On Communication and Collusion

Transcription

1 On Communcaton and Colluson Yu Awaya and Vjay Krshna Unversty of Rochester and Penn State Unversty August 14, 2015 Abstract We study the role of communcaton wthn a cartel. Our analyss s carred out n Stgler s (1964) model of repeated olgopoly wth secret prce cuts. Frms observe nether the prces nor the sales of ther rvals. For a fxed dscount factor, we dentfy condtons under whch there are equlbra wth "cheap talk" that result n near-perfect colluson, whereas all equlbra wthout such communcaton are bounded away from ths outcome. In our model, communcaton mproves montorng and leads to hgher prces and profts. JEL classfcaton: C73, D43 Keywords: Colluson, communcaton, repeated games, prvate montorng 1 Introducton Anttrust authortes vew any nter-frm communcaton wth concern but despte ths, even well-establshed cartels meet regularly. In ther landmark study of a sugarrefnng cartel, Genesove and Mulln (2001) wrte that cartel members met weekly for almost a decade! Harrngton (2006) too reports on the frequent meetngs of cartels n many ndustres: Vtamn A and E (weekly or quarterly), ctrc acd (monthly) and lysne (monthly). Why do cartels fnd t necessary to meet so often? Court documents and nsder accounts reveal that many ssues are dscussed at such gatherngs demand condtons, costs and yes, sometmes even prces. But an mportant reason frms meet s to montor each other s complance wth the collusve agreement. Ths s usually done by exchangng sales data, allowng frms to check that ther market We thank Olver Compte, Joyee Deb, Ed Green and Satoru Takahash for helpful conversatons and varous semnar audences for useful comments. Joseph Harrngton and Robert Marshall provded detaled comments on an earler draft and enhanced our understandng of anttrust ssues. E-mal: YuAwaya@gmal.com E-mal: vkrshna@psu.edu 1

2 shares are n lne wth the agreement. Members of the sostatc-graphte cartel entered ther sales fgures on a pocket calculator that was passed around the table. Only the total was dsplayed, allowng each frm to check ts own market share whle keepng ndvdual sales confdental. Often data are exchanged not n face-to-face meetngs but ndrectly through thrd partes, lke trade asssocatons, consultng frms or statstcal bureaus. The copper plumbng tubes cartel reported ndvdual sales fgures to the World Bureau of Metal Statstcs whch then dssemnated these n aggregate form. Many other cartels for nstance, the amno acds and znc phosphate cartels used trade assocatons for ths purpose. 1 It s clear that n these cases cartel members cannot ascertan each other s sales drectly; otherwse, there would be no need for frms to report these fgures. Selfreported sales numbers, however, are dff cult to verfy and frms have the ncentve to under-report these. Recognzng ths, the lysne cartel even resorted to audtng ts members. Ths was only moderately successful a leadng frm, Ajnomoto, was stll able to hde ts actual sales. Gven that the data provded by frms cannot be verfed what s the pont of havng them report these? Can such "cheap-talk" communcaton really help frms montor each other? In ths paper, we study how unverfable communcaton about past sales can ndeed facltate colluson. We adopt, and adapt, Stgler s (1964) classc olgopoly model of secret prce cuts n a repeated settng frms cannot observe each other s prces nor can they observe each other s sales. Each frm observes only ts own sales and because of demand shocks, both common and dosyncratc, these are nosy sgnals of other frms actons. Absent any communcaton, such mperfect montorng lmts the potental for collusve behavor because and ths was Stgler s pont devatons from a collusve agreement cannot be detected wth confdence. We show, however, that f frms are able to exchange sales nformaton, then collusve outcomes can be sustaned. We study stuatons where the correlaton n frms sales s senstve to prces. Precsely, t s hgh when the dfference n frms prces s small say, when both frms charge close to monopoly prces and decreases when the dfference s large say, because of a unlateral prce cut. 2 For analytc convenence, we suppose that the relatonshp between sales and prces s governed by log-normal dstrbutons, ndependently over tme. We show that n the statstcal envronment outlned above, reports of past sales allow frms to better montor each other. The nformaton communcated s not only unverfable but also payoff rrelevant t has no drect effects on current or future profts. Our man result s 3 THEOREM. For any hgh but fxed dscount factor, when the montorng s nosy 1 Detals of these and other cartel cases can be found n Harrngton (2006) and Marshall and Marx (2012). 2 Ths can arse qute naturally, for example, n a Hotellng-type model wth random transport costs (see Secton 2). 3 A formal statement of the result s n Secton 5. 2

3 but senstve enough, there s an equlbrum wth communcaton whose profts are strctly greater than those from any equlbrum wthout communcaton. The argument underlyng our man result s dvded nto two steps. The frst task s to fnd an effectve bound for the maxmum equlbrum profts that can be acheved n the absence of any communcaton. In Proposton 1 we develop a bound on equlbrum profts by usng a very smple necessary condton a devatng strategy n whch a frm permanently cuts ts prce to an unchangng level should not be proftable. Ths devaton s, of course, rather nave the devatng frm does not take nto account what the other frm knows or does. We show, however, that even ths mnmal requrement can provde an effectve bound when the relatonshp between prces and sales s rather nosy relatve to the dscount factor. For a fxed dscount factor, as sales become ncreasngly nosy, the bound becomes tghter. The second task s to show that the bound developed earler can be exceeded wth communcaton. We drectly construct an equlbrum n whch frms exchange (coarse) sales reports n every perod (see Proposton 2). Frms charge monopoly prces and report ther sales truthfully. As long as the reported sales of the frms whether hgh or low are smlar, monopoly prces are mantaned. If the reported sales are sgnfcantly dfferent say, one frm reports hgh sales whle the other reports low a prce war s trggered. These strateges form an equlbrum because when frms charge monopoly prces, ther sales are hghly correlated and so the lkelhood that ther truthful reports wll agree s also hgh. If a frm cuts ts prce, sales become less correlated and so t cannot accurately predct ts rval s sales. Even f the devatng frm strategcally talors ts report, the lkelhood of an agreement s low. Thus a strategy n whch dfferng sales reports lead to non-cooperaton s an effectve deterrent. In ths way, cheap-talk communcaton allows frms to montor each other more effectvely. We emphasze that the analyss n ths paper s of a dfferent nature than that underlyng the so-called "folk theorems" (see Malath and Samuelson, 2006). These show that for a fxed montorng structure, as players become ncreasngly patent, near-perfect colluson can be acheved n equlbrum. In ths paper, we keep the dscount factor fxed and change the montorng structure n a way that communcaton mproves profts Real-world Cartels Some key features of our model echo aspects of real-world cartel behavor. Frst, montorng s mperfect and dff cult. Second, the purpose of communcaton s to overcome montorng dff cultes and t s unverfable. Thrd, communcaton allows frms to base ther future behavor on relatve sales market shares rather than the absolute level of sales. The dff cultes frms face n montorng each other have been well-documented. Secret prce cuts are hard to track. In the famous Trenton Potteres case, frst-qualty 3

