Self-enforcing collusion in large dynamic markets

The “folk theorem” in game theory implies that any outcome that is better for all players than some single period Nash outcome can be achieved through noncooperative equilibrium in repeated games with discounting. Whether the folk theorem holds for a repeated Cournot oligopoly as the number of firms, N, increases without bound, is investigated. It is shown that the folk theorem holds in the limit iff demand increases at the same rate as the number of firms and the Cournot price sequence is bounded strictly above by the supremum of marginal cost for large N.     

Strategic Complements, Substitutes, and Ambiguity: The Implications for Public Goods

We examine the effect of ambiguity in symmetric games with aggregate externalities. We find that ambiguity will increase/decrease the equilibrium strategy in games with strategic complements/substitutes and positive externalities. These effects are reversed in games with negative externalities. We consider some economic applications of these results to Cournot oligopoly, bargaining, macroeconomic coordination, and voluntary donations to a public good. In particular we show that ambiguity may reduce free-riding. Comparative statics analysis shows that increases in uncertainty will increase donations, to a public good. Journal of Economic Literature C72, D81, H41.     

A characterization of strategic complementarities

I characterize games for which there is an order on strategies such that the game has strategic complementarities. I prove that, with some qualifications, games with a unique equilibrium have complementarities if and only if Cournot best-response dynamics has no cycles; and that all games with multiple equilibria have complementarities. As applications of my results, I show that: (1) generic 2×2 games either have no pure-strategy equilibria, or have complementarities; (2) generic two-player finite ordinal potential games have complementarities.     

The Dynamics of Duopoly Output Games Involving Cournot and Related Best-Response Strategies*

The paper considers a variety of perfect information duopoly games involving the Cournot and related best-response strategies. Firms record their output at the end of each business period. At the beginning of a new period they have to plan their new output; they do so in an attempt to maximize their profits. The importance of discrete time and the resulting time interval between successive outputs is discussed. The difference equations appropriate to the various games are solved exactly, and it is shown that there is always a stable approach to equilibrium at the Cournot point. The convergence of the solutions is discussed.     

The cournot duopoly model reconsidered in the presence of costly communication and enforcement agreements

Recently I proposed a general notion of equilibrium for n-person games which is based on mutual threats and promises (responses), and which leads, in general, to a determinate solution with respect to the order of choice moves. The nature of the solution set in our model is a function of the action sets, the payoff functions, and the cost of ‘communication and enforcement of agreements’ (C&E). The main concern of this paper is to show that the Cournot duopoly solution, for the case of a linear demand function facing two identical firms, can be regenerated by our solution concept under a certain assumption about a high cost C&E. On the other hand, it is shown that under a relatively low cost of C&E the Cournot/Nash solution does not belong to our solution set, because it is Pareto inferior for both duopolists relative to some solutions in our set, among them, the standard monopoly solution.     

Two remarks on Cournot equilibria

We describe two simple results in the theory of Cournot equilibria. The first has to do with the effect of taxation in a Cournot industry and the second describes under what conditions a Cournot equilibrium will implicity maximize an objective function. These results are probably known, but are not well known, and they seem useful enough to be worth spelling out.     

Rationalized subjective equilibria in repeated games

Abstract The purpose of this paper is to provide a new equilibrium concept of repeated games that involve incomplete information ( rationalized subjective equilibrium ) and a related framework ( subjective game ), and to analyse the relationships between players' subjective images of games and the realized outcomes. We demonstrate the relationships among rationalized subjective equilibria, subjective equilibria, and original Nash equilibria by providing several examples. We also present several theorems analogous to the ordinary folk theorem and a simple application using the Cournot duopoly model. JEL Classification: C70, C72
Equilibres subjectifs rationalisés dans des jeux répétés L'objectif de cet article est de proposer un nouveau concept d'équilibre dans les jeux répétés où l'information est incomplète (équilibre subjectif rationalisé) et un cadre d'analyse approprié( jeu subjectif ), ainsi que d'analyser les rapports entre les images subjectives que les acteurs ont des jeux et de leurs résultats. On montre les rapports entre équilibres subjectifs rationalisés, équilibres subjectifs, et équilibres à la Nash à l'aide de divers exemples. On présente aussi plusieurs théorèmes analogues au « ordinary folk theorem >> et une application simple utilisant le modèle de duopole de Cournot.     

