On Tacit versus Explicit Collusion

Transcription

1 On Tacit versus Explicit Collusion Yu Awaya y and Viay Krishna z Penn State University November 3, 04 Abstract Antitrust law makes a sharp distinction between tacit and explicit collusion whereas the theory of repeated games the standard framework for studying collusion does not. In this paper, we study this di erence in Stigler s (964) model of secret price cutting. This is a repeated game with private monitoring since in the model, rms observe neither the prices nor the sales of their rivals. For a xed discount factor, we identify conditions under which there are equilibria under explicit collusion that result in near-perfect collusion pro ts are close to those of a monopolist whereas all equilibria under tacit collusion are bounded away from this outcome. Thus, in our model, explicit collusion leads to higher prices and pro ts than tacit collusion. JEL classi cation: C73, D43 Keywords: Collusion, communication, repeated games, private monitoring Introduction In ruling on an antitrust case in 993, the US Supreme Court clearly stated that tacit collusion the setting of supracompetitive prices without evidence of conspiracy was not in itself unlawful. When there is evidence of explicit collusion, however, the law provides for severe nes, even prison terms. Antitrust law thus makes a sharp distinction between tacit and explicit collusion. In the former, there is no communication between rms, whereas in the latter there is. The theory of repeated games the standard framework for studying collusion does not, however, provide a usti cation for this distinction since in most models communication does not increase We thank Olivier Compte, Joyee Deb and Ed Green for helpful conversations and seminar participants at NYU, Warwick and Indiana for comments. y YuAwayagmail.com z vkrishnapsu.edu See Brooke Group v. Brown & Williamson, 509 US 09, US Supreme Court, June, 993.

2 cartel pro ts. But as Marshall and Marx (0) write: "... repeated [tacit] interaction is not enough in practice, at least not for many rms in many industries. Even for duopolies... explicit collusion was required to substantially elevate prices and pro ts (p. 3)." Harrington (005) points to the shortcomings of economic theory in this regard: "There is a gap between antitrust practice which distinguishes explicit and tacit collusion and economic theory which (generally) does not (p. 6)." In this paper, we study this issue in Stigler s (964) model of secret price cutting, which is a repeated game with private monitoring. Firms cannot observe each other s prices nor can they observe each other s sales. Each rm only observes its own sales and, because of demand shocks, these are imperfect signals of the other rms actions. These signals are noisy in the sense that given the prices, the marginal distribution of a rm s sales is dispersed. At the same time rms sales are correlated. We study situations where this correlation is rather sensitive to prices. Precisely, it is high when the di erence in rms prices is small say, when both rms charge close to monopoly prices and decreases when the di erence is large say, when there is a unilateral price cut. This kind of monitoring structure can arise quite naturally, for example, in a Hotelling-type model with random transport costs (see Section ). For analytic convenience, we suppose that the relationship between sales and prices is governed by log normal distributions. Under tacit collusion, rms base their pricing decisions only on their own history of prices and sales. Because sales are subect to unobserved shocks, it is di cult for rms to detect a rival s price cuts. Under explicit collusion, rms can communicate with each other in every period and pricing decisions are now based on these communications as well as their private histories. The communication is "cheap talk" the rms exchange non-veri able sales reports in every period. Our main result is Theorem For any high but xed discount factor, when the monitoring is noisy but sensitive enough, there is an equilibrium under explicit collusion whose pro ts are strictly greater than those from any equilibrium under tacit collusion. Explicit collusion leads to higher sustainable prices and pro ts because even unveri able communication improves monitoring. In their study of the sugar re ning cartel, based on internal documents, Genesove and Mullin (00), point to the monitoring role of the weekly (!) meetings of the rms. Clark and Houde (04) nd similar evidence in the retail gasoline market in Canada. The exchange of sales gures for monitoring purposes seems to have been key to the functioning of cartels in numerous industries, including citric acid, lysine and graphite electrodes (see Harrington, 005). The argument underlying our main result is divided into two steps. The rst task is to nd an e ective bound for the maximum equilibrium pro ts that can be achieved A formal statement of the result is in Section 5.

3 under tacit collusion. But the model we study is that of a repeated game with private monitoring and there is no known characterization of the set of equilibrium payo s. This is because with private monitoring each rm knows only its own history (of prices and sales) and has to infer its rival s history. Since rms histories are not commonly known, these cannot be used as state variables in a recursive formulation of the equilibrium payo set. Thus we are forced to proceed somewhat di erently. In Proposition we develop a bound on equilibrium pro ts by using a very simple necessary condition a deviating strategy in which a rm permanently cuts its price to an unchanging level should not be pro table. This deviation is, of course, rather naive the deviating rm does not take into account what the other rm knows or does. We show, however, that even this minimal requirement can provide an e ective bound when the relationship between prices and sales is rather noisy relative to the discount factor. For a xed discount factor, as sales become increasingly noisy, the bound becomes tighter. The second task is to show that the bound developed earlier can be exceeded under explicit collusion. This is done by directly constructing an equilibrium in which rms exchange sales reports in every period (see Proposition ). Firms charge monopoly prices and report truthfully. In this case, rms sales are highly correlated and so the likelihood that their reports will agree is also high. If a rm were to cut its price, sales become less correlated and it cannot accurately predict its rival s sales. Even if the deviating rm strategically tailors its report, the likelihood of an agreement is low. Thus a strategy in which di ering sales reports lead to non-cooperation is an e ective deterrent. When the correlation between rms sales is high, the chances of triggering a punishment without a deviation are small and so this equilibrium can achieve high pro ts even for relatively low discount factors. It turns out that noisier sales only make the inference problem for the deviating rm harder and thus decrease the incentive to cut price. The key to our results is that the bound developed in Proposition depends only on the marginal distribution of sales precisely, on how noisy these are and not on the correlation between sales. The equilibrium constructed in Proposition, however, depends on the correlation structure and, as mentioned above, is actually reinforced by noise. Thus we are able to identify conditions under which the bound on tacit collusion is tight while the equilibrium under explicit collusion approximates the monopoly outcome. We emphasize that the analysis in this paper is of a di erent nature than that underlying the so-called "folk theorems" (see Sugaya, 03). These show that for a xed monitoring structure, as players become increasingly patient, near-perfect collusion can be achieved in equilibrium. In this paper, we keep the discount factor xed and change the monitoring structure to drive a wedge between tacit and explicit collusion. Our goal is only to identify some natural circumstances in which this happens we do not attempt a full identi cation of monitoring structures which distinguish between the two. 3

