qjm

12

C 0 (qjm ) = 0. That is, qjm

Hedlund and Oyarzun (2014) argue that social comparison can motivate a similar approach in an environment of technology adoption.

17

is the symmetric collusive output in market j, which uniquely exists under the assumptions here. Notice also that if q < q 0

qjm or qjm

q 0 < q, then

j (q; nq)

<

j (q

0

; nq 0 ). Let

q m = maxj2K qjm . Markets are assumed to be su¢ ciently similar in the sense that q m < q1s . Information and behavior. When a …rm updates behavior, it observes all quantities produced in the previous period and all associated counterfactual pro…ts. I.e., in period t each …rm ij observes qi0 j 0 (t i0 j 0 2 N

1) and counterfactual pro…t

j (qi0 j 0 (t

1); Qj 0 (t

1)), for each

K. Notice in particular that ij observes the pro…t i0 j 0 would have obtained if market

j 0 were identical to market j. Firm ij then chooses a quantity according to IBA applied to the counterfactual pro…ts. While counterfactual pro…ts are obviously not immediately observable, they can be calculated by …rms who (i) understand how inverse demands di¤er across markets and (ii) understand that pro…t equals price by quantity minus cost. Suppose ij observes qi0 j 0 , tionally

jj 0 (Qj 0 )

= Pj (Qj 0 ) Pj 0 (Qj 0 ). The term

jj 0 (Qj 0 )

j 0 (qi0 j 0 ; Qj 0 )

and addi-

equals the di¤erence in price between

markets j and j 0 given the aggregate behavior in j 0 and can be understood as ij’s correct assessment of the ex-ante di¤erence between the own and the observed market. Notice that ij does not need to know the inverse demand functions for this assessment, only their di¤erences. This idea is more plausible if the inverse demands di¤er in a simple way, such as by only a constant (perhaps it is commonly known that j 0 is half the size of j). If the market price Pj 0 (Qj 0 ) is also observable ij can calculate the counterfactual price Pj (Qj 0 ) =

jj 0 (Qj 0 )

+ Pj 0 (Qj 0 ) (i.e.,

the price that would have resulted were j 0 identical to j). Since ij observes

j 0 (qi0 j 0 ; Qj 0 )

both Pj 0 (Qj ) and qi0 j 0 it follows that ij can calculate C(qi0 j 0 ) =

Pj 0 (Qj )qi0 j 0 and

subsequently the counterfactual pro…t

j (qi0 j 0 ; Qj 0 )

j 0 (qi0 j 0 ; Qj 0 )

= Pj (Qj 0 )qi0 j 0

and

C(qi0 j 0 ). This calculation

provides ij with information that is useful in essentially the same way as in the identical markets setting.13 I will make two additional assumptions on …rms’ behavior. First, I assume that there is no inertia, so all …rms update behavior in each time period.14 Second, whenever some …rm 13

Notice that since …rms are not assumed to know the inverse demand or cost function, more sophisticated behavior, such as best responding behavior, is still unfeasible here. 14 Assumptions of no inertia are quite common in the literature, see, e.g., Eshel et al. (1998), Alós-Ferrer (2004) and Bergin and Bernhardt, (2009).

18

observes two or more quantities with the same average counterfactual pro…t, the …rm imitates the quantity used by the largest number of …rms. If this also produces a tie, the …rm randomizes over the tied quantities. Popularity biased imitation was introduced by Ellison and Fudenberg (1993; see also Mengel 2009) and here appears in the weaker form of a tie-breaker. In essence, no inertia and popularity biased tie-breaking imply that …rms behave in a su¢ ciently coordinated way, which will have a stabilizing e¤ect on absorbing states.

4.2

Stochastically stable states

A complication arises since …rms observe counterfactual pro…ts, which leads …rms in di¤erent markets to assess certain situations di¤erently and some polymorphic states (i.e., states at which di¤erent quantities are produced) to become absorbing. For example, if there are two markets and each market j produces qjm , then this state is absorbing since …rms in market 1 observe counterfactual pro…ts

m m 1 (q1 ; nq1 )

m m 1 (q2 ; nq2 ),

>

and vice-versa for market 2. How-

ever, polymorphic states are sensitive to experimentation and not stochastically stable. The reason is that such states must involve quantities close to the symmetric collusive quantities, which implies that there are experimentations available in the direction of qjs for all markets j. Since all stochastically stable states must be monomorphic, identifying the set of stochastically stable states again requires analyzing how experimentation leads the process from one monomorphic state to another.15 For each j 2 K de…ne F j : R2+ ! R by F j (q; q 0 ) =

j (q

0

; (n

1)q + q 0 )

n 1 nk 1

j (q; (n

1)q + q 0 ) +

n(k nk

1) 1

j (q; nq)

.