Learning to collude: An experiment in convergence and equilibrium selection in oligopoly

The paper considers a simple oligopoly model where firms know their own and the average pay-off in the industry. Firms choose decision rules for trading. The theory predicts that there are three types of Nash equilibria in this game (collusive, Cournot and Stackelberg). Our experiments test the selection process. We find that there is clear evidence of convergence to an equilibrium, and whilst both Cournot and collusive outcomes were selected, the collusive equilibrium is more common. The experimental results also give insights into the process of individual learning, confirming that subjects follow aspiration rules rather than reinforcement rules.     

Repeated Games with Almost-Public Monitoring

In repeated games with imperfect public monitoring, players can use public signals to coordinate their behavior, and thus support cooperative outcomes, but with private monitoring, such coordination may no longer be possible. Even though grim trigger is a perfect public equilibrium (PPE) in games with public monitoring, it often fails to be an equilibrium in arbitrarily close games with private monitoring. If a PPE has players' behavior conditioned only on finite histories, then it induces an equilibrium in all close-by games with private monitoring. This implies a folk theorem for repeated games with almost-public almost-perfect monitoring. Journal of Economic Literature Classification Numbers: C72, C73.     

Is a Unique Cournot Equilibrium Locally Stable?

We consider a homogeneous product oligopoly, where the Cournot equilibrium is regular and unique. We show that for a duopoly, a unique Cournot equilibrium is always locally stable. For a “n” firm asymmetric cost oligopoly a unique Cournot equilibrium is locally stable under very general conditions. The sufficient conditions for local stability of a unique Cournot equilibrium are much less restrictive than what the existing literature suggests. For a symmetric cost oligopoly the unique Cournot equilibrium is almost always locally stable, except for a perverse case. Journal of Economic Literature Classification number: L13.     

Essential equilibria in normal-form games

A Nash equilibrium x of a normal-form game G is essential if any perturbation of G has an equilibrium close to x. Using payoff perturbations, we show that for games that are generic in the set of compact, quasiconcave, and generalized payoff secure games with upper semicontinuous sum of payoffs, all equilibria are essential. Some variants of this result are also established.     

Coalition-proof supply function equilibria in oligopoly

In an industry where firms compete via supply functions, the set of equilibrium outcomes is large. If decreasing supply functions are ruled out, this set is reduced significantly, but remains large. Specifically, the set of prices that can be sustained by supply function equilibria is the interval between the competitive price and the Cournot price. In sharp contrast, when the number of firms is above a threshold we identify (e.g., three if demand is linear), only the Cournot outcome can be sustained by a coalition-proof supply function equilibrium.     

Preemption in a real option game with a first mover advantage and player-specific uncertainty

In this paper a two-player real option game with a first-mover advantage is analyzed, where payoffs are driven by a player-specific stochastic state variable. It is shown that there exists an equilibrium which has qualitatively different properties from those in standard real option games driven by common stochastic shocks. The properties of the equilibrium are four-fold: (i) preemption does not necessarily occur, (ii) if preemption takes place, the rent-equalization property holds, (iii) for almost all sample paths it is clear ex-ante which player invests first, and (iv) it is possible that both players invest simultaneously, even if that is not optimal. It is argued from simulations that real option games with a common one-dimensional shock do not provide a good approximation for games with player-specific uncertainty, even if these are highly correlated.     

Strategic interaction between futures and spot markets

We study an oligopolistic industry where firms are able to sell in a futures market at infinitely many moments prior to the spot market. A kind of Folk-theorem is established: any outcome between perfect competition and Cournot can be sustained in equilibrium. We then find that the Cournot outcome can be sustained by a renegotiation-proof equilibrium. However, this is not true for the competitive outcome. Furthermore, only the monopolistic outcome is renegotiation-proof if firms can buy and sell in the futures market. These results suggest, contrary to existing literature, that the introduction of futures markets may have an anti-competitive effect.     