4 products were nvoced as beng of second qualty and so sold at a dscount. In the sugar refnng case, a frm ncluded some shppng expenses n the quoted prce. In each case, other members of the cartel dscovered ths "prce cut" only wth dff culty. Clark and Houde (2014) study the nternal functonng of a gasolne dstrbuton cartel va recordngs of telephone conversatons. They use a "natural experment" a publc announcement of an nvestgaton that led to a cessaton of all communcaton to deduce that hgher prces and profts result when frms communcate. Clark and Houde (2014) fnd that the communcaton helped the members to both coordnate prcng and to montor each other. Harrngton (2006) and Marshall and Marx (2012) study numerous cartels and how these functoned. Agan, regular communcaton for montorng purposes seems to be a common feature of cartels n many ndustres. Often the verty of what s communcated must be nferred. Indeed, Genesove and Mulln (2001, p. 389) wrte that the meetngs of the sugar-refnng cartel often served "as a court n whch an accused frm mght prove ts nnocence, n some cases on factual, n others on logcal, grounds." [Emphass added.] Harrngton (2006) wrtes that the montorng of sales s even more mportant than the montorng of prces. As mentoned above, frms are often nformed only about total ndustry sales va a calculator or a trade assocaton because these are suff cent for frms to calculate ther own market shares. In our model too, a prce war s trggered by relatve sales (market shares) and not by the absolute level of sales. If all frms experence low sales and so shares are not too far off then ths s attrbuted to adverse market condtons and the frms contnue to collude. Implctly, a frm that experences low sales cares about the reason that ts sales are low and not just the fact that they are low. Genesove and Mulln (2001) recount an epsode n whch the presdent of a Western frm threatened a prce war unless t could be convnced that ts low sales were not caused by the actons of Eastern refners. Notce that ths sort of behavor would not be possble wthout communcaton. A frm that only knew ts own sales would be unable to nfer the reason why they were low Other Models of Colluson The applcaton of the theory of repeated games to understand colluson goes back at least to Fredman (1971). Hs model assumes that all past actons are commonly observed wthout any nose. Under such perfect montorng, gven any fxed dscount factor, the set of subgame perfect equlbrum payoffs wth and wthout communcaton s the same there s nothng useful to communcate. Colluson wth mperfect montorng was studed by Green and Porter (1984) n a model of repeated quantty competton wth nosy demand. Montorng s mperfect because frms do not observe each others output but rather only the market prce. Snce all frms observe the same nosy sgnal (the common prce), ths stuaton s referred to as one of publc montorng. Agan gven any fxed dscount factor, the set of profts from (pure strategy) publc perfect equlbra n whch frms strateges depend only on the hs- 4

5 tory of past prces wth and wthout communcaton s also the same. Green and Porter (1984) show that wth suff cently low dscountng, publc perfect equlbra are enough to sustan colluson. 4 Our model s one of prvate montorng dfferent frms observe dfferent nosy sgnals (ther own sales). As we show, n ths envronment, communcaton enlarges the set of equlbra. In both models, the frms are necessarly subject to a "type II" error when colludng prce wars are trggered even f no one has devated. In the Green and Porter (1984) model, frms cannot dstngush between a devaton and an aggregate market shock both lead to lower profts and result n prce wars. In our model, communcaton allows frms to dstngush between the two one leads to lower sales for all whle the other leads to a dscrepancy n sales and market shares. As mentoned above, only the latter trggers a punshment. In our model, communcaton mproves montorng and, by enhancng the possblty of colluson, reduces welfare. Of course, not all communcaton need be welfare reducng. It can play a bengn, even benefcal, role n some crcumstances by allowng frms to share economcally valuable nformaton about demand or cost condtons. Indeed, the courts have long recognzed ths possblty and opned that such nformaton sharng "... can hardly be deemed a restrant of commerce... or n any respect unlawful." 5 Economsts and legal scholars have elaborated ths argument (Shapro, 1986 and Carlton, Gertner and Rosenfeld, 1996). The sharng of prvately known costs va cheap talk s studed n a repeated-game model by Athey and Bagwell (2001). They show that, for moderate dscountng, cheap-talk communcaton ncreases profts. The channel through whch ths occurs s qute dfferent from ours, however. The Athey and Bagwell (2001) model s one of perfect montorng but ncomplete payoff-relevant nformaton (costs). Communcaton allows frms to allocate greater market shares n favor of low cost frms and these cost savngs are the source of almost all of the proft gans. Thus, communcaton actually has socal benefts the resultng producton eff cences ncrease welfare. By contrast, our model s one of mperfect montorng but complete nformaton. Communcaton mproves montorng and ths s the only source of proft gans. Now, communcaton has no socal benefts the resultng hgh prces reduce welfare. Another way to see the dfference s by consderng how a "planner" who can dctate the prces that frms charge would mplement collusve outcomes n the two models. In the Athey and Bagwell (2001) model, such a planner must elct the prvately known cost nformaton n order to mplement maxmally collusve outcomes. The resultng mechansm desgn problem necesstates communcaton. It then seems clear that n the absence of a planner, colludng frms would need to communcate wth each other as well. In our model, however, there s no need to communcate any nformaton to such a planner he/she can smply dctate that frms charge the monopoly prce. In ths case, the fact that n the absence of a planner, maxmally collusve outcomes can only be acheved wth 4 The effect of ntroducng communcaton n such a model has been studed n a paper by Rahman (2014), whch we dscuss later. 5 Areeda and Kaplow (1988, pp ). 5