At any state !(q) …rms in market j imitate an experimentation to q 0 if and only if F j (q; q 0 ) > 0, where the strict inequality is a consequence of the popularity biased tie-breaking rule. The analysis in Section 3 implies that for any j 2 K we have F j (qjs ; q 0 ) < 0 for any q 0 6= qjs and F j (q; q 0 ) > 0 for any q < q 0

qjs and qjs

q0

q. This leads to a certain tension between

the states !(q1s ); :::; !(qks ). The fact that q1s is closer to the symmetric collusive quantity for 15

While, as in the benchmark model, it can be shown that any absorbing set is an absorbing state, the argument is somewhat lengthy, and is omitted for reasons of space. The proof of the main result of this section, however, explicitly shows that no non-singleton absorbing set is stochastically stable.

19

all markets, however, lends !(q1s ) an advantage, which leads the tension to be resolved in its favor. To illustrate the idea, suppose for the moment that q1s < q2s and consider !(q2s ). An experimentation to q 0 = q1s is then imitated by all …rms in market 1 and by no other …rm. Following the experimentation each market j observes counterfactual pro…ts s s j (q2 ; nq2 )

s s j (q1 ; nq1 )

and

and since q m < q1s < q2s all …rms imitate q1s . Hence, a single experimentation su¢ ces

for a transition from !(q2s ) to !(q1s ). A similar argument holds for !(qjs ) with j = 3; 4; ::; k. Consider now !(q1s ) and an experimentation to q 0 = q2s . All …rms except those in market 1 then imitate q 0 . This again leads to counterfactual pro…ts

s s j (q1 ; nq1 )

and

s s j (q2 ; nq2 ),

so all …rms

again imitate q1s and the process returns to !(q1s ). The argument, however, requires no inertia, since otherwise any subset of …rms in Knf1g may follow the experimentation, which might induce …rms in market 1 to imitate q2s . For a similar reason, experimentations to arbitrary q 0 > q1s require popularity biased tie-breaking to ensure that …rms in a market j such that F j (q1s ; q 0 ) = 0 do not randomize between q1s and q 0 . Thus, no inertia and popularity biased tie-breaking coordinate …rms’ behavior. This has a stabilizing e¤ect on all absorbing states and is su¢ cient to make !(q1s ) uniquely stable against single experimentations. Since !(q1s ) is also comparatively easy to enter, this state is uniquely stochastically stable. Proposition 3 Suppose there is no inertia and that ties are broken in favor of the most popular quantities. If

is su¢ ciently small, then !(q1s ) is the unique stochastically stable state of the

model of sophisticated imitation across asymmetric markets. In comparison to the case of identical markets the outcome is less competitive when sophisticated …rms imitate across asymmetric markets. In the long run, …rms tend to produce the smallest quantity (across inverse demands) that would be stable under identical markets. Intuitively, this follows since markets behave fairly independently, which reveals the bene…ts of more collusive quantities. As before, …rms follow experimentations toward quantities that perform well against a weighted average of the pro…ts of the status quo quantity. Di¤erent markets, however, have di¤erent opinions with respect to which experimentations perform well in this sense. The market willing to follow experimentations to the most collusive quantities reveals the bene…t of such quantities to other markets. If, for example, inverse demands di¤er by a constant, the smallest market leads the other markets toward more collusive behavior. 20

