Poisson.Cournot Games

We construct a Cournot model with uncertainty in the number of firms in the industry. We model such an uncertainty as a Poisson game and characterize the set of equilibria after deriving novel properties of the Poisson distribution. When marginal costs are zero, the number of equilibria increases with the expected number of firms (n) and for n ≥ 3 every equilibrium exhibits overproduction relative to the model with deterministic population size. For a fixed n, overproduction is robust to sufficiently small marginal costs. The set of equilibria can be Pareto ranked. If n ≥ 3, even the expected consumer surplus induced by the lowest quantity equilibrium is larger than the consumer surplus in the model without population uncertainty

Le Modèle de Cournot avec entrées aléatoires de firmes *

International audienceLe nom de Cournot est aujourd’hui mondialement connu et, à l’instar de celui de Walras, il n’est aucune personne qui se soit un tant soit peu intéressée à la discipline économique qui ne l’ait, à un titre ou à un autre, aperçu au moins une fois. Ses apports majeurs concernent au moins : 1/ la fonction de demande, 2/ l’élasticité, 3/ le coût marginal, 4/ le monopole, 5/ l’oligopole, et 6/ la concurrence. De plus, même si la réception de son œuvre économique ne fut pas simple, rapide et directe, comme le démontre notamment Sigot, 2005, il est actuellement presque unanimement considéré, a minima, comme l’un des pères fondateurs de l’application de la méthode mathématique à l’analyse économique, de l’analyse en termes d’équilibre partiel et d’équilibre général, du paradigme néo-classique, ainsi qu’un précurseur du concept d’équilibrenon-coopératif de Nas

Convergence of Large Atomic Congestion Games

We consider the question of whether, and in what sense, Wardrop equilibria provide a good approximation for Nash equilibria in atomic unsplittable congestion games with a large number of small players. We examine two different definitions of small players. In the first setting, we consider a sequence of games with an increasing number of players where each player's weight tends to zero. We prove that all (mixed) Nash equilibria of the finite games converge to the set of Wardrop equilibria of the corresponding nonatomic limit game. In the second setting, we consider again an increasing number of players but now each player has a unit weight and participates in the game with a probability tending to zero. In this case, the Nash equilibria converge to the set of Wardrop equilibria of a different nonatomic game with suitably defined costs. The latter can also be seen as a Poisson game in the sense of Myerson (1998), establishing a precise connection between the Wardrop model and the empirical flows observed in real traffic networks that exhibit stochastic fluctuations well described by Poisson distributions. In both settings we give explicit upper bounds on the rates of convergence, from which we also derive the convergence of the price of anarchy. Beyond the case of congestion games, we establish a general result on the convergence of large games with random players towards Poisson games.Comment: 34 pages, 3 figure

On Dynamic Games with Randomly Arriving Players

International audienceWe consider a dynamic game where additional players (assumed identical , even if there will be a mild departure from that hypothesis) join the game randomly according to a Bernoulli process. The problem solved here is that of computing their expected payoff as a function of time and the number of players present when they arrive, if the strategies are given. We consider both a finite horizon game and an infinite horizon, discounted game. As illustrations , we discuss some examples relating to oligopoly theory (Cournot, Stackelberg, cartel)