Common Knowledge and Consensus with Noisy Communication

Parikh and Krasucki (1990, JET 52) have suggested in an informal manner that a consensus does not require common knowledge. Weyers (1992, CORE DP 9228) has proved that their model does not permit such a conclusion, and that a more general one has to be constructed. Heifetz (1996, JET 70) has given an example with three agents, inspired by computer science works, which illustrates the intuition of the first authors, i.e. where a consensus is obtained without common knowledge of it. We propose a general setting of noisy communication to confirm this result. We show that for any non public and noisy communication, no event can become common knowledge if it was not at the beginning, but that under some assumptions a consensus and arbitrary high levels of interactive knowledge are achievable. A minimal example is given, with two agents and two states. Nevertheless, for public and noisy communication, some results on common knowledge are obtained, depending on the richness of available language. We apply our results to describe some conditions that ensure or prevent epistemic conditions for Nash equilibrium. In general, non public and noisy communication is not sufficient for the conjectures to form, during time, a Nash equilibrium, even if the game and mutual rationality are mutually known. However, with only two agents, or with a noisy and public communication protocol, sufficient conditions are given for the conjectures to form a Nash equilibrium in a finite number of communication periods.

Common Knowledge and Interactive Behaviors: A Survey

This paper surveys the notion of common knowledge taken from game theory and computer science. It studies and illustrates more generally the effects of interactive knowledge in economic and social problems. First of all, common knowledge is shown to be a central concept and often a necessary condition for coordination, equilibrium achievement, agreement, and consensus. We present how common knowledge can be practically generated, for example, by particular advertisements or leadership. Secondly, we prove that common knowledge can be harmful, essentially in various cooperation and negotiation problems, and more generally when there are con icts of interest. Finally, in some asymmetric relationships, common knowledge is shown to be preferable for some players, but not for all. The ambiguous welfare effects of higher-order knowledge on interactive behaviors leads us to analyze the role of decentralized communication in order to deal with dynamic or endogenous information structures.Interactive knowledge, common knowledge, information structure, communication.

Multistage communication with and without verifiable types

We survey the main results on strategic information transmission, which is often referred to as ``persuasion" when types are verifiable and as ``cheap talk" when they are not. In the simplest ``cheap talk'' model, an informed player sends a single message to a receiver who makes a decision. The players' utilities depend on the sender's information and the receiver's decision, but not on the sender's message. Furthermore, the messages that are available to the sender do not depend on his true information. As is well-known, such a unilateral ``cheap talk" can affect the sender's decision at equilibrium. In a more general model, both players can exchange simultaneous costless messages during several stages before the final decision. The utility functions are unchanged. Multistage conversation allows the players to reach more equilibrium outcomes, which possibly Pareto dominate the original ones. More precisely, the set of equilibrium outcomes of long cheap talk games is fully characterized; it increases with the number of communication stages and can become even larger if no deadline is imposed. Concentrating on cheap talk is not appropriate if the informed player can influence the decision maker by producing unfalsifiable documents. In order to capture this possibility formally, one assumes that the informed player's set of messages depends on his private information. The literature has mostly dealt with unilateral persuasion. But multistage, bilateral communication enables the players to reach more equilibrium outcomes in the case of verifiable types as in the case of unverifiable ones. Equilibria of long persuasion games are fully characterized when information can be certified at any precision level.Cheap talk; certification; incomplete information; information transmission; jointly controlled lotteries; verifiable types

Information Design in Large Anonymous Games

We consider anonymous Bayesian cost games with a large number of players, i.e., games where each player aims at minimizing a cost function that depends on the action chosen by the player, the distribution of the other players' actions and an unknown parameter. We study the nonatomic limit versions of these games. In particular, we introduce the concepts of correlated and Bayes correlated Wardrop equilibria, which extend the concepts of correlated and Bayes correlated equilibria to nonatomic games. We prove that (Bayes) correlated Wardrop equilibria are indeed limits of action flow distributions induced by (Bayes) correlated equilibria of the game with a large finite set of small players. For nonatomic games with complete information admitting a convex potential, we show that the set of correlated Wardrop equilibria is the set of probability distributions over Wardrop equilibria. Then, we study how to implement optimal Bayes correlated Wardrop equilibria and show that in games with a convex potential, every Bayes correlated Wardrop equilibrium can be fully implemented.Comment: 53 page

Transmission stratégique de l'information et certification (Strategic Information Transmission and Certification)

Nous présentons une synthèse des principaux modèles de transmission stratégique de l'information. Dans une première partie, nous étudions les jeux dits de "cheap-talk", c'est-à-dire les jeux de communication directe dans lesquels les messages sont gratuits et non vérifiables. Nous analysons ensuite les jeux dits de "persuasion", dans lesquels les joueurs sont capables de certifier leurs informations. Dans la dernière partie, nous montrons dans différents cadres comment plusieurs étapes de communication bilatérales et la certification retardée de l'information permettent d'étendre de manière significative l'ensemble des résultats d'équilibre, même si un seul individu détient des informations privées. L'article propose des caractérisations géométriques des équilibres, de nombreux exemples et des applications. (This paper surveys the main models of strategic information transmission. First, we study cheap talk games, i.e., direct communication games in which messages are costless and unverifiable. Then, we analyze persuasion games, in which players' information is certifiable. Finally, we show in various settings how multiple stages of bilateral communication and delayed information certification may significantly expand the set of equilibrium outcomes, even if only one individual is privately informed. Several geometric equilibrium characterizations, examples and applications are examined along the paper.)cheap talk, certification, credibility, incomplete information, information transmission, jointly controlled lotteries, verifiable types

Long Persuasion Games

This paper characterizes geometrically the set of all Nash equilibrium payoffs achievable with unmediated communication in persuasion games, i.e., games with an informed expert and an uninformed decisionmaker in which the expert's information is certifiable. The first equilibrium characterization is provided for unilateral persuasion games, and the second for multistage, bilateral persuasion games. As in Aumann and Hart (2003), we use the concepts of diconvexification and dimartingale. A leading example illustrates both geometric characterizations and shows how the expert, whatever his type, can increase his equilibrium payoff compared to all equilibria of the unilateral persuasion game by delaying information certification.Cheap talk, communication, diconvexification, dimartingale, disclosure of certifiable information, jointly controlled lotteries, long conversation, persuasion, verifiable types