On Globally Optimal Punishments in the Repeated Cournot Game

Delbono, Flavio; Lambertini, Luca Working Paper On Globally Optimal Punishments in the Repeated Cournot Game Quaderni - Working Paper DSE, No. 1091 Provided in Cooperation with: University of Bologna, Department of Economics Suggested Citation: Delbono, Flavio; Lambertini, Luca (2016) : On Globally Optimal Punishments in the Repeated Cournot Game, Quaderni - Working Paper DSE, No. 1091, Alma Mater Studiorum - Università di Bologna, Dipartimento di Scienze Economiche (DSE), Bologna, https://doi.org/10.6092/unibo/amsacta/5483 This Version is available at: https://hdl.handle.net/10419/159929 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc/3.0/ ISSN 2282-6483 On Globally Optimal Punishments in the Repeated Cournot Game Flavio Delbono Luca Lambertini Quaderni - Working Paper DSE N°1091 On Globally Optimal Punishments in the Repeated Cournot Game Flavio Delbono#and Luca Lambertinix Department of Economics, University of Bologna # Piazza Scaravilli 2, 40126 Bologna, Italy; ‡[email protected] § Strada Maggiore 45, 40125 Bologna, Italy; [email protected] December 20, 2016 Abstract We challenge the global optimality of one-shot punishments in in- …nitely repeated games with discounting. Speci…cally, we show that the stick-and-carrot punishment à la Abreu (1986) may not be globally optimal. We prove our result by investigating tacit collusion in the in…nite repetition of a linear Cournot game. We illustrate the existence of the stick-and-carrot globally optimal punishment for large cartels, and fully characterise it. Then, we show that for mall cartels, global optimality may be reached only with two-period punishments. Keywords: cartel stability, implicit collusion, repeated games JEL Classi…cation: C73, L13 1 1 Introduction While it is still an empirical question of major importance to establish if (and how) enterprises do collude, the theory of supergames has certainly strengthened our understanding of how cartels may be shaped by oligopolistic interaction. The term supergame was likely coined by Luce and Rai¤a (1957) when examining repetitions of the prisoner’s dilemma. Actually , according to Aumann (1981), “it has been known since the middle to late 1960s that any individually rational payo¤ vector (i.e., not smaller than a payo¤ that the player can guarantee himself) can be supported as a Nash equilibrium outcome in an in…nitely repeated game where there is no discounting . . . , and it is this result that he dubs the Folk Theorem”(Friedman 1986, p. 103). However, when switching to impatient players, the strategies forming a subgame perfect equilibrium (SPE) in supergames without discounting do not necessarily work. In a pioneering paper, Friedman (1971) proved that every feasible payo¤ that Pareto dominates a Nash equilibrium of the stage game is a SPE payo¤ of the in…nite repetition of the stage game, if players are patient enough. Friedman’s punishment take the simple form of reversion to the Nash equilibrium of the stage game forever (grim strategy). This line of research has been furthered by Abreu (1986). He identi…es a class of (pure) strategies that support the Folk Theorem in repeated games with discounting. Such strategies allow one to …nd SPE where any individually rational outcome can be achieved. Moreover, Abreu (1986) has characterized a two-phase punishment as an optimal symmetric punishment which is more severe than Nash reversion; such a two-phase punishment consists of a stick-and-carrot strategy. Fudenberg and Maskin (1986) have then extended Abreu’s results also to games of incomplete information. In Shapiro’s (1989) words: “Within the class of symmetric punishments, Abreu proves that the optimal punishment has a simple, two-phase strategy: immediately following 2 the defection, each …rm participates in a ‘price war’by producing a higher output than previously; but immediately thereafter all …rms return to their optimal, tacitly-collusive output levels. It is striking that, when optimally punishing a defector, the industry returns after only a single period to the most collusive sustainable con…guration. Abreu describes these types of punishments as o¤ering a stick and a carrot; apparently, the carrot (returning to collusion) is necessary to make the stick (the one-period price war) both credible and as