4 In our model, rms share sales information and this allows them to better monitor each other. There are, of course, other pieces of information that may be communicated to facilitate the cartel. Firms may coordinate on market shares after exchanging privately known cost information (as in Athey and Bagwell, 00 and Escobar and Toikka, 03). Another view is that communication allows the cartel to coordinate on one among many repeated game equilibria (see Green, Marshall and Marx, 03). But there is no formal "meta-theory" of how players coordinate on a single equilibrium. Our explanation of the gains from communication does not rely on equilibrium selection. We exhibit an equilibrium under explicit collusion that dominates all equilibria under tacit collusion. Related literature There is a vast literature on repeated games under di erent monitoring assumptions. Under perfect monitoring, given any xed discount factor, the set of perfect equilibrium payo s with and without communication is the same. Under public monitoring, again given any xed discount factor, the set of (public) perfect equilibrium payo s with and without communication is also the same. Thus, in these settings there is no di erence between tacit and explicit collusion. The reason, of course, is that all relevant information is commonly known and so there is nothing useful to communicate. Compte (998) and Kandori and Matsushima (998) study repeated games with private monitoring when there is communication among the players. In this setting, they show that the folk theorem holds any individually rational and feasible outcome can be approximated as the discount factor tends to one. This line of research has been pursued by others as well, in varying environments (see Fudenberg and Levine (007) and Obara (009) among others). Aoyagi s (00) work is, in particular, closely related because he also considers a secret price cutting model with a similar monitoring structure and communication. 3 He shows that monopoly outcomes can be approximated as the discount factor tends to one. Harrington and Skrzypacz (0) also study explicit collusion but allow for transfers. All of these papers thus show that communication is su cient for cooperation. But as Kandori and Matsushima (998) recognize, One thing which we did not show is the necessity of communication for a folk theorem (p. 648, their italics)." In a remarkable paper, Sugaya (03) shows the surprising result that in very general environments, the folk theorem holds without any communication. Thus, in fact, communication is not necessary for a folk theorem. The analysis of repeated games with private monitoring is known to be di cult and more so if communication is absent. Although Sugaya s result was preceded by folk theorems for some limiting cases where the monitoring was almost perfect (or almost public), the fact that such a result holds even when monitoring is of very low quality is quite unexpected. An important component of Sugaya s proof is that players implicitly communicate via 3 Zheng (008) explores a similar monitoring structure in the context of general symmetric games. 4

5 their actions. 4 Thus, he shows that with enough time, there is no need for explicit communication. In a di erent vein, Awaya (04a) studies the prisoners dilemma with private monitoring and shows that for a xed discount rate, there exist environments in which without communication, the only equilibrium is the one-shot equilibrium whereas with communication, almost perfect cooperation can be sustained. This paper is a precursor to the current one. Key to our result is a method of bounding the set of payo s under tacit collusion. In a recent paper, Pai, Roth and Ullman (04) also provide a bound on the equilibrium payo s that is e ective when monitoring quality is low. The measure of monitoring quality used by Pai et al. is based on how the oint distribution of the private signals is a ected by players actions. But the bound obtained by them applies to the payo s from explicit as well as from tacit collusion, and so does not help in distinguishing between the two. In contrast, our measure, and hence the bound in Proposition, is based solely on the marginal distributions and not on any correlation between players signals (sales). On the other hand, the equilibrium construction in Proposition relies primarily on the properties of the oint distribution. The fact that correlation can vary while keeping the marginal distributions xed is key to our main result. A method developed by Cherry and Smith (0) is also unable to distinguish between tacit and explicit collusion. The monitoring structure we study was introduced by Aoyagi (00) and then also explored in Zheng (008) and Awaya (04a). These papers all assume that the correlation between signals depends on actions in a particular way it is high when players take similar actions and low when they do not. We also follow Aoyagi (00) in postulating the way that rms communicate. But our Proposition, which constructs an equilibrium with communication, is very di erent in nature from Aoyagi s result. In his paper, the monitoring structure is xed and an equilibrium is constructed for discount factors tending to one. In our result, the discounting is held xed and an equilibrium is constructed for noisy but correlated monitoring structures. As noted above, because of Sugaya s (03) result, the rst exercise is unable to show that communication is necessary for collusion which is, of course, the goal of this paper. In a recent paper, Rahman (04) shows how communication can overcome the "no collusion" result of Sannikov and Skrzypacz (007) in a repeated Cournot oligopoly with public monitoring. But since the latter result concerns only equilibria in public strategies, this again does not establish the necessity of communication for collusion, only its su ciency. Moreover, the communication considered by Rahman (04) is either mediated or veri able. In our work, as well as in the papers mentioned above, communication is unmediated and unveri able. Our paper provides a theoretical basis for distinguishing between tacit and explicit collusion. There is strong empirical evidence in support of this distinction that comes 4 This idea was used by Hörner and Olzewski (006) to prove a folk theorem with almost perfect monitoring. 5