6 communcaton s our man fndng. We beleve that we are the frst to establsh that cheap-talk communcaton about payoff rrelevant nformaton can ad colluson. The remander of the paper s organzed as follows. The next secton outlnes the nature of the market. Secton 3 analyzes the repeated game wthout communcaton whereas Secton 4 does the same wth. The fndngs of the earler sectons are combned n Secton 5 to derve the man result. We also calculate explctly the gans from communcaton n an example wth lnear demands. Omtted proofs are collected n an Appendx. 2 The Market There are two symmetrc frms n the market, labelled 1 and 2. The frms produce dfferentated products at a constant cost, whch we normalze to zero. Each frm sets a prce p P = [0, p max ], for ts product and gven the prces set by the frms, ther sales are stochastc. Prces affect the jont dstrbuton of sales as follows. Frst, they affect expected sales n the usual way an ncrease n p decreases frm s expected sales and ncreases frm j s expected sales. Second, they affect how correlated are the sales of the two frms the more smlar are the two frms prces, the hgher s the correlaton n sales. To facltate the analyss, we wll suppose that gven the two frms prces p = (p 1, p 2 ), ther sales (Y 1, Y 2 ) are jontly dstrbuted accordng to a bvarate log normal densty f (y 1, y 2 p). Precsely, at prces p the log sales (ln Y 1, ln Y 2 ) have a bvarate normal densty wth means µ 1 (p), µ 2 (p) and varance-covarance matrx of the form Σ (p) = σ 2 [ 1 ρ (p) ρ (p) 1 Note that we are assumng that the varance of log sales s unaffected by prces. 6 We now specfy the exact manner n whch prces affect the dstrbuton of sales. 2.1 Expected Sales The functon µ (p, p j ) determng the expected log sales of frm s assumed to be a contnuous functon that s decreasng n p and ncreasng n p j. The frms are symmetrc so that µ = µ j. Note that the frst argument of µ s always frm s s own prce and the second s ts compettor s prce. Because sales are log normally dstrbuted, the expected sales of frm at prce p are E[Y p] = exp ( µ (p) σ2) In what follows, t wll be convenent to denote the expected sales E[Y Q (p, p j ). The functon Q s then frm s expected demand functon. ] p] by 6 A heteroskedastc specfcaton n whch the varance ncreased wth the mean log sales can be easly accommodated. 6

7 ln Y 2 p 1 = p 2 µ µ 2 p 1 < p 2 µ µ 1 ln Y 1 Fgure 1: A depcton of the assumed stochastc relatonshp between prces and log sales va the contours of the resultng normal denstes. When frms charge the same prce, ther log sales have the same mean and hgh correlaton. If frm 1 cuts ts prce, ts mean log sales rse whle those of ts rval fall. Correlaton also falls. 2.2 Correlaton of Sales We wll suppose the correlaton between frms (log) sales s hgh when they charge smlar prces and low when they charge dssmlar prces. Ths s formalzed as: ASSUMPTION. There exsts ρ 0 (0, 1) and a symmetrc functon γ (p 1, p 2 ) [0, 1] such that ρ = ρ 0 γ (p 1, p 2 ) and γ satsfes the followng condtons: (1) for all p, γ (p, p) = 1; and (2) for all p 1 p 2, γ/ p 1 > 0 and 2 γ/ p and so, γ s an ncreasng and convex functon of p 1. Note that ρ/ p 1 = ρ 0 γ/ p 1, and so for fxed γ, an ncrease n ρ 0 represents an ncrease n the senstvty of the correlaton to prces. Some smple examples of the functons γ that satsfy the assumpton are: γ (p 1, p 2 ) = mn (p 1, p 2 ) / max (p 1, p 2 ) and γ (p 1, p 2 ) = 1/(1 + p 1 p 2 ). Fgure 1 s a schematc llustraton of the stochastc relatonshp between prces and sales. Ths knd of correlaton structure s qute natural n many settngs. Consder, for example, a symmetrc Hotellng-type market n whch the frms are located at 7

8 dfferent ponts on the lne and consumers have dentcal but random "transport" costs. Frst, consder the case where the two frms charge very smlar prces. In ths case, ther sales are smlar roughly, they splt the market no matter what the realzed transport costs are. In other words, when frms charge smlar prces, ther sales are hghly correlated. Next, consder the case where the two frms prces are dssmlar, say, frm 1 s prce s much lower than frm 2 s prce. When transport costs are hgh, consumers are not that prce senstve and so the frms realzed sales are rather smlar. But when transports costs are low, consumers are rather prce senstve and so frm 1 s sales are much hgher than frm 2 s sales. In other words, when frms charge dssmlar prces, the correlaton between ther sales s low. Of course, the same knd of reasonng apples f we substtute search costs for transport costs. Another settng n whch the knd of correlaton structure postulated above arses s the "random utlty" model for dscrete choce, wdely used as the bass of many emprcal ndustral organzaton studes. In such a specfcaton, consumers utlty s assumed to have both common and dosyncratc components and the common component plays a role not unlke that of transport costs n the Hotellng model. Agan, consder frst the case where frms charge smlar prces. Now, regardless of the realzaton of the common component, ther sales are smlar and so hghly correlated. But when the frms charge dfferent prces, how close ther sales are depends on the realzaton of the common component of utlty. When the common component s hgh, prces are not that mportant for consumers and so the sales of the two frms wll be relatvely smlar. When the common component s low, however, prce dfferences become mportant, the lower-prced frm wll experence much hgher sales than ts rval, and so the two frms sales wll be dssmlar. Overall, ths means that the correlaton between ther sales s lower. 2.3 One-shot Game Frms maxmze ther expected profts: π (p, p j ) = p Q (p, p j ) and we suppose that π s strctly concave n p. 7 Let G denote the one-shot game where the frms choose prces p and p j and the expected profts are gven by π (p, p j ). Under the assumptons made above, there exsts a symmetrc Nash equlbrum (p N, p N ) of G and let π N be the resultng profts of a frm. 8 Suppose that (p M, p M ) s the unque soluton to the monopolst s problem: max p,p j π (p, p j ) 7 In terms of the prmtves, ths s guaranteed f µ s suff cently concave. 8 If the one-shot game has multple symmetrc Nash equlbra, let (p N, p N ) denote the one wth the lowest equlbrum profts. 8

