Nicolas Vieille

Two-player stochastic games II: The case of recursive games

This paper contains the second step in the proof of existence of equilibrium payoffs for two-player stochastic games. It deals with the case of positive absorbing recursive games

Two-player stochastic games I: A reduction

This paper is the first step in the proof of existence of equilibrium payoffs for two-player stochastic games with finite state and action sets. It reduces the existence problem to the class of so-called positive absorbing recursive games. The existence problem for this class is solved in a subsequent paper.

Deterministic multi-player Dynkin games

A multi-player Dynkin game is a sequential game in which at every stage one of the players is chosen, and that player can decide whether to continue the game or to stop it, in which case all players receive some terminal payoff. We study a variant of this model, where the order by which players are chosen is deterministic, and the probability that the game terminates once the chosen player decides to stop may be strictly less than one. We prove that a subgame-perfect e-equilibrium in Markovian strategies exists. If the game is not degenerate this e-equilibrium is actually in pure strategies.

Informational externalities and emergence of consensus

We study a general model of dynamic games with purely informational externalities. We prove that eventually all motives for experimentation disappear, and provide the exact rate at which experimentation decays. We also provide tight conditions under which players eventually reach a consensus. These results imply extensions of many known results in the literature of social learning and getting to agreement.

Correlated equilibrium in stochastic games

We study the existence of uniform correlated equilibrium payoffs in stochastic games. The correlation devices that we use are either autonomous (they base their choice of signal on previous signals, but not on previous states or actions) or stationary (their choice is independent of any data and is drawn according to the same probability distribution at every stage). We prove that any n-player stochastic game admits an autonomous correlated equilibrium payoff. When the game is positive and recursive, a stationary correlated equilibrium payoff exists. Journal of Economic Literature Classification Numbers: C72, C73.

Stochastic Games with a Single Controller and Incomplete Information

We study stochastic games with incomplete information on one side, in which the transition is controlled by one of the players. We prove that if the informed player also controls the transitions, the game has a value, whereas if the uninformed player controls the transitions, the max-min value as well as the min-max value exist, but they may differ. We discuss the structure of the optimal strategies, and provide extensions to the case of incomplete information on both sides.

Continuous-time games of timing

We address the question of existence of equilibrium in general timing games with complete information. Under weak assumptions, any two-player timing game has a Markov subgame perfect var epsilon-equilibrium, for each var epsilon>0. This result is tight. For some classes of games (symmetric games, games with cumulative payoffs), stronger existence results are established.

Aggregation of multiple prior opinions

Experts are asked to provide their advice in a situation of uncertainty. They adopt the decision makerʼs utility function, but each has a potentially different set of prior probabilities, and so does the decision maker. The decision maker and the experts maximize the minimal expected utility with respect to their sets of priors. We show that a natural Pareto condition is equivalent to the existence of a set Λ of probability vectors over the experts, interpreted as possible allocations of weights to the experts, such that (i) the decision makerʼs set of priors is precisely all the weighted-averages of priors, where an expertʼs prior is taken from her set and the weight vector is taken from Λ; (ii) the decision makerʼs valuation of an act is the minimal weighted valuation, over all weight vectors in Λ, of the expertsʼ valuations.

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.

The MaxMin value of stochastic games with imperfect monitoring

Abstract.We study finite zero-sum stochastic games in which players do not observe the actions of their opponent. Rather, at each stage, each player observes a stochastic signal that may depend on the current state and on the pair of actions chosen by the players. We assume that each player observes the state and his/her own action. We prove that the uniform max-min value always exists. Moreover, the uniform max-min value is independent of the information structure of player 2. Symmetric results hold for the uniform min-max value.