menacing as possible”(Shapiro, 1989, p. 368, italics added). Abreu (1986) also provides conditions under which the symmetric two-phase punishment is globally optimal. In this paper we challenge such conditions showing that stick-and-carrot punishments may not be globally optimal and more than a single period is then required to enforce a collusive path. As a workhorse we employ the textbook version of a linear Cournot model as in Abreu’s (1986, p. 206) example. We know that: (i) global optimality of punishments requires that “in continuation equilibria …rms earn zero pro…ts”(Shapiro, 1989, p. 369); (ii) condition (i) is granted by minmax strategies after deviation from the cartel, and (iii) minmax strategies are nor subgame perfect in variable-sum games as ours. Hence, the central issue we are going to tackle deals with the existence of subgame perfect punishments capable of reproducing the same critical threshold of the discount factor as under minmax strategies. We prove that, for small cartels, stick-and-carrot is not a globally optimal strategy and global optimality can be reached only via two-period punishments. This result belies the validity of Abreu’s (1986) Theorems 18 and 19. For large cartels, instead, a stick-and-carrot punishment is globally optimal as it entails the same critical threshold of the discount factor as under minmax. Irrespective of cartel size, his example is mistaken. The paper is organized as follows. In Section 2 we present the setup and establish the benchmark for the threshold of the discounted factor under 3 grim or minmax strategies. In Sections 3 and 4 we get into Abreu’s analysis and investigate the optimality of stick-and-carrot punishments. In Section 5 we show under which conditions, for small cartels, global optimality may be obtained by means of a non stationary punishment lasting two periods. Section 6 concludes. 2 Setup Consider a market for a homogeneous good, served by N= 2; :::n identical single-product …rms, endowed with the same technology. Let the market exist over discrete time t= 0;1;2; :::1:All …rms share the same intertemporal preferences, measured by the time-invariant discount factor 2[0;1]. In each period, the inverse market demand function is p=a n X i=1 qi(1) where parameter a > 0. The cost function of …rm iis Ci=cqiwith c2[0; a). Accordingly, the individual pro…ts are i= a n X i=1 qic!qi(2) 2.1 Grim trigger and minmax strategies Consider …rst Friedman’s (1971) version of the grim trigger strategies, where collusion is sustained by the threat of an in…nite reversion to the CournotNash equilibrium of the constituent game. We brie‡y summarise this result here. Assume perfect tacit collusion with cartel members setting the output vector so as to maximise joint pro…ts  = Pn i=1 i:The resulting individual collusive output level is qC= (ac)=(2n);granting an individual pro…t 4 C= (ac)2=(4n). The unilateral deviation against the n1loyal cartel members is qDqC=(ac) (n+ 1) 4n(3) delivering deviation pro…ts DqC=(ac)2(n+ 1)2 16n2(4) During the in…nite Nash reversion, the per-period Cournot-Nash pro…ts are N= (ac)2=(n+ 1)2:Collusion is stable i¤ DC DNF=(n+ 1)2 n(n+6)+1 (5) where Fis concave and monotonically increasing in n; with F= 9=17 for n= 2 and limn!1 F= 1. Treating nas a continuous variable, @F @n =4 (n21) [n(n+ 6) + 1]2>0(6) The intuitive message, which has been incorporated in the acquired view on these matters, is that, in the collusive outcome “the per period and per …rm pro…t is a decreasing function of n. A large number of …rms reduces the pro…t per …rm and thus the cost of being punished for undercutting. In contrast, the short run gain from undercutting the monopoly price slightly... increases with n... In this sense market concentration facilitates tacit collusion” (Tirole, 1988, p. 248). If instead the in…nite punishment consists in each player reverting to minmax one another through qm= (ac)=n, the per-period individual payo¤ during the punishment phase is nil and the threshold of the discount factor ensuring the stability of tacit collusion becomes DC Dm=(n1)2 (n+ 1)2< F8n(7) 5 where subscript mmnemonics for minmax, with m= 1=9for n= 2 and limn!1 m= 1. That is, minmax strategies appear to be more e¢ cient than Nash ones in stabilising the cartel. However, since the constituent game is a variable-sum one, it is well known that the use of minmax strategies does not produce subgame perfection. Yet, the threshold of the discount factor mdelivered by minmax strategies identi…es the benchmark to be reproduced using globally optimal punishments which must meet the additional requirement of subgame perfection. 