9 and let π M be the resultng profts per frm. We assume that monopoly prcng (p M, p M ) s not a Nash equlbrum. For techncal reasons we wll also assume that a frm s expected sales are bounded away from zero. 3 Colluson wthout Communcaton Let G δ (f) denote the nfntely repeated game n whch frms use the dscount factor δ < 1 to evaluate proft streams. Tme s dscrete. In each perod, frms choose prces p and p j and gven these prces, ther sales are realzed accordng to f as descrbed above. As n Stgler (1964), each frm observes only ts own realzed sales y ; t observes nether j s prce p j nor j s sales y j. We wll refer to f as the montorng structure. Let h t 1 = ( p 1, y 1, p 2, y 2,..., p t 1 after t 1 perods of play and let H t 1 ), y t 1 denote the prvate hstory observed by frm denote the set of all prvate hstores of frm. In perod t, frm chooses ts prces p t knowng h t 1 and nothng else. A strategy s for frm s a collecton of functons (s 1, s 2,...) such that s t : H t 1 (P ), where (P ) s the set of dstrbutons over P. Thus, we are allowng for the possblty that frms may randomze. Of course, snce H 0 s null, s 1 (P ). A strategy profle s s smply a par of strateges (s 1, s 2 ). A Nash equlbrum of G δ (f) s strategy profle s such that for each, the strategy s s a best response to s j. The man result of ths secton provdes an upper bound to the jont profts of the frms n any Nash equlbrum of the repeated game wthout communcaton. 9 The task s complcated by the fact that there s no known characterzaton of the set of equlbrum payoffs of a repeated game wth prvate montorng. Because the players n such a game observe dfferent hstores each frm knows only ts own past prces and sales such games lack a straghtforward recursve structure and the knds of technques avalable to analyze equlbra of repeated games wth publc montorng (as n the work of Abreu, Pearce and Stacchett, 1990) cannot be used here. Instead, we proceed as follows. Suppose we want to determne whether there s a Nash equlbrum of G δ (f) such that the sum of frms dscounted average profts are wthn ε of those of a monopolst, that s, 2π M. If there were such an equlbrum, then both frms must set prces close to the monopoly prce p M often (or equvalently, wth hgh probablty). Now consder a secret prce cut by frm 1 to p, the statc best response to p M. Such a devaton s proftable today because frm 2 s prce s close to p M wth hgh probablty. How ths affects frm 2 s future actons depends on the qualty of montorng, that s, how much frm 1 s prce cut affects the dstrbuton of 2 s sales. If the qualty of montorng s poor, frm 1 can keep on devatng to p wthout too much fear of beng punshed. In other words, a frm has a proftable devaton, contradctng that there were such an equlbrum. Ths reasonng shows that the resultng bound on Nash equlbrum profts depends 9 Of course, the bound so derved apples to any refnement of Nash equlbrum as well. 9

10 on three factors: (1) the trade-off between the ncentves to devate and eff cency n the one-shot game 10 ; (2) the qualty of the montorng, whch determnes whether the short-term ncentves to devate can be overcome by future actons; and, of course (3) the dscount rate. We consder each of these factors n turn. 3.1 Incentves versus Eff cency n the One-shot Game Defne, as above, p = arg max p π (p, p M ), the statc best-response to p M. Let α (P 1 P 2 ) be a jont dstrbuton over frms prces. We want to fnd an α such that () the sum of the expected profts from α s wthn ε of 2π M ; and () t mnmzes the (sum of) the ncentves to devate to p. To that end, for ε 0, defne subject to Ψ (ε) mn α [π (p, α j ) π (α)] (1) π (α) 2π M ε where α j denotes the margnal dstrbuton of α over P j. The functon Ψ measures the trade-off between the ncentves to devate (to the prce p) and frms profts. Precsely, f the frms profts are wthn ε of those of a monopolst, then the total ncentve to devate s Ψ (ε). It s easy to see that Ψ s (weakly) decreasng. Two other propertes of Ψ also play an mportant role. Frst, Ψ s convex because both the objectve functon and the constrant are lnear n the choce varable α. Second, lm ε 0 Ψ (ε) > 0. To see ths, note that Ψ (0) > 0 because at ε = 0, the only feasble soluton to the problem above s (p M, p M ) and by assumpton, ths s not a Nash equlbrum. Fnally, the fact that Ψ s contnuous at ε = 0 follows from the Berge maxmum theorem. Snce (p N, p N ) s feasble for the program defnng Ψ when ε = 2π M 2π N, t follows that Ψ (2π M 2π N ) 0. We emphasze that Ψ s completely determned by the one-shot game G. Defne the nverse of Ψ by Ψ 1 (x) = sup {ε : Ψ (ε) = x} The bound we develop below wll depend on Ψ Qualty of Montorng Consder two prce pars p = (p 1, p 2 ) and p = (p 1, p 2) and the resultng dstrbutons of frm s sales: f ( p) and f ( p ). If these two dstrbutons are close together, then t wll be dff cult for frm to detect the change from p to p. Thus, the qualty 10 By "eff cency" we mean how eff cent the cartel s n achevng hgh profts and not "socal eff cency." 10

11 of montorng can be measured by the maxmum "dstance" between any two such dstrbutons. In what follows, we use the so-called total varaton metrc to measure ths dstance. Snce f s symmetrc, the qualty of montorng s the same for both frms. Defnton 1 The qualty of a montorng structure f s defned as η = max f ( p) f ( p ) p,p T V where f s the margnal of f on Y and g h T V denotes the total varaton dstance between g and h. 11 It s mportant to note that the qualty of montorng depends only on the margnal dstrbutons f ( p) over s sales and not on the jont dstrbutons of sales f ( p). In partcular, the fact that the margnal dstrbutons f ( p) and f ( p ) are close η s small does not mply that the underlyng jont dstrbutons f ( p) and f ( p ) are close. Because of symmetry, η s the same for both frms. When f ( p) s a bvarate log normal, η can be explctly determned as η = 2Φ ( µmax 2σ ) 1 (2) where Φ s the cumulatve dstrbuton functon of a unvarate standard normal and µ max = max p,p ln Q (p, p j ) ln Q (p, p j) s the maxmum possble dfference n log expected sales. As σ ncreases, η decreases and goes to zero as σ becomes arbtrarly large. 3.3 A Bound on Profts The man result of ths secton, stated below, develops a bound on Nash equlbrum profts when there s no communcaton. An mportant feature of the bound s that t s ndependent of any correlaton between frms sales and depends only on the margnal dstrbuton of sales. Proposton 1 In any Nash equlbrum of the repeated game wthout communcaton, the average profts ( ) π 1 + π 2 2π M Ψ 1 4πη δ2 (3) 1 δ where π = max pj π (p, p j ) and η s the qualty of montorng The total varaton dstance between two denstes g and h on X s defned as g h T V = g (x) h (x) dx. X 11

12 π 2 (π M, π M ) Ψ 1 (π N, π N ) 0 π 1 Fgure 2: The set of feasble profts of the two frms and the poston of the nocommuncaton bound s depcted. The bound les between monopoly and one-shot Nash profts and ts sze depends on the demand structure, the dscount factor and the montorng qualty. Before embarkng on a formal proof of Proposton 1 t s useful to outlne the man deas (see Fgure 3 for an llustraton). A necessary condton for a strategy profle s to be an equlbrum s that a devaton by frm 1 to a strategy s 1 n whch t always charges p not be proftable. Ths s done n two steps. Frst, we consder a fcttous stuaton n whch frm 1 assumes that frm 2 wll not respond to ts devaton. The hgher the equlbrum profts, the more proftable would be the proposed devaton n the fcttous stuaton ths s exactly the effect the functon Ψ captures n the oneshot game and Lemma A.1 shows that Ψ captures the same effect n the repeated game as well. Second, when the montorng s poor η s small frm 2 s actons cannot be very responsve to the devaton and so the fcttous stuaton s a good approxmaton for the true stuaton. Lemma A.5 measures precsely how good ths approxmaton s and qute naturally ths depends on the qualty of montorng and the dscount factor. Notce that profts both from the canddate equlbrum and from the play after the devaton are evaluated n ex ante terms. Observe that f we fx the qualty of montorng η and let the dscount factor δ approach one, then the bound becomes trval (snce lm δ 1 Ψ ( 1 4πηδ 2 / (1 δ) ) = 0) and so s consstent wth Sugaya s (2013) folk theorem. On the other hand, f we fx the dscount factor δ and decrease the qualty of montorng η, the bound converges to 2π M Ψ 1 (0) < 2π M and s effectve. One may reasonably conjecture that f 12

