FORGES Francoise

We consider a sender-receiver game, in which the sender has finitely many types and the receiver's decision is a real number. We assume that utility functions are concave, single-peaked and single-crossing. After the cheap talk phase, the receiver makes a decision, which requires the sender's approval to be implemented. Otherwise, the sender "exits". At a perfect Bayesian equilibrium without exit, the receiver must maximize his expected utility subject to the participation constraints of all positive probability types. This necessary condition may not hold at the receiver's prior belief, so that a non-revealing equilibrium may fail to exist. Similarly, a fully revealing equilibrium may not exist either due to the sender's incentive compatibility conditions.We propose a constructive algorithm that always achieves a perfect Bayesian equilibrium without exit.

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We consider a sender-receiver game with an outside option for the sender. After the cheap talk phase, the receiver makes a proposal to the sender, which the latter can reject. We study situations in which the sender’s approval is crucial to the receiver. We show that a partitional, (perfect Bayesian Nash) equilibrium exists if the sender has only two types or if the receiver’s preferences over decisions do not depend on the type of the sender as long as the latter participates. The result does not extend: we construct a counter-example (with three types for the sender and type-dependent affine utility functions) in which there is no mixed equilibrium. In the three type case, we provide a full characterization of (possibly mediated) equilibria.

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We consider generalized sender–receiver games in which the sender also has an action to choose, but this action is payoff-relevant only to himself. We study “cooperate and talk” equilibria (CTE) in which, before sending his message, the sender can commit to delegate his decision to the receiver. CTE are beneficial to the receiver (with respect to no communication) and unlike the equilibria of the plain cheap talk game, preserve him from afterwards regret. While existence of CTE cannot be guaranteed in general, a sufficient condition is that the receiver has a “uniform punishment decision” against the sender.

Starting from a Bayesian game that accounts for the interaction of agents in the absence of any cooperative agreement, we review a few scenarios that allow players to constrain their decisions through commitments. We compare the effects of these different scenarios and suggest some links with contract theory.

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We define feasible, posterior individually rational solutions for two-person Bayesian games with a single informed player. Such a solution can be achieved by direct signalling from the informed player and requires approval of both players after the signal has been sent. Without further assumptions on the Bayesian game, a solution does not necessarily exist. We show that, if the uninformed player has a “uniform punishment strategy” against the informed one, the existence of a solution follows from the existence of Nash equilibrium in infinitely repeated games with lack of information on one side. We also consider the extension of the result when both players have private information.

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We show that in a standard symmetric Cournot duopoly with unknown demand, the optimal information disclosure policy of an informed benevolent planner is to fully inform one of the duopolists and disclose no information to the other one. We discuss possible extensions of the result.

We consider two-person undiscounted and discounted infinitely repeated games in which every player privately knows his own payoffs (private values). Under a further assumption (existence of uniform punishment strategies), the Nash equilibria of the Bayesian infinitely repeated game without discounting are payoff-equivalent to tractable, completely revealing, equilibria. This characterization does not apply to discounted games with sufficiently patient players. We show that in a class of public good games, the set of Nash equilibrium payoffs of the undiscounted game can be empty, while limit (perfect Bayesian) Nash equilibrium payoffs of the discounted game, as players become increasingly patient, do exist. These equilibria share some features with the ones of two-sided reputation models.

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We identify a natural counterpart of the standard GARP for demand data in which goods are all indivisible. We show that the new axiom (DARP, for "discrete axiom of revealed preference") is necessary and sufficient for the rationalization of the data by a well-behaved utility function. Our results complement the main finding of Polisson and Quah, Am. Econ. J.: Micro. 5(1) p.28-34 (2013), who rather minimally modify the original consumer problem with indivisible goods so that the standard GARP still applies.

The folk theorem characterizes the (subgame perfect) Nash equilibrium payoffs of an undiscounted or discounted infinitely repeated game - with fully informed, patient players - as the feasible individually rational payoffs of the one-shot game. To which extent does the result still hold when every player privately knows his own payoffs ? Under appropriate assumptions (private values and uniform punishments), the Nash equilibria of the Bayesian infinitely repeated game without discounting are payoff equivalent to tractable, completely revealing, equilibria and can be achieved as interim cooperative solutions of the initial Bayesian game. This characterization does not apply to discounted games with sufficiently patient players. In a class of public good games, the set of Nash equilibrium payoffs of the undiscounted game can be empty, while limit (perfect Bayesian) Nash equilibrium payoffs of the discounted game, as players become infinitely patient, do exist. These equilibria share some features with the ones of multi-sided reputation models.

We define feasible, posterior individually rational solutions for two-person Bayesian games with a single informed player. Such a solution can be achieved by direct signalling from the informed player and requires approval of both players after the signal has been sent. Without further assumptions on the Bayesian game, a solution does not necessarily exist. We show that, if the uninformed player has a "uniform punishment strategy"against the informed one, the existence of a solution follows from the existence of Nash equilibrium in infinitely repeated games with lack of information on one side. We consider the extension of the result when both players have private information. We are grateful to.

