Olivier Gossner

Positive value of information in games

Abstract.We exhibit a general class of interactive decision situations in which all the agents benefit from more information. This class includes as a special case the classical comparison of statistical experiments à la Blackwell.

The folk theorem for finitely repeated games with mixed strategies

This paper proves a Folk Theorem for finitely repeated games with mixed strategies. To obtain this result, we first show a similar property for finitely repeated games with terminal payoffs.

Simple bounds on the value of a reputation

We introduce entropy techniques to study the classical reputation model in which a long-run player faces a series of short-run players. The long-run players actions are possibly imperfectly observed. We derive explicit lower and upper bounds on the equilibrium payoffs to the long-run player.

How to play with a biased coin

We characterize the maxmin of repeated zero-sum games in which player one plays in pure strategies conditional on the private observation of a fixed sequence of random variables. Meanwhile we introduce a definition of a strategic distance between probability measures, and relate it to the standard Kullback distance.

Secret Correlation in Repeated Games with Imperfect Monitoring

We characterize the maximum payoff that a team can guarantee against another in a class of repeated games with imperfect monitoring. Our result relies on the optimal tradeoff for the team between optimization of stage payoffs and generation of signals for future correlation.

Empirical Distributions of Beliefs Under Imperfect Observation

Let (xn)n be a process with values in a finite set X and law P, and let yn f(xn) be a function of the process. At stage n, the conditional distribution pn P(xnx1,,xn1), element of (X), is the belief that a perfect observer, who observes the process online, holds on its realization at stage n. A statistician observing the signals y1,,yn holds a belief enP(pnx1,,xn) () on the possible predictions of the perfect observer. Given X and f, we characterize the set of limits of expected empirical distributions of the process (en) when P ranges over all possible laws of (xn)n.

On the complexity of coordination

Many results on repeated games played by finite automata rely on the complexity of the exact implementation of a coordinated play of length n. For a large proportion of sequences, this complexity appears to be no less than n. We study the complexity of a coordinated play when allowing for a few mismatches. We prove the existence of a constant C such that if (m ln m)/n ≥ C, for almost any sequence of length n, there exists an automaton of size m that achieves a coordination ratio close to 1 with it. Moreover, we show that one can take any constant C such that C > e|X| ln |X|, where |X| is the size of the alphabet from which the sequence is drawn. Our result contrasts with Neyman (1997) that shows that when (m ln m)/n is close to 0, for almost no sequence of length n there exists an automaton of size m that achieves a coordination ratio significantly larger 1/|X| with it.

Impermanent types and permanent reputations

We study the impact of unobservable stochastic replacements for the long-run player in the classical reputation model with a long-run player and a series of short-run players. We provide explicit lower bounds on the Nash equilibrium payoffs of a long-run player, both ex-ante and following any positive probability history. Under general conditions on the convergence rates of the discount factor to one and of the rate of replacement to zero, both bounds converge to the Stackelberg payoff if the type space is sufficiently rich. These limiting conditions hold in particular if the game is played very frequently.

Entropy and the Value of Information for Investors

Consider any investor who fears ruin facing any set of investments that satisfy no-arbitrage. Before investing, he can purchase information about the state of nature in the form of an information structure. Given his prior, information structure is more informative than information structure if whenever he rejects at some price, he also rejects at that price. We show that this complete informativeness ordering is represented by the decrease in entropy of his beliefs, regardless of his preferences, initial wealth or investment problem. It is also shown that no prior-independent informativeness ordering based on similar premises exists.

Informational cascades elicit private information

We introduce cheap talk in a dynamic investment model with information externalities. We first show how social learning adversely affects the credibility of cheap talk messages. Next, we show how an informational cascade makes truth-telling incentive compatible. A separating equilibrium only exists for high-surplus projects. Both an investment subsidy and an investment tax can increase welfare. The more precise the senders information, the higher her incentives to truthfully reveal her private information.