13 there were "zero montorng" n the lmt, that s, f η 0, then no colluson would be possble. But n fact 2π N < 2π M Ψ 1 (0) so that even wth zero montorng, Proposton 1 does not rule out the presence of collusve equlbra. Ths s consstent wth the fndng of Awaya (2014b). 12 Proof. (Of Proposton 1) We argue by contradcton. Suppose that G δ (f) has an equlbrum, say s, whose average total profts 13 π 1 (s) + π 2 (s) exceed the bound on the rght-hand sde of (3). If we wrte ε (s) 2π M π 1 (s) π 2 (s), then ths s equvalent to Ψ (ε(s)) > 4πη 1 δ Gven the strategy profle s, defne δ2 α t j = E s [s t j(h t 1 j )] (P j ) where the expectaton s defned by the probablty dstrbuton over t 1 jont hstores (h t 1, h t 1 j ) determned by s. Note that α j depends on the strategy profle s and not just s j. Let α j = (α 1 j, α 2 j,...) denote the strategy of frm j n whch t plays α t j n perod t followng any t 1 perod hstory. The strategy α j replcates the ex ante dstrbuton of prces p j resultng from s but s non-responsve to hstores. Let s denote the strategy of frm n whch t plays p wth probablty one followng any hstory. In the appendx we show that the functon Ψ, whch measures the ncentve-eff cency trade-off n the one-shot game can also be used to measure ths trade-off n the repeated game as well. Essentally, the convexty of Ψ ensures that the best dynamc ncentves are n fact statonary the argument resembles a consumpton smoothng result. Formally, from Lemma A.1 [π (s, α j ) π (s)] Ψ (ε(s)) (4) Second, from Lemma A.5 we have for = 1, 2 π (s, s j ) π (s, α j ) 2πη δ2 1 δ (5) whch follows from the fact that the total varaton dstance between the dstrbuton of j s sales nduced by (s, s j ) and (s, α j ) n any one perod does not exceed η (Lemma A.3) and, as a result, the dstance between dstrbuton of j s t-perod sales hstores does not exceed tη (Lemma A.4). A smple calculaton then shows that the dfference n payoffs does not exceed the rght-hand sde of (5). 12 Awaya (2014b) constructs an example n whch there s zero montorng the margnal dstrbuton of every player s sgnals s the same for all acton profles but, nevertheless, there are non-trval equlbra. 13 We use π (s) to denote the dscounted average payoffs from the strategy profle s as well as the payoffs n the one-shot game. 13

14 Combnng (4) and (5), we have (π (s, s j ) π (s)) = (π (s, α j ) π (s)) + (π (s, α j ) π (s)) (π (s, s j ) π (s, α j )) π (s, s j ) π (s, α j ) Ψ (ε(s)) 4πη δ2 1 δ whch s strctly postve. But ths means that at least one frm has a proftable devaton, contradctng the assumpton that s s an equlbrum. Ths completes the proof. In a recent paper, Pa, Roth and Ullman (2014) also provde a bound on equlbrum payoffs that s effectve when montorng s poor. They derve necessary condtons for an equlbrum by consderng "one-shot" devatons n whch a player cheats n one perod and then resumes equlbrum play. Pa et al. s bound s based on how the jont dstrbuton of the prvate sgnals s affected by players actons. In games wth prvate montorng, "one-shot" devatons affect the devatng players belefs about the other player s sgnals thereafter and so affect hs subsequent (optmal) play. The optmal play, qute naturally, explots any correlaton n sgnals. The devaton we consder a permanent prce cut s rather nave but has the feature that future play, whle suboptmal, s straghtforward. Ths renders unnecessary any conjectures about the future behavor of the other frm and so our bound does not depend on any correlaton between sgnals (sales); t s based solely on the margnal dstrbutons. When we consder communcaton n the next secton, we wll explot the correlaton of sgnals to construct an equlbrum whose profts exceed our bound (Proposton 2 below). The fact that correlaton can vary whle keepng the margnal dstrbutons fxed s key to solatng the effects of communcaton. The bound obtaned by Pa, et al. (2014) apples to both forms of colluson wth and wthout communcaton and cannot dstngush between the two. Sugaya and Woltzky (2015) also provde a bound but one that s based on entrely dfferent deas. For a fxed dscount factor, they ask whch montorng structure yelds the hghest equlbrum profts. Ths, of course, then dentfes a bound that apples across all montorng structures. But ths method, whle qute general, agan cannot dstngush between the two settngs. 4 Colluson wth Communcaton We now turn to a stuaton n whch frms can, n addton to settng prces, communcate wth each other n every perod, sendng one of a fnte set of messages to each other. The sequence of actons n any perod s as follows: frms set prces, 14