3 The supergame with optimal punishments Abreu (1986) aims at …nding a one-shot punishment strategy possessing the properties of being (i) subgame perfect, (ii) more e¢ cient than Friedman’s (1971) Nash reversion, and (iii) globally optimal. Since the stick is harsher than the Nash strategy, this requires a condition of its own for incentive compatibility about the implementation of the punishment itself. Moreover, since the severity of the punishment may drive the resulting punishment pro…ts below zero, even if for a single period, one has to control for the nonnegativity of the discounted ‡ow of pro…ts over the continuation of the game from the punishment period to doomsday. All of this requires the following conditions to hold in a symmetric subgame perfect equilibrium: DqCCCP(8) DqPPCP(9) P+C 1 X t=1 t=P+C 10(10) considering, for the moment, DqP>0:For the moment, we shall suppose it is. This amounts to saying that we start examining optimal (but not 6 necessarily globally optimal) punishments, with conditions (8-10) referring to the setup in Abreu (1986) up to his Lemma 17, p. 204. Inequality (8) must be satis…ed for the collusive path to be stable. Inequality (9) must hold for …rms to implement the optimal punishment qP; delivering the punishment payo¤ P; DqPbeing the pro…ts generated by the optimal deviation qqPfrom the punishment qP. Condition (10) is the participation constraint whereby the discounted continuation payo¤ cannot be negative. Sticking to the assumption of full collusion, the two unknowns to be determined are the critical threshold of the discount factor and the intensity of the punishment qP: If all …rms adopt the punishment, the per-…rm punishment pro…ts are P=anqPcqP(11) while the unilateral deviation from qPis qqP=ac(n1) qP 2>08qP20;ac n1(12) where, for future reference, we may de…ne (ac)=(n1) qP. Whenever the best reply in (12) is indeed positive, the pro…ts granted by optimally deviating from the punishment are DqP=a(n1) qPc2 4(13) Now, solving (8-9) w.r.t. and qP, one obtains (n+ 1)2 16nA qPqP A(ac) (3n1) 2n(n+ 1) (14) with @qP A=@n < 0;for the intensity of the punishment is diluted as the number of cartel members increases. In correspondence of the lower bound of qPin 7 which proves that imposing D(qP 1)=0replicates the above (unacceptable) result whereby the individual output in the …rst period of the punishment is qP mand then …rms revert to the cartel production in the second period of the presumed punishment. Hence, this approach cannot be pursued. The second approach consists in considering D(qP 1) = a(n1) qP 1c2 4>0(37) together with a non-stationary punishment, as in Lambertini and Sasaki (2002). The related system of inequalities is D(qC)C 2 X t=1 tCP(qP t)(38) D(qP 1)P(qP 1) 2 X t=1 tCP(qP t)(39) P(qP 1) + P(qP 2) + 2C 10(40) Condition (38) reformulates the constraint concerning the stability of the collusive path, while (39) ensures players’incentive compatibility about the implementation of a biperiodal punishment. The third condition prevents players quitting the supergame. In all of them, P(qP t) = a2qP tcqP t(41) and, in (39-40), D(qP t) = ac(n1) qP t2 4; t = 1;2(42) The system (38-40) delivers four solutions w.r.t. the triple qP 1; qP 2; ; 14 of which only one is acceptable: qP 1qP A(ac) (3n1) 2n(n+ 1) qP 2bqP 2(ac) [2n(n1) + (1 + n(n6)) pn] 4n2(n1) m(n1)2 (n+ 1)2 (43) On the basis of (43), one can easily establish qP>(ac) (3n1) 2n(n+ 1) >(ac) [2n(n1) + (1 + n(n6)) pn] 4n2(n1) (44) for all n2[2;5] ;i.e., the punishment is decreasing over time and the constraint about the the optimal deviation from the punishment is respected. Moreover, plugging the triple qP 1=qP A; qP 2=bqP 2;  =minto (40), one …nds that the constraint ensuring the …rms’participation to the continuation of the supergame is indeed satis…ed at the margin, i.e., (40) holds as an equality. Moreover, controlling for the price in the two punishment periods, it turns out that it is strictly positive for n= 2;3;while for n= 4;5;a su¢ cient condition for its non-negativity is a2(c; 7c]:To see this, observe that the