6 from the study of cartels in di erent industries. Genesove and Mullin (00) examine this question by looking at a cartel in the sugar re ning industry and nd strong support that higher prices and pro ts emerge when rms communicate. Clark and Houde (04) nd the same to be true in the retail gasoline market in Canada. The same conclusion has been reached in laboratory experiments as well, by Fonseca and Normann (0) and Cooper and Kühn (04) among others. The remainder of the paper is organized as follows. The next section outlines the nature of the market. Section 3 analyzes the repeated game under tacit collusion whereas Section 4 does the same when collusion is explicit. The ndings of the earlier sections are combined in Section 5 to derive the main result. In Section 6 we calculate explicitly the gains from communication in an example with linear demands. Omitted proofs are collected in an Appendix. The market There are two rms in the market, labelled and : The rms produce di erentiated products at a constant cost, which we normalize to zero. Each rm sets a price p i P i = [0; p max ], for its product and given the pair of prices p and p, the sales Y and Y are stochastic. Prices a ect sales via two channels. First, they a ect expected sales in the usual way an increase in p decreases rm s expected sales and increases rm s expected sales. Second, they a ect how correlated are the sales of the two rms in a manner speci ed below. As in Aoyagi (00), sales are more correlated when the di erence in rms prices is small. Expected demand. The expected demand of rm i is determined as follows: E [Y i p ; p ] = Q i (p i ; p ) () where Q i is a continuous function that is decreasing in p i and increasing in p : We will suppose that the rms are symmetric so that Q i = Q. Note that the rst argument of Q i is always the rm s own price and the second is its competitor s price. The expected pro t of rm i is then i (p i ; p ) = p i Q i (p i ; p ) and we suppose that i is strictly concave in p i. Let G denote the one-shot game where the rms choose prices p i and p and the pro ts are given by i (p i ; p ) : Under the assumptions made above, there exists a symmetric Nash equilibrium (p N ; p N ) of the resulting one-shot game and let N be the resulting pro ts of a rm. 5 5 If the one-shot game has multiple symmetric Nash equilibria, let (p N ; p N ) denote the one with the lowest equilibrium pro ts. 6

7 ln Y 6 r r - ln Y Figure : Monitoring Structure Suppose that (p M ; p M ) is the unique solution to the monopolist s problem: max p i ;p P i (p i ; p ) i and let M be the resulting pro ts per rm. We assume that monopoly pricing (p M ; p M ) is not a Nash equilibrium. For technical reasons we will also assume that a rm s expected sales are bounded away from zero. Distribution of sales. We will suppose that given prices (p ; p ), the two rms sales (Y ; Y ) are ointly distributed according to a bivariate log normal density f (y ; y p ; p ); equivalently, the log sales (ln Y ; ln Y ) are ointly normally distributed. Prices a ect the means i of the normal distribution as well as the correlation coe cient ; but, for simplicity, do not a ect the (identical) variance. 6 Speci cally, i = ln Q i (p i ; p ) so that () holds. 7 The correlation between rms (log) sales is high when they charge similar prices and low when their prices are dissimilar. 6 A heteroskedastic speci cation in which the variance increased with the mean log sales can be easily accommodated. 7 Recall that if ln Y is normally distributed with mean and variance ; then E [Y ] = exp + : 7

8 For our purposes, a bene t of the (log) normal speci cation is that the variance and correlation parameters can vary independently. Figure is a schematic illustration of such a monitoring structure. When both rms charge the same price, their (log) sales have the same mean and a high correlation, depicted by the narrower contours of the resulting normal density. When prices are di erent, say rm charges a lower price, then the (log) sales have di erent means > and low correlation, now depicted by the wider contours. This is formalized as: Assumption There exists 0 (0; ) and a symmetric function (p ; p ) [0; ] such that = 0 (p ; p ) and satis es the following conditions: () for all p; (p; p) = ; and () for all p p ; =p > 0 and =p > 0 and so, is an increasing and convex function of p : 8 Note that =p = 0 =p ; and so for xed ; an increase in 0 represents an increase in the sensitivity of the correlation to prices. This kind of correlation structure can result, for example, in a symmetric Hotellingtype market in which consumers have identical but random "transport costs". When rms charge similar prices, their sales are similar no matter what the realized transport costs are. In other words, when rms charge similar prices, their sales are highly correlated. When rms charge dissimilar prices, their sales are again similar if the realized transport costs are high because consumers are not so price sensitive. But their sales are quite dissimilar if the realized costs are low because now consumers are very price sensitive. In other words, when rms charge dissimilar prices, the correlation between their sales is low. Of course, the same kind of reasoning applies if we substitute search costs for transport costs. 3 Tacit collusion Let G (f) denote the in nitely repeated game with private monitoring in which rms use the discount factor < to evaluate pro t streams. Time is discrete. In each period, rms choose prices p i and p and given these prices, their sales are realized according to f as described above. As in Stigler (964), each rm i observes only its own realized sales y i ; it observes neither s price p nor s sales y : We will refer to f as the monitoring structure. Let h t i = p i ; yi ; p i ; yi ; :::; p t i ; y t i denote the private history observed by rm i after t periods of play and let H t i denote the set of all private histories of rm i: In period t; rm i chooses its prices p t i knowing h t i and nothing else. A strategy s i for rm i is a collection of functions (s i ; s i ; :::) such that s t i : H t i! (P i ) : Of course, since Hi 0 is null, s i (P i ) : We will denote by s t i p i h t i the 8 Some examples satisfying the assumption are (p ; p ) = min (p ; p ) = max (p ; p ) and (p ; p ) = = + p p : 8