5

Concluding remarks

The analysis reveals that the moderation of competition that occurs when …rms imitate both rivals and non-rivals is substantially more general than indicated by existing literature and also extends to a setting in which sophisticated …rms imitate across asymmetric markets. A new …nding is that the number of markets (i.e., the extent to which …rms imitate non-rivals) has a clear impact on the long-run outcome. In particular, the outcome becomes less competitive as the number of markets increases and …rms observe a larger proportion of non-rivals. This prediction may be suitable for empirical testing, including experimental tests of the behavioral hypothesis of imitation across markets under IBA. While the policy implications of these results should not be overemphasized, some points associated with the publication of information about …rm performance are worth mentioning. Vega-Redondo’s (1997) result suggests that making information about …rm performance publicly available might increase competition. Huck et al. (2000) argue along these lines and provide experimental data con…rming that information about the performance of rivals in a single market intensi…es competition (see also Huck et al. 1999). The results of the present paper imply that matters are less de…nite and should be quali…ed. First, if there are segmented markets and …rms imitate across these markets, the publication of market information may have a more limited e¤ect on competition. Second, publishing information is less likely to be e¤ective if the number of segmented markets is large. Third, the publication of market information is more likely to be e¤ective if such information can be made locally available in each market, thus avoiding the negative e¤ect on competition of imitation across markets. As in most of the literature on imitation in oligopolies, the focus in this paper is on quantity competition. An interesting direction for future research would be to extend the analysis to price competition. While the imitation of non-rivals here leads to a tension between relative and absolute performance and between the Nash equilibrium and perfectly competitive outcomes, di¤erent forces are likely to be present in price competition. Ania (2008) shows that in price competition evolutionary stable quantities are frequently Nash equilibria, and Alós-Ferrer, Ania and Schenk-Hoppé (2000) show that imitation dynamics re…ne the set of Nash equilibria. Future research might analyze whether these results are robust if …rms imitate across markets 21

or whether such an analysis would provide new results.

6

Appendix (proofs)

Proof. (Lemma 2) Let hq (q 0 ) be the function obtained by …xing q in F (q; q 0 ). Of course, hq (q) = 0 for all q. Step 1. There exists a unique quantity q s such that h0qs (q s ) = 0, and q s 2 (q c ; q w ). Proof. Note that h0q (q 0 ) = P 0 ((n g(q) := h0q (q) = P 0 (nq)(1

1)q + q 0 )(q 0

q) + P ((n

1)q + q 0 )

)q + P (nq) C 0 (q). Then g 0 (q) = nP 00 (nq)(1

)q + P 0 (nq)(1 + n

) C 00 (q 0 ) < 0. Finally, g(q c ) = P 0 ((nq c )q c +P (nq c ) C 0 (q c ) P 0 ((nq c ) q c = and g(q w ) = P 0 ((nq w )(1

)q w + P (nq w )

C 0 (q w ) = P 0 ((nq w )(1

C 0 (q 0 ). Let

P 0 ((nq c ) q c > 0

)q w < 0. By continuity

there is a unique quantity q s such that g(q s ) = h0qs (q s ) = 0, and q s 2 (q c ; q w ). Step 2. h0qs (q 0 ) > 0 for all q 0 < q s and h0qs (q 0 ) < 0 for all q 0 > q s . Proof. First, for q 0 > q s we have h00qs (q 0 ) = P 00 ((n q0)

1)q s + q 0 )(q 0

q s ) + 2P 0 ((n

C 00 (q s ) < 0. Next, h0qs (q 0 ) > 0 for q 0 < q s , since q s < q w and therefore P ((n

1)q s +

1)q s + q 0 )

C 0 (q 0 ) > 0. Step 1 and 2 together with hqs (q s ) = 0 imply hqs (q 0 ) < 0 for all q 0 6= q s . Step 1 and hq (q) = 0 for all q imply that for any q 6= q s , there is some q 0 such that hq (q 0 ) > 0. Proof. (Lemma 3) Let fq0 (q) be the function obtained by …xing q 0 in F (q; q 0 ). Step 1 : fq0 s (q s ) = 0. Proof. First, fq0 0 (q) = (n 1)P 0 ((n 1)q +q 0 )(q 0 (1

)P (nq) + C 0 (q). Hence, fq0 s (q s ) =

P ((n 1)q +q 0 ) (1

q)

P 0 ((nq s )q s (1

)

)nP 0 (nq)q

P (nq s ) + C 0 (q s ) =

g(q s ) = 0.

Step 2. fq0 s (q) > 0, for all q > q s and fq0 s (q) < 0 for all q < q s . Proof. Suppose q 1)P 0 ((n

1)q + q s )

(1

Then F12 (q; q s ) = (n

q s = . Then fq00s (q) = (n )n2 P 00 (nq)q 1)P 00 ((n

1)q + q s )(q s

1)2 P 00 ((n

q)

2 (n

)nP 0 (nq) + C 00 (q) > 0. Suppose q 2 (q s ; q s = ).