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According to a minimalist version of Afriat's theorem, a consumer behaves as a utility maximizer if and only if a feasibility matrix associated with his choices is cyclically consistent. An essential experiment consists of observed consumption bundles (x_1,., x_n) and a feasibility matrix \alpha. Starting with a standard experiment, in which the economist has access to precise budget sets, we show that the necessary and sufficient condition for the existence of a utility function rationalizing the experiment, namely, the cyclical consistency of the associated feasibility matrix, is equivalent to the existence, for any budget sets compatible with the deduced essential experiment, of a utility function rationalizing them (and typically depending on them). In other words, the conclusion of the standard rationalizability test, in which the economist takes budget sets for granted, does not depend on the full specification of the underlying budget sets but only on the essential data that these budget sets generate. Starting with an essential experiment (x_1,., x_n. alpha) only, we show that the cyclical consistency of alpha, together with a further consistency condition involving both (x_1,., x_n) and alpha, guarantees the existence of a budget representation and that the essential experiment is rationalizable almost robustly, in the sense that there exists a single utility function which rationalizes at once almost all budget sets which are compatible with (x_1,., x_n. alpha). The conditions are also trivially necessary.

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Alvin Roth and Lloyd Shapley were awarded the 2012 Royal Bank of Sweden Prize in Economic Science in memory of Alfred Nobel for their work on the centralized organization of certain economic markets, which depend on matching agents of two distinct types (students and schools, for example). Shapley co-authored, with David Gale, the seminal paper in the field, which proposed an algorithm for achieving stable matching. Roth led the redesign of the U.S. hospital intern assignment process and the design of a kidney transplant market. After reporting on these contributions, we also discuss Shapley's seminal role in game theory.

This is a brief survey describing some of the recent progress and open problems in the area of cooperative games with incomplete information. We discuss exchange economies, cooperative Bayesian games with orthogonal coalitions, and issues of cooperation in non-cooperative Bayesian games.

The article shows how Jean-François Mertens contributed to the development of game theory and microeconomics as we know it today. Over the course of about eighty papers, the topics Mertens covered range from the formulation of Bayesian decision making in games with incomplete information to the foundations of "cost-benefit" analysis of long-run economic policies and include, among others, the selection of Nash equilibria in non-cooperative games, the existence of solutions for stochastic games, and the extension of Shapley's value in cooperative games with infinitely many negligible agents.

Alvin Roth and Lloyd Shapley were awarded the 2012 Royal Bank of Sweden Prize in Economic Science in memory of Alfred Nobel for their work on the centralized organization of certain economic markets, which depend on matching agents of two distinct types (students and schools, for example). Shapley co-authored, with David Gale, the seminal paper in the field, which proposed an algorithm for achieving stable matching. Roth led the redesign of the hospital internship assignment process in the United States and the design of a market related to kidney transplantation. After reporting on these contributions, we also discuss Shapley's seminal role in game theory.

The set of all Bayesian–Nash equilibrium payoffs that the players can achieve by making conditional commitments at the interim stage of a Bayesian game coincides with the set of all feasible, incentive compatible and interim individually rational payoffs of the Bayesian game. Furthermore, the various equilibrium payoffs, which are achieved by means of different commitment devices, are also the equilibrium payoffs of a universal, deterministic commitment game.

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We show that essentially every communication equilibrium of any finite Bayesian game with two players can be implemented as a strategic form correlated equilibrium of an extended game, in which before choosing actions as in the Bayesian game, the players engage in a possibly infinitely long (but in equilibrium almost surely finite), direct, cheap talk.

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The correlated equilibrium is a game theoretic solution concept. It was proposed by Aumann (1974, 1987) in order to capture the strategic correlation opportunities that the players face when they take into account the extraneous environment in which they interact. The notion is illustrated in Section II. A formal definition is given in Section III. The correlated equilibrium also appears as the appropriate solution concept if preplay communication is allowed between the players. As shown in Section IV, this property can be given several precise statements according to the constraints imposed on the players communication, which can go from plain conversation to exchange of messages through noisy channels. Originally designed for static games with complete information, the correlated equilibrium applies to any strategic form game. It is geometrically and computationally more tractable than the better known Nash equilibrium. The solution concept has been extended to dynamic games, possibly with incomplete information. As an illustration, we define in details the communication equilibrium for Bayesian games in Section V.

We study the issue of collusion in auctions. We begin by presenting a model of first-price auctions under full information and non-symmetric direct externalities. We follow a non-cooperative approach by studying the bargaining process that describes the formation of a cartel. We show that in the presence of direct externalities the formation of the grand coalition is not guaranteed, by proposing an example of an auction in which a small coalition forms in equilibrium. We continue by studying the stability (in the core sense) of coalitions in Bayesian games. We show that any coalitional equilibrium is incentive-free without loss of generality. We thus apply the notion of stability to joint auction procedures without direct externalities, establishing (especially) the stability of the grand coalition. With direct externalities in full information we show that the grand coalition (as well as a smaller coalition) can become unstable. We end by examining the notion of stability in second price auctions with direct externalities in incomplete information. We identify a class of manageable equilibria in these auctions for any given form of collusion. Finally, in this model, we demonstrate the instability of the grand coalition in the presence of direct externalities, again identifying direct externalities as an obstacle to cooperation.