9 probability that rm i sets a price p i following the private history h t i : Thus, we are allowing for the possibility that rms may randomize. A strategy pro le s is simply a pair of strategies (s ; s ). A sequential equilibrium of G (f) is strategy pro le s such that for each i and every private history h t i such that the continuation strategy of i following h t i ; denoted by s i h t ; is a best response to E[s i h t h t i ]: Since f has full support, the set of sequential equilibrium outcomes (price paths) is the same as the set of Nash equilibrium outcomes (see Mailath and Samuelson, 006, p. 396). We remind the reader that as yet there is no communication between the rms. 3. Equilibrium under tacit collusion The purpose of this subsection is to provide an upper bound to the oint pro ts of the rms in any equilibrium under tacit collusion. The task is complicated by the fact that there is no known characterization of the set of equilibrium payo s of a repeated game with private monitoring. Because the players in such a game observe di erent histories each rm knows only its own past prices and sales such games lack a straightforward recursive structure and the kinds of techniques available to analyze (public perfect) equilibria of repeated games with public monitoring (see Abreu, Pearce and Stacchetti, 990) cannot be used here. Instead, we proceed as follows. Suppose we want to determine whether there is an equilibrium of G (f) such that the sum of rms discounted average pro ts are within " of those of a monopolist, that is, M : If there were such an equilibrium, then both rms must set prices close to the monopoly price p M often (or equivalently, with high probability). Now consider a secret price cut by rm to p, the static best response to p M : Such a deviation is pro table today because rm s price is close to p M with high probability. How this a ects rm s future actions depends on the quality of monitoring, that is, how much rm s price cut a ects the distribution of s sales. If the quality of monitoring is poor, rm can keep on deviating to p without too much fear of being punished. In other words, a rm has a pro table deviation, contradicting that there were such an equilibrium. This reasoning shows that the bound on equilibrium pro ts depends on three factors of the market: () the trade-o between the incentives to deviate and e ciency in the one-shot game 9 ; () the quality of the monitoring, which determines whether the short-term incentives to deviate can be overcome by future actions; and, of course (3) the discount factor. We consider each of these factors in turn. Incentives versus e ciency in the one-shot game. De ne, as above, p = arg max pi i (p i ; p M ) ; the static best-response to p M : Let (P P ) be a oint 9 By "e ciency" we mean how e cient the cartel is in achieving high pro ts and not "social e ciency." 9

10 distribution over rms prices. We want to nd an such that (i) the sum of the expected pro ts from is within " of M ; and (ii) it minimizes the (sum of) the incentives to deviate to p: To that end, for " 0; de ne subect to (") min P [ i (p; ) i ()] () i P i () M " i where denotes the marginal distribution of over P : The function measures the trade-o between the incentives to deviate (to the price p) and rms pro ts. Precisely, if the rms pro ts are within " of those of a monopolist, then the total incentive to deviate is (") : It is easy to see that is (weakly) decreasing. Lemma A. establishes that it is convex and satis es lim "!0 (") > 0: Since (p N ; p N ) is feasible for the program de ning when " = M N ; it follows that ( M N ) 0: We emphasize that is completely determined by the one-shot game G. De ne by (x) = sup f" : (") = xg (3) Quality of monitoring. Consider two price pairs p = (p ; p ) and p 0 = (p 0 ; p 0 ) and the resulting distributions of rm i s sales: f i ( p) and f i ( p 0 ) : If these two distributions are close together, then it will be di cult for rm i to detect the change from p to p 0 : Thus, the quality of monitoring can be measured by the "distance" between the two distributions. In what follows, we use the so-called total variation metric to measure this distance. De nition The quality of a monitoring structure f is de ned as where f i is the marginal of f on Y i and kg between g and h. 0 = max p;p 0 kf i ( p) f i ( p 0 )k T V hk T V denotes the total variation distance It is important to note that the quality of monitoring depends only on the marginal distributions f i ( p) over i s sales and not on the oint distributions of sales f ( p) : In particular, the fact that the marginal distributions f i ( p) and f i ( p 0 ) are close is small does not imply that the underlying oint distributions f ( p) and f ( p 0 ) are close. When f ( p) is a bivariate log normal, can be explicitly determined as = max (4) R 0 The total variation distance between two densities g and h on X is de ned as kg hk T V = g (x) h (x) dx: X 0

11 6 s ( M ; M ) s ( N ; N ) 0 - Figure : Bound on Pro ts from Tacit Collusion where is the cumulative distribution function of a univariate standard normal and max = max p;p 0 ln Qi (p i ; p ) ln Q i p 0 i; p 0 is the maximum possible di erence in log expected sales. As increases, decreases and goes to zero as becomes arbitrarily large. 3.. A bound on tacit collusion The main result of this section develops a bound on equilibrium pro ts under tacit collusion. An important feature of the bound is that it is independent of any correlation between rms sales and depends only on the marginal distribution of sales. Proposition In any equilibrium of G (f) ; the repeated game without communication, the pro ts + M 4 where = max p i (p; p ) and is the quality of monitoring. Before embarking on a formal proof of Proposition it is useful to outline the main ideas (see Figure for an illustration). A necessary condition for a strategy pro le s to be an equilibrium is that a deviation by rm to a strategy s in which it always charges p not be pro table. This is done in two steps. First, we consider a ctitious situation in which rm assumes that rm will not respond to its deviation. The higher the equilibrium pro ts, the more pro table would be the proposed deviation in