2(1

1)q + q s )(q s

q) + (n

1

)P 0 ((n

1)q + q s ) < 0.

Combining Step 1 in the proof of Lemma 2 with Step 1 here, F1 (q; q) > 0 for q > qs . Hence, fq0 s (q) = F1 (q; q s ) > F1 (q; q) > 0. Suppose q < q s . Then F12 (q; q s ) < 0 and F1 (q; q) < 0. Hence, fq0 s (q) = F1 (q; q s ) < F1 (q; q) < 0. 22

Since F (q s ; q s ) = 0, we thus have fqs (q) > 0 for all q. Lemma 2 implies uniqueness. Proof. (Corollary 1) As above, let hq (q 0 ) = F (q; q 0 ). Suppose q < q s . The proof of Lemma 2 implies h0q (q) > 0, h00q (q 0 ) < 0 for any q 0

q and

by Lemma 3 hq (q s ) > 0. Therefore, if q < q 0 < q s then F (q; q 0 ) = hq (q 0 ) > 0. Since additionally h0q (q 0 ) > 0 for q 0 < q, if q 0 < q < q s then F (q; q 0 ) = hq (q 0 ) < 0. Suppose q s < q. The proof of Lemma 2 implies h0q (q) < 0 and h00q (q 0 ) < 0 for any q 0 > q and therefore if q s < q < q 0 then F (q; q 0 ) = hq (q 0 ) < 0. Consider q s < q 0 < q. The proof of Lemma 3 implies F1 (q; q) > 0 and F12 (q; q 0 ) < 0 for q < q 0 = and hence F1 (q; q 0 ) > F1 (q; q) > 0 if q s < q 0 < q < q 0 = . Since F11 (q; q 0 ) > 0 if q

q 0 = we obtain F (q; q 0 ) > 0 if q s < q 0 < q.

Proof. (Proposition 1) By Lemma 2 !(q s ) cannot be left by a single experimentation and by Lemma 3 !(q s ) can be entered from any absorbing state by a single experimentation. In Ellison’s (2000) terms, !(q s ) therefore has radius greater than 1 and coradius 1. By Theorem 1 in Ellison (2000), !(q s ) is therefore uniquely stochastically stable. Proof. (Remark 1) Di¤erentiating P 0 (nq s )(1 @q s = @k @q s = @n

)q s + P (nq s )

C 0 (q s ) = 0 implicitly

P 0 (nq s )q s n(n 1) <0 g 0 (q s ) (nk 1)2 P 00 (nq s )(1 )(q s )2 + P 0 (nq s )q s (1 g 0 (q s )

@ ) @n

where g(q) is de…ned in the proof of Lemma 2 and g 0 (q s ) < 0. Note that and hence,

@q s @n

< 0. Finally, limk!1 P 0 (nq)(1

)q + P (nq)

@ @n

=

k 1 (nk 1)2

C 0 (q) = P 0 (nq)q + P (nq)

2 (0; 1) C 0 (q)

and hence limk!1 q s (k; n) = q c . Proof. (Proposition 2) Set

=

and write U (q 0 ; (q; :::; q)) = F (q; q 0 ) + (1

a symmetric Nash equilibrium q^ 2 arg maxq0 F (^ q ; q 0 ) + (1 Notice that the proof of Lemma 2 does not depend on

) (q; nq). At

) (^ q ; n^ q ) = arg maxq0 F (^ q ; q 0 ). being any particular function of n

and k and therefore implies that the unique solution of the maximization problem is de…ned by P 0 (n^ q )(1

)^ q + P (n^ q)

C 0 (^ q ) = 0.

Proof. (Proposition 3) For any absorbing set A 23

kn

, let S(A) be the set of quantities

produced in A. Since at any state all …rms belonging to the same market imitate the same quantity with positive probability, any absorbing set contains a state at which all …rms in the kn

same market produce the same quantity. For any absorbing sets A; A0

I write A ! A0

if a single experimentation leads the process from A to A0 with positive probability. Suppose j \ [q m ; q1s ]j > k

1, which holds for

su¢ ciently small. kn

Step 1. For any absorbing set A1

there is a …nite sequence (Ar )sr=1 such that each

Ar is an absorbing set and A1 ! A2 ! ::: ! As = !(q) for some q 2 . kn

Proof. Suppose that in period t the system is in some state x 2 A1

such that all

…rms belonging to the same market produce the same quantity and such that A1 is absorbing. q m , then