Perfectly competitive markets as the limits of cournot markets

A perfectly competitive, partial equilibrium market for a single homogeneous good with a (bounded) continuum of infinitesimal firms is considered. Cost functions are essentially unrestricted and are allowed to vary smoothly across firms. A sequence (net) of Cournot markets (each with a finite number of firms) which converge smoothly to the perfectly competitive limit in terms of both the inverse demand functions and the distributioon of firm technologies is introduced and it is shown that all markets sufficiently far along the sequence have a Cournot equilibrium and all the Cournot equilibria converge to the perfectly competitive equilibrium of the limit market.     

Repeated games with one-memory

We study the extent to which equilibrium payoffs of discounted repeated games can be obtained by 1-memory strategies. We establish the following in games with perfect (rich) action spaces: First, when the players are sufficiently patient, the subgame perfect Folk Theorem holds with 1-memory. Second, for arbitrary level of discounting, all strictly enforceable subgame perfect equilibrium payoffs can be approximately supported with 1-memory if the number of players exceeds two. Furthermore, in this case all subgame perfect equilibrium payoffs can be approximately supported by an ε-equilibrium with 1-memory. In two-player games, the same set of results hold if an additional restriction is assumed: Players must have common punishments. Finally, to illustrate the role of our assumptions, we present robust examples of games in which there is a subgame perfect equilibrium payoff profile that cannot be obtained with 1-memory. Thus, our results are the best that can be hoped for.     

INTERTEMPORAL COURNOT AND WALRAS EQUILIBRIA: AN ILLUSTRATION*

In an intertemporal general equilibrium framework, we compare a Cournot equilibrium to the Walras equilibrium. The Cournot agents trade and invest less than the Walras agents. This generates an inefficiency that does not vanish as the number of Cournot agents tends to infinity. A larger number of strategic Cournot agents implies that the amount of trade (relative to their aggregate consumption) increases (i.e., it moves towards the Walrasian amount), but their investment (relative to the stock) decreases (i.e., it moves away from the Walrasian amount). "S'il y avait 3, 4, ...n producteurs en concurrence [...] la valeur de p, qui en résulte, diminuerait indéfiniment par l'accroissement indéfini du nombre n,”A. A. Cournot (1838, p. 63)."     

Stackelberg versus Cournot Equilibrium

We reconsider Stackelberg's classical critique of the Cournot duopoly, in the framework of endogenous timing for two-player games. For quantity duopoly we provide different sets of minimal conditions, directly on the demand and cost functions, yielding respectively the simultaneous and the two sequential modes of play. While our findings essentially confirm the predominance of the former, they also indicate that the latter is natural under some robust but restrictive conditions. No extraneous assumptions (such as concavity, existence, or uniqueness of equilibria…) are needed, and the analysis makes crucial use of the basic results from the theory of supermodular games.Journal of Economic LiteratureClassification Numbers: B21, C72, D43, L13.     

Competition and innovation: an experimental investigation

The paper analyzes the effects of more intense competition on firms’ investments in process innovations. More intense competition corresponds to an increase in the number of firms or a switch from Cournot to Bertrand competition. We carry out experiments for two-stage games, where R&D investment choices are followed by product market competition. An increase in the number of firms from two to four reduces investments, whereas a switch from Cournot to Bertrand increases investments, even though theory predicts a negative effect in the four-player case. The results arise both in treatments in which both stages are implemented and in treatments in which only one stage is implemented. However, the positive effect of moving from Cournot to Bertrand competition is more pronounced in the former case.     

A note on evolutionary stability of Bertrand equilibrium

It is shown that the equilibrium notion of an evolutionary stable strategy (ESS) does have predictive power for standard models of Bertrand competition. This is in contrast to a recent claim by Qin and Stuart (1997). The claim is based on the observation that the solution concept ESS behaves discontinuously when finite (discrete) action games approach an infinite (continuous) action game in the limit. Furthermore, it is argued that from a model-theoretic point of view evolutionary stability in prices (i.e. in the Bertrand model) is quite different from evolutionary stability in quantities (i.e. in the Cournot model).