12 the ctitious situation this is exactly the e ect the function captures in the oneshot game and Lemma A. shows that captures the same e ect in the repeated game as well. Second, when the monitoring is poor is small rm s actions cannot be very responsive to the deviation and so the ctitious situation is a good approximation for the true situation. Lemma A.6 measures precisely how good this approximation is and quite naturally this depends on the quality of monitoring and the discount factor. It is usual to derive necessary conditions for an equilibrium by considering "oneshot" deviations in which a player cheats in one period and then resumes equilibrium play. But in games with private monitoring, such deviations a ect the deviating players beliefs about the other player s signals thereafter and so a ect his subsequent (optimal) play. The deviation we consider a permanent price cut is rather naive but has the feature that future play, while suboptimal, is straightforward. Notice that pro ts both from the candidate equilibrium and from the play after the deviation are evaluated in ex ante terms. Observe that if we x the quality of monitoring and let the discount factor approach one, then the bound becomes trivial (since lim! 4 = 0) and so is consistent with the folk theorem. On the other hand, if we x the discount factor and decrease the quality of monitoring ; the bound converges to M (0) < M and is e ective. One may reasonably conecture that if there were "zero monitoring" in the limit, that is, if! 0, then no collusion would be possible. But in fact N < M (0) so that even with zero monitoring, Proposition does not rule out the presence of collusive equilibria. This is consistent with the nding of Awaya (04b). Proof of Proposition. We argue by contradiction. Suppose that G (f) has an equilibrium, say s, whose average pro ts 3 (s)+ (s) exceed M 4 : If we write " = M (s) (s) ; then this is equivalent to < (") : 4 Given the strategy pro le s; de ne t = E s s t h t (P ) where the expectation is de ned by the probability distribution over t oint histories h t i ; h t determined by s: Note that depends on the strategy pro le s and not ust s. Let = ; ; ::: denote the strategy of rm in which it plays t in period t following any t period history. The strategy replicates the ex ante distribution of prices p resulting from s but is non-responsive to histories. Pai et al. (04) consider such a deviation. Awaya (04b) constructs an example in which there is zero monitoring the distribution of a player s signals is the same for all action pro les but, nevertheless, there are non-trivial equilibria. 3 We use i (s) to denote the discounted average payo s from the strategy pro le s as well as the payo s in the one-shot game.

13 Let s i denote the strategy of rm i in which it plays p with probability one following any history. From Lemma A. From Lemma A.6 we have for i = ; and X ( i (s i ; s ) i (s)) i P [ i (s i ; ) i (s)] (") i i (s i ; s ) i (s i ; ) X i (s i ; s ) i 4 + (") i (s i ; ) + X i ( i (s i ; ) i (s)) which is strictly positive. But this means that at least one rm has a pro table deviation, contradicting the assumption that s is an equilibrium. 4 Explicit collusion We now turn to a situation in which rms can explicitly collude. By this we mean that prior to choosing prices in any period, rms can communicate with each other, sending one of a nite set of messages to each other. The sequence of actions in any period is as follows: rms set prices, receive their private sales information and then simultaneously send messages to each other. Messages are costless the communication is "cheap talk" and are transmitted without any noise. The communication is unmediated. Formally, there is a nite set of messages M i for each rm. A t period private history of rm i now consists of the complete list of its own prices and sales as well as the list of all messages sent and received. Thus a private history is now of the form h t i = p i ; y i ; m i ; m and the set of all such histories is denoted by H t i : A strategy for rm i is now a pair (s i ; r i ) where s i = (s i ; s i ; :::), the pricing strategy, and r i = (ri ; ri ; :::) ; the reporting strategy, are collections of functions: s t i : H t i! (P i ) and ri t : H t i P i Y i! (M i ) : t (f) : Sequen- Call the resulting in nitely repeated game with communication G com tial equilibrium is de ned as before. = 3

14 4. Equilibrium strategies We will now identify some properties of the monitoring structure f that will allow the rms to achieve near-perfect collusion, that is, the sum of their pro ts will be close to those of a monopolist. The log-normal monitoring structure has two parameters the variance of log-sales and the correlation between log-sales, 0, when the rms charge identical prices. Of course, the discount factor is key parameter as well. Monopoly pricing will be sustained using a grim trigger pricing strategy together with a threshold sales-reporting strategy in a manner rst identi ed by Aoyagi (00). Since the price set by a competitor is not observable, the trigger will be based on the communication between rms, which is observable. The communication itself consists only of reporting whether one s sales were "high" above a commonly known threshold or "low". Firms start by setting monopoly prices and continue to do so as long as the two sales reports agree both rms report "high" or both report "low". Di ering sales reports trigger permanent non-cooperation as a punishment. Speci cally, consider the following strategy (s i ; ri ) in the repeated game with communication where there are only two possible messages H ("high") and L ("low"). The pricing strategy s i is: In period ; set the monopoly price p M. In any period t > ; if in all previous periods, the reports of both rms were identical (both reported H or both reported L), set the monopoly price p M ; otherwise, set the Nash price p N. The communication strategy r i is: In any period t ; if the price set was p i = p M, then report H if log sales ln y t i M ; otherwise, report L: In any period t ; if the price set was p i 6= p M, then report H if ln yi t i + M ; otherwise, report L: ( M = ln Q i (p M ; p M ) ; i = ln Q i (p i ; p M ) and = ln Q (p M ; p i ) :) Denote by (s ; r ) the resulting strategy pro le. We will establish that if rms are patient enough and the monitoring structure is noisy ( is high) but correlated ( 0 is high), then the strategies speci ed above constitute an equilibrium. But before doing this, it is useful to calculate the lifetime average pro ts if rms follow the proposed strategies. 4

