Polynomial Games Research Articles

Shapley Value of Homogeneous Cooperative Games

The paper gives a description of the integral representation of the Shapley value for polynomial cooperative games. This representation obtained using the so-called Shapley functional. The relationship between the proposed version of the Shapley value and the polar forms of homogeneous polynomial games is analyzed for both a finite and an infinite number of participants. Special attention is paid to certain classes of homogeneous cooperative games generated by products of non-atomic measures. A distinctive feature of the approach proposed is the systematic use of extensions of polynomial set functions to the corresponding measures on symmetric powers of the original measurable spaces.

On the Core and Shapley Value for Regular Polynomial Games

On the Core and Shapley Value for Regular Polynomial Games

Double Oracle Algorithm for Computing Equilibria in Continuous Games

Many efficient algorithms have been designed to recover Nash equilibria of various classes of finite games. Special classes of continuous games with infinite strategy spaces, such as polynomial games, can be solved by semidefinite programming. In general, however, continuous games are not directly amenable to computational procedures. In this contribution, we develop an iterative strategy generation technique for finding a Nash equilibrium in a whole class of continuous two-person zero-sum games with compact strategy sets. The procedure, which is called the double oracle algorithm, has been successfully applied to large finite games in the past. We prove the convergence of the double oracle algorithm to a Nash equilibrium. Moreover, the algorithm is guaranteed to recover an approximate equilibrium in finitely-many steps. Our numerical experiments show that it outperforms fictitious play on several examples of games appearing in the literature. In particular, we provide a detailed analysis of experiments with a version of the continuous Colonel Blotto game.

Characterizing the value functions of polynomial games

We provide a characterization of the set of real-valued functions that can be the value function of some polynomial game. Specifically, we prove that a function u:R→R is the value function of some polynomial game if and only if u is a continuous piecewise rational function.

Note on unique Nash equilibrium in continuous games

This note studies whether any set of finitely supported mixed strategies can be represented as the unique Nash equilibrium of a game. This note shows that if strategy spaces are metric spaces containing infinitely many points, then any set of finitely supported mixed strategies can be represented as the unique Nash equilibrium to a separable game. If the strategy spaces are additionally subsets of Euclidean space with infinitely many cluster points, then any set of finitely supported mixed strategies can be represented as the unique Nash equilibrium to a polynomial game.

Note on Unique Nash Equilibrium in Continuous Game

This note studies whether any set of finitely supported mixed strategies can be represented as the unique Nash equilibrium of a game. This note shows that if strategy spaces are metric spaces containing infinitely many points, then any set of finitely supported mixed strategies can be represented as the unique Nash equilibrium to a separable game. If the strategy spaces are additionally subsets of Euclidean space with infinitely many cluster points, then any set of finitely supported mixed strategies can be represented as the unique Nash equilibrium to a polynomial game.

Welfare-maximizing correlated equilibria using Kantorovich polynomials with sparsity

We provide motivations for the correlated equilibrium solution concept from the game-theoretic and optimization perspectives. We then propose an algorithm that computes $${\varepsilon}$$ -correlated equilibria with global-optimal (i.e., maximum) expected social welfare for normal form polynomial games. We derive an infinite dimensional formulation of $${\varepsilon}$$ -correlated equilibria using Kantorovich polynomials, and re-express it as a polynomial positivity constraint. We exploit polynomial sparsity to achieve a leaner problem formulation involving sum-of-squares constraints. By solving a sequence of semidefinite programming relaxations of the problem, our algorithm converges to a global-optimal $${\varepsilon}$$ -correlated equilibrium. The paper ends with two numerical examples involving a two-player polynomial game, and a wireless game with two mutually-interfering communication links.

Computation of Correlated Equilibrium with Global-Optimal Expected Social Welfare

In this paper, we propose an algorithm which computes the correlated equilibrium with global-optimal (i.e., maximum) expected social welfare for single stage polynomial games. We first derive tractable primal/dual semidefinite programming (SDP) relaxations for an infinite-dimensional formulation of correlated equilibria. We give an asymptotic convergence proof, which ensures solving the sequence of relaxations leads to solutions that converge to the correlated equilibrium with the highest expected social welfare. Finally, we give a dedicated sequential SDP algorithm and demonstrate it in a wireless application with numerical results.

