We present a simple primal-dual algorithm for computing approximate Nash equilibria in two-person zero-sum sequential games with incomplete information and perfect recall (like Texas Hold’em poker). Our algorithm only performs basic iterations (i.e matvec multiplications, clipping, etc., and no calls to external first-order oracles, no matrix inversions, etc.) and is applicable to a broad class...

Zero-sum stochastic games provide a formalism to study competitive sequential interactions between two agents with diametrically opposing goals and evolving state. A solution to such games with discrete state was presented by Littman (Littman, 1994). The continuous state version of this game remains unsolved. In many instances continuous state solutions require nonlinear optimisation, a problem...

Laboratory subjects repeatedly played one of two variations of a simple two-person zero-sum game of ‘‘hide and seek’’. Three puzzling departures from the prescriptions of equilibrium theory are found in the data: an asymmetry related to the player’s role in the game; an asymmetry across the game variations; and positive serial correlation in subjects’ play. Possible explanations for these depar...

We investigate a two-player zero-sum stochastic differential game in which the players have an asymmetric information on the random payoff. We prove that the game has a value and characterize this value in terms of dual solutions of some second order Hamilton-Jacobi equation. Key-words : stochastic differential game, asymmetric information, viscosity solution. A.M.S. classification : 49N70, 49L...

We show that even when the information structure is independent of the state of nature, the value of the n-stage zero-sum game with incomplete information is not necessarily monotonic with respect to the length of the game. More precisely, we give an example of such an n-stage game in which v~>v~<v~.

In this document we provide a couple of exercises that help understand the basic concepts of game theory, including pure and mixed strategies, zero sum and non-zero sum games, and saddle-point and Nash solutions. The definition for concepts and the ideas for these exercises were taken in part from [1, 2]. Before we start, let us introduce some notation. We are going to assume that we have only ...

An infinite two-player zero-sum game with a Borel winning set, in which the opponent’s actions are monitored eventually but not necessarily immediately after they are played, admits a value. The proof relies on a representation of the game as a stochastic game with perfect information, in which Nature operates as a delegate for the players and performs the randomizations for them.

We prove a generalization of von Neumann’s minmax theorem to the class of separable multiplayer zerosum games, introduced in [Bregman and Fokin 1998]. These games are polymatrix—that is, graphical games in which every edge is a two-player game between its endpoints—in which every outcome has zero total sum of players’ payoffs. Our generalization of the minmax theorem implies convexity of equili...

In this paper, we initiate a study on optimal control problem for stochastic differential games under generalized expectation via backward stochastic differential equations and partial information. We first prove a sufficient maximum principle for zero-sum stochastic differential game problem. And then extend our approach to general stochastic differential games (nonzero–sum games), and obtain ...