15 4.. Optimality of communication strategy Suppose rm follows the strategy (s ; r ) and until this period, both have made identical sales reports. Recall that a punishment will be triggered only if the reports disagree. Thus, rm will want to maximize the probability that its report agrees with that of rm. Since rm is following a threshold strategy, it is optimal for rm to do so as well. If rm adopts a threshold of such that it reports H when its log sales exceed ; and L when they are less than ; the probability that the reports will agree is Pr [ln Y < ; ln Y < M ] + Pr [ln Y > ; ln Y > M ] (5) The optimal reporting threshold is (see Lemma A.7) (p ) = + ( M ) (6) If rm deviated and cut its price to p < p M ; then clearly the expected (log) sales of the two rms are such that > M > : Thus, (p ) > > M ; which says, as expected, that once rm cuts its price and so experiences stochastically higher sales it should optimally under-report relatively to the equilibrium reporting strategy r. On the other hand, if rm did not deviate and set a price p M, then the (6) implies that it is optimal for it to use a threshold of M = (p M ) as well. We have thus established that if rm plays according to (s ; r ) ; then following any price p that rm sets, the communication strategy r is optimal. The optimality of the proposed pricing strategy s depends crucially on the probability of triggering the punishment and we now establish how this is a ected by the extent of a price cut. Thus, if rm sets a price of p ; the probability that its report will be the same as that of rm (and so the punishment will not be triggered) is given by (p ) Z (p ) Z M (z ; z ; ) dz dz + Z (p ) Z M (z ; z ; ) dz dz (7) where (z ; z ; 0 ) is a standard bivariate normal density with correlation coe cient 0 (0; ) : 4 Note that while (p ),, and the correlation coe cient depend on p ; is independent of p : Observe also that for any p p M ; (p ) M where denotes the cumulative distribution function of the standard univariate normal. This is because rm could always adopt a communication strategy in which after a deviation to p < p M ; it always reports L independently of its own sales, e ectively setting (p ) = : This guarantees that rm s report will be 4 The standard (with both means 0 and both variances ) bivariate normal density is (z ; z ; ) = exp ( ) z + z z z p 5

16 the same as rm s report with a probability equal to M : Since M ; M : This means that the probability of detecting a deviation is less than one-half. 4.. Equilibrium pro ts If both set prices p M and follow the proposed reporting strategy, the probability that their reports will agree is ust (p M ) ; obtained by setting (p ) = = = M and = 0 in (7). Sheppard s formula for the cumulative of a bivariate normal (see Tihansky, 97) implies that (p M ) = arccos ( 0) which is increasing in 0 and converges to as 0 goes to : The lifetime average pro t resulting from the proposed strategies is given by ( ) M + [ (p M ) + ( (p M )) N ] = (8) and it is easy to see that for any xed ; lim 0! = M 4..3 Equilibrium with communication Proposition There exists a such that for all > ; once and 0 are large enough, then (s ; r ) constitutes an equilibrium of G com (f) ; the repeated game with communication. Proof. Suppose that in all previous periods, both rms have followed the proposed strategies and their reports have agreed. If rm deviates to p < p M in the current period, it gains (p ) = ( ) (p ; p M ) + [ (p ) + ( (p )) N ] (9) where is de ned in (8). Thus, 0 (p ) = ( ) p (p ; p M ) + 0 (p ) [ N ] We will show that when is large enough, for all p ; 0 (p ) > 0: Since (p M ) = 0; this will establish that a deviation to a price p < p M is not pro table. Now observe that from Lemma A.8, lim! 0 (p ) = ( ) p (p ; p M ) + p 0 (p ;p M ) 0 p (p ; p M ) [ N ] ( ) p (p M ; p M ) + p 0 (0;p M ) 0 p (0; p M ) [ N ] 6

17 where the last inequality follows from the fact that since is concave in p ; p (p ; p M ) > p (p M ; p M ) and the fact that (p ; p M ) is increasing and convex in p : Let be the solution to ( ) p (p M ; p M ) + p (0;p M ) p (0; p M ) [ N ] = 0 (0) which is ust the right-hand side of the inequality above when 0 =. Such a exists since p (p M ; p M ) is nite and, by assumption, p (0; p M ) is strictly positive. Notice that for any > ; the expression on the left-hand side is strictly positive. Now observe that p 0 (;p M ) 0 p (0; p M ) [ N ] is increasing and continuous in 0 (recall that is increasing in 0 ). Thus, given any > ; there exists a 0 () such that for all 0 = 0 () ( ) (p M ; p M ) + p p 0 (0;p M ) 0 p (0; p M ) [ N ] = 0 Note that 0 () is a decreasing function of and for any 0 > 0 () ; the left-hand side is strictly positive. A deviation by rm to a price p > p M is clearly unpro table. This completes the proof. Aoyagi (00) was the rst to introduce threshold reporting strategies. He shows that for a given monitoring structure ( 0 and xed) as the discount factor goes to one, these strategies constitute an equilibrium. The idea as in all the "folk theorems" is that even when the probability of a deviation being detected is low, if players are patient enough, future punishments are a su cient deterrent even if they are distant. In contrast, Proposition shows that for a given discount factor ( high but xed), as 0 goes to one and goes to in nity, there is an equilibrium with high pro ts. Its logic, however, is di erent from that underlying the "folk theorems". Here the punishment power derives not from the patience of the players; rather it comes from the noisiness of the monitoring. A deviating rm will then nd it very di cult to predict its rival s sales and hence, even it "lies" optimally, a deviation is very likely to trigger a punishment. 5 Gains from communication Proposition shows that the pro ts from any equilibrium under tacit collusion cannot exceed 7