Notice …rst that if q^ := min S(A1 )

q ; n^ q) j (^

>

j (q; nq)

for all q 2 S(A1 ) and

j 2 K. In t + 1 all …rms therefore imitate q^ and hence A1 = !(^ q ). Suppose instead q^ < q m . Let k^ > 0 be the number of markets producing q^ in t. In t + 1, let some …rm ij such that qij (t) = q^ experiment to some q 0 2 (q m ; q1s ]nS(A1 ), which is possible since by assumption j \ [q m ; q1s ]j > k j (q

0

; (n

1)^ q + q0)

j (q

0

; (n

1)^ q + q0)

"

1. Then for any j 2 K

n 1 nk^ 1

q ; (n j (^

n(k^ 1)^ q + q0) + nk^

n 1 nk 1

q ; (n j (^

1)^ q + q0) +

where the …rst inequality follows since follows since q^ < q 0

1)^ q + q0) <

q ; (n j (^

n(k nk

1) 1 1) 1

q ; n^ q) j (^

#

q ; n^ q) j (^ q ; n^ q) j (^ and k^

= F j (^ q; q0) > 0

k and the second

qjs for all j 2 K. Since for all markets the average counterfactual pro…t

of q^ is strictly smaller than that of q 0 we have q^ 62 fqij (t + 2)gij2N leads the system to an absorbing set A2 such that S(A2 )

K.

Eventually imitation

(S(A1 ) [ fq 0 g)nf^ q g and therefore

min S(A2 ) > q^. By repeating the argument an absorbing set As such that min S(As ) is eventually reached. By the previous paragraph As = !(q) for some q

qm

q m . Since each

transition required only one experimentation A1 ! A2 ! ::: ! As = !(q). Step 2. For any absorbing set A1

kn

there is a …nite sequence (Ar )sr=1 such that each

Ar is an absorbing set and A1 ! A2 ! ::: ! As = !(q1s ). Proof. It is su¢ cient to show !(q) ! !(q1s ) for any q 6= q1s . First, suppose q < q1s . Then F j (q; q1s ) > 0 for all j 2 K since q < q1s

qjs for all j 2 K and hence !(q) ! !(q1s ). Suppose 24

that in t the system is in state !(q) for some q > q1s and that some …rm experiments to q1s in t + 1. In t + 2 all …rms in markets in K 0 = fj 2 K : F j (q; q1s ) > 0g = 6 ? imitate q1s and all remaining …rms choose q. If K 0 = K we are …nished. Otherwise, since for any j 2 K we have qjm < q1s < q and therefore

s s j (q1 ; nq1 )

>

j (q; nq),

all …rms imitate q1s in t + 3 and hence

!(q) ! !(q1s ). Step 3. It is not possible to leave !(q1s ) by a single experimentation. Proof. First, for any j 2 K we have q1s

qjs and therefore F j (q1s ; q 0 ) < 0 for any q 0 < q.

Hence, no experimentation q 0 < q1s can be imitated. Suppose that in t the system is in state !(q1s ) and that some …rm experiments to q 0 > q1s in t + 1. Let K 0 = fj 2 K : F j (q1s ; q 0 ) > 0g and notice that K 0 6= K. In t + 1 all …rms in markets in K 0 imitate q 0 and no …rm in KnK 0 imitates q 0 . Since for any j 2 K we have qjm < q1s < q 0 and therefore

s s j (q1 ; nq1 )

>

j (q

0

; nq 0 )

all …rms imitate q1s in t + 2 and we are back to !(q1s ). In Ellison’s (2000) terms, Step 2 and 3 imply, respectively, that !(q1s ) has modi…ed coradius 1 and radius greater than 1. By Theorem 2 in Ellison (2000), !(q1s ) is therefore uniquely stochastically stable. Acknowledgements: I thank an associate editor and two referees for constructive feedback and Carlos Alós-Ferrer, José Apesteguia, Antonio Morales, Joerg Oechssler, Carlos Oyarzun, Adam Sanjurjo, Luis Ubeda and Amparo Urbano and seminar participants at the University of Alicante, Stockholm School of Economics, ADRES 2010 and ASSET 2010 for helpful comments and suggestions. Financial support from the Spanish Ministry of Innovation and Science, with reference BES-2008-008040, is gratefully acknowledged.

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