15 receve ther prvate sales nformaton and then smultaneously send messages to each other. Messages are costless the communcaton s "cheap talk" and are transmtted wthout any nose. The communcaton s unmedated. Formally, there s a fnte set of messages M for each frm and that each M contans at least two elements. A t 1 perod prvate hstory of frm now conssts of the complete lst of ts own prces and sales as well as the lst of all messages sent and receved. Thus a prvate hstory s now of the form h t 1 = ( p τ, y τ, m τ, m τ j and the set of all such hstores s denoted by H t 1. A strategy for frm s now a par (s, r ) where s = (s 1, s 2,...), the prcng strategy, and r = (r 1, r 2,...), the reportng strategy, are collectons of functons: s t : H t 1 (M ). ) t 1 τ=1 (P ) and r t : H t 1 P Y Call the resultng nfntely repeated game wth communcaton G com δ (f). A se- (f) s a strategy profle (s, r) such that for each and quental equlbrum of G com δ every prvate hstory h t 1 (s, r ) h t 1, the contnuaton strategy of followng h t 1, denoted by h t 1 ]. 14, s a best response to E[(s j, r j ) h t 1 j 4.1 Equlbrum Strateges Monopoly prcng wll be sustaned usng a grm trgger prcng strategy together wth a threshold sales-reportng strategy n a manner frst dentfed by Aoyag (2002). Snce the prce set by a compettor s not observable, the trgger wll be based on the communcaton between frms, whch s observable. The communcaton tself conssts only of reportng whether one s sales were "hgh" above a commonly known threshold or "low". Frms start by settng monopoly prces and contnue to do so as long as the two sales reports agree both frms report "hgh" or both report "low". Dfferng sales reports trgger permanent non-cooperaton as a punshment. Specfcally, consder the followng strategy (s, r ) n the repeated game wth communcaton where there are only two possble messages H ("hgh") and L ("low"). The prcng strategy s s: In perod 1, set the monopoly prce p M. In any perod t > 1, f n all prevous perods, the reports of both frms were dentcal (both reported H or both reported L), set the monopoly prce p M ; otherwse, set the Nash prce p N. The communcaton strategy r s: 14 Sequental equlbrum s usually defned only for games wth a fnte set of actons and sgnals. We are applyng t to a game wth a contnuum of actons and sgnals. But ths causes no problems n our model because sgnals (sales) have full support and belefs are well-defned. 15

16 In any perod t 1, f the prce set was p = p M, then report H f log sales ln y t µ M ; otherwse, report L. In any( perod t) 1, f the prce set was p p M, then report H f ln y t µ + 1 µm µ ρ j ; otherwse, report L. (µ M = ln Q (p M, p M ) 1 2 σ2, µ = ln Q (p, p M ) 1 2 σ2 and µ j = ln Q j (p M, p ) 1 2 σ2.) Denote by (s, r ) the resultng strategy profle. We wll establsh that f frms are patent enough and the montorng structure s nosy (σ s hgh) but correlated (ρ 0 s hgh), then the strateges specfed above consttute an equlbrum. We begn by showng that the reportng strategy r s ndeed optmal Optmalty of Reportng Strategy Suppose frm 2 follows the strategy (s 2, r 2) and untl ths perod, both have made dentcal sales reports. Recall that a punshment wll be trggered only f the reports dsagree. Thus, frm 1 wll want to maxmze the probablty that ts report agrees wth that of frm 2. Snce frm 2 s followng a threshold strategy, t s optmal for frm 1 to do so as well. If frm 1 adopts a threshold of λ such that t reports H when ts log sales exceed λ, and L when they are less than λ, the probablty that the reports wll agree s Pr [ln Y 1 < λ, ln Y 2 < µ M ] + Pr [ln Y 1 > λ, ln Y 2 > µ M ] whch, for normally dstrbuted varables, s λ µ 1 σ µ M µ 2 σ φ (z 1, z 2 ; ρ) dz 2 dz 1 + λ µ 1 σ µ M µ 2 σ φ (z 1, z 2 ; ρ) dz 2 dz 1 where φ (z 1, z 2 ; ρ) s a standard bvarate normal densty wth correlaton coeff cent ρ (0, 1). 16 Maxmzng ths wth respect to λ results n the the optmal reportng threshold: λ (p 1 ) = µ ρ (µ M µ 2 ) (6) where µ 1, µ 2 and ρ are evaluated at the prce par (p 1, p M ). If frm 1 devates and cuts ts prce to p 1 < p M, then clearly the expected (log) sales of the two frms wll be such that µ 1 > µ M > µ 2. Thus, λ (p 1 ) > µ 1 > µ M, whch 15 The strateges descrbed above have been dubbed sem-publc by Compte (1998) and others. All actons depend only on past publc sgnals that s, the communcaton. The communcaton tself also depends on current prvate sgnals. 16 The standard (wth both means equal to 0 and both varances equal to 1) bvarate normal densty s φ (z 1, z 2 ; ρ) = exp( (z z 2 2 2ρz 1 z 2 )/2(1 ρ 2 )) 2π 1 ρ 2. 16

17 1 β(p 1 ) 1 2 p M p 1 Fgure 3: Ths fgure shows the probablty that sales reports agree after frm 1 cuts ts prce and reports optmally. Ths probablty les strctly between one-half and one. It s hghest at the monopoly prce and decreases wth the extent of the prce cut. says, as expected, that once frm 1 cuts ts prce and so experences stochastcally hgher sales t should optmally under-report relatve to the equlbrum reportng strategy r 1. On the other hand, f frm 1 does not devate and sets a prce p M, then (6) mples that t s optmal for t to use a threshold of µ M = λ (p M ) as well. We have thus establshed that f frm 2 plays accordng to (s 2, r 2), then followng any prce p 1 that frm 1 sets, the communcaton strategy r 1 s optmal. The optmalty of the proposed prcng strategy s 1 depends crucally on the probablty of trggerng the punshment and we now establsh how ths s affected by the extent of a prce cut. If frm 1 sets a prce of p 1, the probablty that ts report wll be the same as that of frm 2 (and so the punshment wll not be trggered) s gven by β (p 1 ) (7) λ(p 1) µ 1 σ µ M µ 2 σ φ (z 1, z 2 ; ρ) dz 2 dz 1 + λ(p 1 ) µ 1 σ µ M µ 2 σ φ (z 1, z 2 ; ρ) dz 2 dz 1 Note that whle λ (p 1 ), µ 1, µ 2 and the correlaton coeff cent ρ depend on p 1, σ does not. Observe also that for any p 1 p M, β (p 1 ) Φ ( µ M µ 2 ) σ 1 where Φ denotes the 2 cumulatve dstrbuton functon of the standard unvarate normal. In other words, the probablty of detectng a devaton s less than one-half. Ths s because frm 1 could always adopt a reportng strategy n whch after a devaton to p 1 < p M, t always says L ndependently of ts own sales, effectvely settng λ (p 1 ) =. Ths guarantees that frm 1 s report wll be the same as frm 2 s report wth a probablty equal to Φ ( µ M µ 2 ) σ. Fgure 3 depcts the functon β when ρ = ρ0 /(1 + p 1 p 2 ). If both follow the proposed prcng and reportng strateges, the probablty that ther reports wll agree s just β (p M ), obtaned by settng λ (p 1 ) = µ 1 = µ 2 = µ M 17