price levels in the two punishment periods are pqP 1=c(3n1) a(n3) 2 (n+ 1) pqP 2=c12pn6n+ 2n3=2+n2a1+2pn6n2n3=2+n2 4pn(n1) (45) Whereas pqP 2>0for all n2[2;5] ; p qP 1may become negative at n= 4;5 because its partial derivative w.r.t. ais negative for n= 4;5. The foregoing discussion can be summarised in the following: Proposition 4 For n2[2;5] ;the globally optimal punishment requires two periods and its intensity is decreasing over time. It is admissible for all 15 a2(c; 7c]:The resulting critical threshold of the discount factor for the stability of full collusion is the same as under in…nite reversion to the minmax strategies, m;at which v() = 0. On the basis of Propositions 3-4, we can formulate Theorem 5 Assume a2(c; 7c]. In the linear Cournot supergame, there exists a globally optimal punishment path minimising the value of the stability threshold of the discount factor for all n2. For any cartel size, the discounted payo¤ ‡ow generated by the continuation game following the initial deviation from the cartel path is nil. However, the structure of the globally optimal punishment depends on cartel size: the punishment is one-shot (stick-and-carrot) only for n6, in which case it requires the optimal deviation payo¤ from the punishment to be nil, without any restriction on market size; for smaller cartels, a2(c; 7c]is required and the optimal deviation payo¤ from the punishment must not be nil. The punishment must be distributed over two periods along which its severity is decreasing. Now recall Abreu’s (1986, p. 205) claim “Consider two-phase punishments. The only way v() can equal zero and (x1; x2)be a P.E. is if (x1)=0...”as reported above. Our Theorem proves that the attainment of the lowest threshold of the discount factor mdoes require v() = 0, but not, in general, (x1) = 0:Indeed, for small cartels, this is not the case because the stick-and-carrot punishment scheme does not work, global optimality requiring a two-period punishment. The intuitive explanation relies upon the fact that the slice from cartel participation decreases with cartel size. Hence, the punishment required to stabilise a small cartel has to be more severe than for large cartels. This, when global optimality is looked for, entails an extension of the punishment span. 16 6 Concluding remarks In this paper, we have proved that global optimality in the in…nite repetition of a linear Cournot game may not be granted by one-shot stick-and-carrot strategy. Our conclusions contrast strikingly with those of Abreu (1986). We show that globally optimal subgame perfect strategies depend on the number of …rms. For ‘large’cartels, the globally optimal one-shot punishment exists and reproduces the same critical threshold of the discount factor as under minmax strategies. For ‘small’cartels, global optimality cannot be implemented via one-shot punishments, but only by means of a two-period punishment. Whatever is the number of …rms, the example in Abreu (1986) delivers thresholds of the discount factor which cannot be globally optimal as they do not coincide with that delivered by the reversion to the minmax strategy. The source of this problem is that the requirement that the payo¤ ‡ow generated by the continuation of the supergame be nil makes two of the three incentive compatibility constraints coincide. Our …ndings, which we have derived from the simplest possible framework, might prelude to a revisitation of the large literature investigating the impact of product di¤erentiation on cartel stability2using Friedman’s (1971) folk theorem and the representative consumer with a preference for variety as in Singh and Vives (1984). 2See Deneckere (1983), Majerus (1988), Ross (1992), Rothschild (1992), Lambertini (1997) and Albæk and Lambertini (1998), inter alia. 17 References [1] Abreu, D.J. (1986), “Extremal Equilibria of Oligopolistic Supergames”, Journal of Economic Theory,39, 191-225. [2] Abreu, D.J. (1988), “On the Theory of In…nitely Repeated Games with Discounting”, Econometrica,56, 383-96. [3] Albæk, S. and L. Lambertini (1998), “Collusion in Di¤erentiated Duopolies Revisited”, Economics Letters,59, 305-8. [4] Aumann, R.J. (1981), “Survey of Repeated Games”, in R. J. Aumann et al. (eds), Essays in Game Theory, Mannheim, Bibliographisches Institut. [5] Deneckere, R. 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