18 M 6 0 t Explicit Tacit bound M (0) () - Figure 3: Gains from Communication M 4 whereas Proposition provides conditions under which there is an equilibrium under explicit collusion with pro ts (as de ned in (8)). The two results together lead to the formal version of the result stated in the introduction. Let be determined as in (0). Theorem For any > ; there exist ( () ; 0 ()) such that for all (; 0 ) ( () ; 0 ()) there is an equilibrium under explicit collusion with total pro ts such that > M 4 As 0!,! M and as!,! 0. Thus, in the limit the di erence in pro ts between explicit and tacit collusion is at least (0) > 0: The workings of the main result can be seen in Figure 3, which is drawn for the case of linear demand (see the next section for details). First, notice that the pro ts from explicit collusion depend on 0 and not on (but has to be su ciently high to guarantee that the suggested strategies form an equilibrium). The bound on pro ts from tacit collusion, on the other hand, depends on and not on 0 : For small values of ; the bound is ine ective and says only that these do not exceed oint monopoly pro ts. As increases, the bound becomes tighter but at intermediate levels the pro ts from the equilibrium under explicit collusion do not exceed the bound. Once > () ; explicit collusion results in strictly higher pro ts than tacit collusion. In the limit, the bound is M (0). 8

19 6 Linear demand In this section, we illustrate the workings of our results when (expected) demand is linear. Suppose that 5 Q i (p i ; p ) = max (A bp i + p ; ) where A > 0 and b > : For this speci cation, the monopoly price p M = A= (b ) and monopoly pro ts M = A =4 (b ) : There is a unique Nash equilibrium of the one-shot game with prices p N = A= (b ) and pro ts N = A b= (b ) : A rm s best response if the other rm charges the monopoly price p M is p = A (b ) =4b (b ) : The highest possible pro t that rm can achieve when charging a price of p is = (p; p M ) = A (b ) =6b (b ) : It remains to specify how the correlation between the rms log sales is a ected by prices. In this example we adopt the following speci cation: = 0 + p p which, of course, satis es Assumption. Then, recalling (), it may be veri ed that for " [0; M =b ] (") = " + A A p (b ) (b ) p " 8b(b ) which is achieved at equal prices. Note that (0) = M =b (b ) and (0) = M =b : Finally, from (4) = max where max = ln Q (0; p M ) ln Q (p M ; 0) : A numerical example Suppose A = 0 and b = : Let = 0:7 and 0 = 0:95: For these parameters, M = 3600; = 4050 and max = 5:9: Also, the pro ts from the equilibrium under explicit collusion, = 354 (approximately). Figure 3 depicts the bound on pro ts from tacit collusion as a function of using Proposition. For low values of (approximately = 60 or lower), the bound is ine ective it equals M and as! ; converges to M (0) = As shown, the pro ts under explicit collusion exceed the bound when > () = 00 (approximately). Thus, in this example the gains from communication are approximately 450 in the limit. Note that this gain is approximately 56% of the potential gain M N. Figure 4 veri es that the strategies (s ; r ) constitute an equilibrium a deviation to any p < p M is unpro table as (p ) < 0 (as de ned in (9)). This is veri ed for 5 This speci cation of "linear" demand is used because ln 0 is not de ned. 9

20 0 p p M (p ) Figure 4: Unpro table Deviations = 60 and, of course, the same strategies remain an equilibrium for higher values of : 7 Conclusion We have provided theoretical support for the idea that communication facilitates greater collusion. Where do the gains from communication come from? When the monitoring is poor, under tacit collusion, a price cut cannot be detected with any con dence and this is the basis of the bound developed in Proposition. Under explicit collusion, however, the probability that a price cut will trigger a punishment is signi cant relative to the short-term gains. Thus, communication reduces the type II error associated with imperfect monitoring and this is the driving force behind the main result. We conecture that the fact that communication facilitates greater collusion holds quite generally beyond the circumstances we have identi ed in this paper that the monitoring quality be low and this in turn requires that sales be rather volatile. We view our result as only a rst step towards distinguishing between the two forms of collusion and recognize its limitations. First, we did not identify the best equilibrium under explicit collusion; we only constructed an equilibrium. This equilibrium was based on very simple grim trigger strategies and these, because of their unforgiving nature, are known to perform badly. Moreover, the communication strategies are not very good at detecting deviations the probability that a price cut will be trigger a punishment is less than one-half. Second, the upper bound on pro ts under tacit collusion provided here bites only when the monitoring quality is rather poor. The development of better payo bounds for repeated games with private monitoring remains a challenge. 0