18 and ρ = ρ 0 n (7). Sheppard s formula for the cumulatve of a bvarate normal (see Thansky, 1972) mples that β (p M ) = 1 π arccos ( ρ 0) (8) whch s ncreasng n ρ 0 and converges to 1 as ρ 0 goes to 1. Importantly, ths probablty does not depend on σ. 4.3 Optmalty of Prcng Strategy We have argued above that gven that the other frm follows the prescrbed strategy, the reportng strategy r 1 s an optmal response for frm 1. To show that the strateges (s, r ) consttute an equlbrum, t only remans to show that the prcng strategy s 1 s optmal as well. Ths s verfed next. Proposton 2 There exsts a δ such that for all δ > δ, once σ and ρ 0 are large enough, then (s, r ) consttutes an equlbrum of G com δ (f), the repeated game wth communcaton. Proof. Frst, note that the lfetme average proft π resultng from the proposed strateges s gven by (1 δ) π M + δ [β (p M ) π + (1 β (p M )) π N ] = π (9) Next, suppose that n all prevous perods, both frms have followed the proposed strateges and ther reports have agreed. If frm 1 devates to p 1 < p M n the current perod, t gans 1 (p 1 ) = (1 δ) π 1 (p 1, p M ) + δ [β (p 1 ) π + (1 β (p 1 )) π N ] π (10) where π s defned n (9). Thus, 1 (p 1 ) = (1 δ) π 1 p 1 (p 1, p M ) + δβ (p 1 ) [π π N ] We wll show that when σ s large enough, for all p 1, 1 (p 1 ) > 0. Snce 1 (p M ) = 0, ths wll establsh that a devaton to a prce p 1 < p M s not proftable. Now observe that from Lemma A.6, lm σ 1(p 1 ) = (1 δ) π 1 1 p 1 (p 1, p M ) + δ π (1 δ) π 1 1 p 1 (p M, p M ) + δ 1 ρ ρ γ 2 0 γ(p 1,p M ) 2 0 π 1 ρ ρ γ 2 0 γ(0,p M ) 2 0 p 1 (p 1, p M ) [π π N ] p 1 (0, p M ) [π π N ] where the last nequalty uses the fact that snce π 1 s concave n p 1, π 1 p 1 (p 1, p M ) > π 1 p 1 (p M, p M ) and the fact that γ (p 1, p M ) s ncreasng and convex n p 1. Let σ (δ) be such that for all σ > σ (δ), the nequalty above holds. 18

19 Let δ be the soluton to (1 δ) π 1 1 p 1 (p M, p M ) + δ γ π 1 γ(0,p M ) 2 p 1 (0, p M ) [π π N ] = 0 (11) whch s just the rght-hand sde of the nequalty above when ρ 0 = 1. Such a δ exsts snce π 1 ρ p 1 (p M, p M ) s fnte and, by assumpton, p 1 (0, p M ) s strctly postve. Notce that for any δ > δ, the expresson on the left-hand sde s strctly postve. Now observe that 1 π 1 ρ ρ γ 2 0 γ(1,p M ) 2 0 p 1 (0, p M ) [π π N ] s ncreasng and contnuous n ρ 0 (recall that π s ncreasng n ρ 0 ). Thus, gven any δ > δ, there exsts a ρ 0 (δ) such that for all ρ 0 = ρ 0 (δ) (1 δ) π 1 1 p 1 (p M, p M ) + δ π 1 ρ ρ γ 2 0 γ(0,p M ) 2 0 p 1 (0, p M ) [π π N ] = 0 Note that ρ 0 (δ) s a decreasng functon of δ and for any ρ 0 > ρ 0 (δ), the left-hand sde s strctly postve. A devaton by frm 1 to a prce p 1 > p M s clearly unproftable. Ths completes the proof of Proposton 2. Aoyag (2002) was the frst to use threshold reportng strateges n a correlated envronment. He shows that for a gven montorng structure (ρ 0 and σ fxed) as the dscount factor δ goes to one, such strateges consttute an equlbrum. The dea as n all "folk theorems" s that even when the probablty of detecton s low, f players are patent enough, future punshments are a suff cent deterrent. In contrast, Proposton 2 shows that for a gven dscount factor (δ hgh but fxed), as ρ 0 goes to one and σ goes to nfnty, there s an equlbrum wth hgh profts. Its logc, however, s dfferent from that underlyng the "folk theorems" wth communcaton, as for nstance, n the work of Compte (1998) and Kandor and Matsushma (1998). In these papers, sgnals are condtonally ndependent and, n equlbrum, players are ndfferent among the messages they send. Kandor and Matsushma (1998) also show that wth correlaton, strct ncentves for "truth-tellng" can be provded as s also true n our constructon. But the key dfference between our work and these papers s that n Proposton 2 the punshment power derves not from the patence of the players; rather t comes from the nosness of the montorng. A devatng frm fnds t very dff cult to predct ts rval s sales and hence, even t "les" optmally, a devaton s very lkely to trgger a punshment. The equlbrum constructed n Proposton 2 reles on usng the correlaton of sgnals to "check" frms reports. Ths s remnscent of the logc underlyng the fullsurplus extracton results of Crémer and McLean (1988) but, of course, n our settng there s no mechansm desgner who can commt to arbtrarly large punshments to nduce truth-tellng. 19

20 In our model, frms communcate smultaneously nether frm knows the other s report pror to ts own and ths feature s crucal to the equlbrum constructon. It s the uncertanty about what the other frm wll say that dscplnes frms behavor. Often trade assocatons help cartels exchange sales data n a manner that s mmcked by the smultaneous reportng n our equlbrum. Cartel members make confdental sales reports to the assocaton whch, as mentoned above, dssemnates aggregate sales data to the cartel. Indvdual sales fgures are not shared. 17 Ther confdentalty s usually preserved by thrd partes. The pocket calculator ploy mentoned n the ntroducton also serves the same purpose. Fnally, the "grm trgger" strateges specfed above are unforgvng n that a sngle dsagreement at the communcaton stage trggers a permanent reverson to the one-shot Nash equlbrum. Snce dsagreement occurs wth postve probablty even f nether frm cheats, ths means that n the long-run, colluson nevtably breaks down. Real-world cartels do punsh transgressors but these punshments are not permanent. The equlbrum strateges we have used can easly be amended so that the punshment phase s not permanent and that, after a pre-specfed number of perods, say T, frms return to monopoly prcng. Specfcally, ths means that the equlbrum profts π are now defned by (1 δ) π M + δ [ β (p M ) π + (1 β (p M )) (( 1 δ T +1) π N + δ T +1 π )] = π nstead of (9). For any δ > δ (as defned by (11)), there exsts a T large enough so that the forgvng strategy consttutes an equlbrum as well. Note that π > π. 5 Gans from Communcaton Proposton 1 shows that the profts from any equlbrum wthout communcaton cannot exceed ( ) 2π M Ψ 1 4πη δ2 1 δ whereas Proposton 2 provdes condtons under whch there s an equlbrum wth communcaton that wth profts 2π (as defned n (9)). From (8) t follows that as ρ 0 1, β (p M ) 1. Now from (9) t follows that as ρ 0 1, π π M. Combnng these facts leads to the formal verson of the result stated n the ntroducton. Let δ be determned as n (11). Theorem 1 For any δ > δ, there exst (σ (δ), ρ 0 (δ)) such that for all (σ, ρ 0 ) (σ (δ), ρ 0 (δ)) there s an equlbrum wth communcaton wth total profts 2π such that ( ) 2π > 2π M Ψ 1 4πη δ2 1 δ 17 The sharng of aggregate data s not llegal (see the dscusson of the Maple Floorng case n Areeda and Kaplow, 1988). 20

21 2π M ρ 0 2π Communcaton No communcaton 2π M Ψ 1 (0) σ(δ) σ Fgure 4: An llustraton of the man result. The no-communcaton bound decreases wth σ, the standard devaton of log sales, but s ndependent of ρ 0, ther correlaton. The profts from the constructed equlbrum wth communcaton are, on the other hand, ndependent of σ, but ncrease wth ρ 0. Thus, for large σ and ρ 0, the profts wth communcaton exceed those from any equlbrum wthout. In the lmt, as ρ 0 1 and σ, the left-hand sde of the nequalty goes to 2π M and the rght-hand sde goes 2π M Ψ 1 (0) snce η 0. Thus, the dfference n profts between the two regmes s at least Ψ 1 (0) > 0. The workngs of the man result can be seen n Fgure 4, whch s drawn for the case of lnear demand (see the next secton for detals). Frst, notce that the profts wth communcaton depend on ρ 0 and not on σ (but σ has to be suff cently hgh to guarantee that the suggested strateges form an equlbrum). The bound on profts wthout communcaton, on the other hand, depends on σ and not on ρ 0. For small values of σ, the bound s neffectve and says only that these do not exceed jont monopoly profts. As σ ncreases, the bound becomes tghter but at ntermedate levels the profts from the equlbrum wth communcaton do not exceed the bound. Once σ > σ (δ), communcaton strctly ncreases profts. 5.1 Example wth Lnear Demand We now llustrate the workngs of our results when (expected) demand s lnear. 21

22 Suppose that 18 Q (p, p j ) = max (A bp + p j, 1) where A > 0 and b > 1. For ths specfcaton, the monopoly prce p M = A/2 (b 1) and monopoly profts π M = A 2 /4 (b 1). There s a unque Nash equlbrum of the one-shot game wth prces p N = A/ (2b 1) and profts π N = A 2 b/ (2b 1) 2. A frm s best response f the other frm charges the monopoly prce p M s to charge p = A (2b 1) /4b (b 1). The hghest possble proft that frm 1 can acheve when chargng a prce of p s π = π 1 (p, p M ) = A 2 (2b 1) 2 /16b (b 1) 2. It remans to specfy how the correlaton between the frms log sales s affected by prces. In ths example we adopt the followng specfcaton: ρ = ρ p 1 p 2 whch, of course, satsfes the assumpton made n Secton I. Then, recallng (1), t may be verfed that for ε [0, π M /2b 2 ] ( Ψ (ε) = ε + A A 2 2 (b 1) (2b 1) ) ε 8b(b 1) 2 whch s acheved at equal prces. Note that Ψ (0) = π M /2b (b 1) and Ψ 1 (0) = π M /2b 2. Fnally, from (2) ( ) η = 2Φ µmax 1 2σ where µ max = ln Q 2 (0, p M ) ln Q 2 (p M, 0). As a numercal example, suppose A = 120 and b = 2. Let δ = 0.7 and ρ 0 = For these parameters, π M = 3600, π = 4050 and µ max = Also, the profts from the equlbrum wth communcaton, π = 3524 (approxmately). Fgure 4 depcts the bound on profts wthout communcaton as a functon of σ usng Proposton 1. For low values of σ (approxmately σ = 60 or lower), the bound s neffectve t equals 2π M and as σ, converges to 2π M Ψ 1 (0). As shown, the profts wth communcaton exceed the bound when σ > σ (δ) = 200 (approxmately). Fgure 5 verfes that the strateges (s, r ) consttute an equlbrum a devaton to any p 1 < p M s unproftable as (p 1 ) < 0 (as defned n (10)). Ths s verfed for σ = 60 and, of course, the same strateges reman an equlbrum when σ s hgher. 5.2 On the Necessty of Communcaton Theorem 1 dentfes crcumstances n whch communcaton s necessary for colluson n a prvate montorng settng. That t s suff cent has been ponted out n many 18 Ths specfcaton of "lnear" demand s used because ln 0 s not defned. 22

23 0 p p M (p 1 ) Fgure 5: A depcton of the net gans from devatng from the prescrbed strateges wth communcaton. Small prce cuts are not that proftable but are hard to detect. Bgger prce cuts are more proftable but easer to detect. Ths trade-off accounts for the non-monotoncty of net gans. studes, begnnng wth Compte (1998) and Kandor and Matsushma (1998) and then contnung wth Fudenberg and Levne (2007), Zheng (2008) and Obara (2009). Of partcular nterest s the work of Aoyag (2002) and Harrngton and Skrzypacz (2011) n olgopoly settngs. Under varous assumptons, all of these conclude that the folk theorem holds any ndvdually ratonal and feasble outcome can be approxmated as the dscount factor tends to one. But as Kandor and Matsushma (1998) recognze, One thng whch we dd not show s the necessty of communcaton for a folk theorem (p. 648, ther talcs)." Indeed when players are arbtrarly patent, communcaton s not necessary for colluson, as a remarkable paper by Sugaya (2013) shows. He establshes the surprsng result that n very general envronments, the folk theorem holds wthout any communcaton. An mportant component of Sugaya s proof s that players mplctly communcate va ther actons. Thus, he shows that wth enough tme, there s no need for drect communcaton. In a recent paper, Rahman (2014) shows how communcaton can overcome the "no colluson" result of Sannkov and Skrzypacz (2007) n a repeated Cournot olgopoly wth publc montorng. But snce the latter result concerns only equlbra n publc strateges, ths agan does not establsh the necessty of communcaton for colluson, only ts suff cency. Moreover, the communcaton consdered by Rahman (2014) s ether medated or verfable. In our work, as well as n the papers mentoned above, communcaton s unmedated and unverfable. The restrcton to publc strateges s also a feature of the analyss of Athey and Bagwell (2001) n ther model of colluson wth ncomplete nformaton about costs. Communcaton s used to sustan colluson but as shown by Hörner and Jamson (2008), once the restrcton to publc strateges s removed, communcaton s no 23