21 A Appendix A. Tacit collusion The rst lemma derives some simple properties of the function. This function delineates the trade-o between e ciency and incentives in the one-shot game and is central to the bound for equilibrium payo s of the repeated game developed below. Lemma A. is non-increasing, convex and satis es lim "!0 (") = (0) > 0: Proof. The fact that is non-increasing follows trivially from its de nition. To see that is convex, note that Z i () = i (p i ; p ) d (p i ; p ) Suppose 0 is the solution to the program above for " = " 0 and similarly, suppose 00 is the solution for " = " 00 : Then since the constraint is a linear function of ; for any [0; ], 0 + ( ) 00 is feasible for " = " 0 + ( ) " 00 : The convexity of now follows since the obective function is also linear in : The fact that (") converges to (0) as "! 0 follows from the Berge Maximum Theorem. To see that (0) > 0; note that for " = 0 the only feasible solution to the optimization problem de ning is (p M ; p M ) : Since this is not a Nash equilibrium, (0) > 0: A.. Non-responsive strategies The induced ex ante distribution over P in period t induced by a strategy pro le s is t (s) = E s s t h t (Pi ) () Given a strategy pro le s; recall that denotes the strategy of rm i in which it plays t (s) in period t following any t period history. The strategy replicates the ex ante distribution of prices resulting from s but is non-responsive to histories. The following lemma shows that the function, which determines the incentives versus e ciency trade-o in the one-shot game, embodies the same trade-o in a repeated setting if the non-deviating player follows a non-responsive strategy. It shows that to minimize the average incentive to deviate while achieving average pro ts within " of M one should split the incentive evenly across periods. The lemma resembles an intertemporal "consumption smoothing" argument (recall that is convex). Lemma A. (Smoothing) For any strategy pro le s whose pro ts are greater than M "; P [ i (s i ; (s)) i (s)] (") i

22 where s i denote the strategy of rm i in which it plays p with probability one following any history. Proof. De ne " (t) = M E s P i i s t h t as the di erence between the sum of e cient pro ts M and the sum of expected pro ts in period t: Now clearly ( ) P t= t " (t) ": Then, P [ i (s i ; (s)) i (s)] i X = E s "( ) t P i p; p # t i p t i; p t i ( ) X t= t t= (" (t)) where the rst equality follows from the fact that the induced distribution over prices p t is the same under (s i ; ) as it is under s: The second inequality follows from the de nition of. Now note that Lemma A. guarantees that a solution to the problem subect to min ( ) f"(t)g X t= t (" (t)) P ( ) t " (t) " t= is to set " (t) = " for all t. Thus, we have that ( ) X t (" (t)) (") t= A.. Weak monitoring For a xed strategy pair (s ; s ), let t be the induced probability distribution over rm s private histories h t H t = (P Y ) t R t. Similarly, let t be the probability distribution over s private histories that results from the strategy pair (s i ; s ) : 6 We wish to determine the total variation distance between t and t. 6 Recall that s i denotes the strategy of rm i in which it sets p with probability one following any history.

23 The total variation distance between two distributions G and G over R n is equal to G GT = sup V E ['] E ['] () k'k where E and E denote the expectations with respect to the distribution G and G; respectively and the supremum is taken over all measurable functions ' with sup norm k'k. Note that the de nition in () is equivalent to the one in De nition. See, for instance, Levin, Peres and Wilmer (009). As a rst step, we decompose the total variation distance between two probability distributions into the distance between their marginals and that between their conditionals. Lemma A.3 Given two distributions G and G over R m R n ; G G T V G X G T X V + sup G Y X G T Y X V x where G X is the marginal distribution of G on R m and G Y X ( x) is the conditional distribution of G on R n given X = x (and similarly for G). Proof. In what follows we denote by E all expectations with respect to G and by E; all expectations with respect to G: Given any function ' : R m R n! [ ; ] ; we have E ['] E ['] = E X EY X ['] E X EY X ['] EX EY X ['] E X EY X ['] + EX EY X ['] E X EY X ['] sup EX ['] E X ['] + E EY X X ['] E Y X ['] k'k GX G T X V + E GY X X G T Y X V GX G T X V + sup GY X G T Y X V x and so G GT V = sup E ['] k'k E ['] G X G X T V + sup x G Y X G Y X T V Next we show that given a history h t ; the total variation distance between the two conditional distributions cannot exceed the monitoring quality. Lemma A.4 For any h t ; t h t t h t T V 3

24 Proof. Let S h t denote the distribution of prices that s strategy s induces after s private history h t : Similarly, let S i h t i denote the distribution of prices that i s strategy induces after i s private history h t i : Let S b i h t = E S i h t i h t denote s expectation about i s distribution of prices, given = Si b p i h t S p h t denote the (recall that s own history h t : Finally, let S b p h t oint distribution of prices that expects given s private history h t rms choices are independent). Now, for any ' : P Y! [ ; ] P Z Z 'd t h t 'd t h t P Y P Y Z Z Z = 'df (y p) ds p h t dsi b p i h t P i P Y Z Z 'df (y p; p ) ds p h t P Y Z Z Z Z = 'df (y p) d b S p h t P Y P Z Z = ' [f (y p) f (y p; p )] dy d b S p h t P Y Z " Z # 'f (y p) 'f (y p; p ) dy d b S p h t P Y Z ds b p h t = P P i P 'df (y p; p ) ds b p h t Y where the second equality follows from the fact that Z Z Z Z Z 'df (y p; p ) ds p h t = 'df (y p; p ) ds p h t dsi b p i h t Y Y and the second inequality follows from the de nition of total variation. Thus, t h t t Z Z h t T = sup 'd t h t 'd t h t V k'k P Y P Y Combining the preceding two results we obtain 4