Structure of extreme correlated equilibria: a zero-sum example and its implications

We exhibit the rich structure of the set of correlated equilibria by analyzing the simplest of polynomial games: the mixed extension of matching pennies. We show that while the correlated equilibrium set is convex and compact, the structure of its extreme points can be quite complicated. In finite games the ratio of extreme correlated to extreme Nash equilibria can be greater than exponential in the size of the strategy spaces. In polynomial games there can exist extreme correlated equilibria which are not finitely supported; we construct a large family of examples using techniques from ergodic theory. We show that in general the set of correlated equilibrium distributions of a polynomial game cannot be described by conditions on finitely many moments (means, covariances, etc.), in marked contrast to the set of Nash equilibria which is always expressible in terms of finitely many moments.

Welfare-Maximizing Correlated Equilibria with an Application to Wireless Communication

AbstractThe set of correlated equilibria is convex and contains all Nash equilibria as special cases. Thus, the social welfare-maximizing correlated equilibrium is amenable to convex analysis and offers social welfare that is at least as good as the game's best performing Nash equilibria. We employ robust semidefinite programming (SDP) for computing the social welfare-maximizing correlated equilibria in static polynomial games, giving rise to a dedicated sequential SDP algorithm, the first of this type that can cope with multivariate strategy sets. We apply this algorithm to a wireless communication problem, where two mutually-interfering transmitters and receivers maximize their channel capacities.

Semidefinite programming for min–max problems and games

We consider two min–max problems (1) minimizing the supremum of finitely many rational functions over a compact basic semi-algebraic set and (2) solving a 2-player zero-sum polynomial game in randomized strategies with compact basic semi-algebraic sets of pure strategies. In both problems the optimal value can be approximated by solving a hierarchy of semidefinite relaxations, in the spirit of the moment approach developed in Lasserre (SIAM J Optim 11:796–817, 2001; Math Program B 112:65–92, 2008). This provides a unified approach and a class of algorithms to compute Nash equilibria and min–max strategies of several static and dynamic games. Each semidefinite relaxation can be solved in time which is polynomial in its input size and practice on a sample of experiments reveals that few relaxations are needed for a good approximation (and sometimes even for finite convergence), a behavior similar to what was observed in polynomial optimization.

Separable and low-rank continuous games

In this paper, we study nonzero-sum separable games, which are continuous games whose payoffs take a sum-of-products form. Included in this subclass are all finite games and polynomial games. We investigate the structure of equilibria in separable games. We show that these games admit finitely supported Nash equilibria. Motivated by the bounds on the supports of mixed equilibria in two-player finite games in terms of the ranks of the payoff matrices, we define the notion of the rank of an n-player continuous game and use this to provide bounds on the cardinality of the support of equilibrium strategies. We present a general characterization theorem that states that a continuous game has finite rank if and only if it is separable. Using our rank results, we present an efficient algorithm for computing approximate equilibria of two-player separable games with fixed strategy spaces in time polynomial in the rank of the game.

Tomoyuki Yamakami. Polynomial games and determinacy, Annals of pure and applied logic, vol. 80 (1996), pp. 1–16.

Tomoyuki Yamakami. Polynomial games and determinacy, Annals of pure and applied logic, vol. 80 (1996), pp. 1–16.

Polynomial games and determinacy

Two-player, zero-sum, non-cooperative, blindfold games in extensive form with incomplete information are considered in this paper. Any information about past moves which players played is stored in a database, and each player can access the database. A polynomial game is a game in which, at each step, all players withdraw at most a polynomial amount of previous information from the database. We show resource-bounded determinacy of some kinds of finite, zero-sum, polynomial games whose pay-off sets are computable by non-deterministic polynomial-time function-oracle Turing machines. We call a pay-off set F -determined if, for any polynomial game G associated with the given pay-off set, either player has a winning strategy which is in F for any subgames of G. We show that there exists an FP-strongly-determined pay-off set which is computed by an exponential-time oracle Turing machine, where FP is the set of polynomial-time computable functions. We also discuss several relationships between the determinacy of polynomial games and recursion-theoretic properties for the classes co-NP and co-NEXP. We show that the polynomial version of the axiom of choice holds under some assumption of polynomial determinacy for a pay-off set which is polynomial-time computable with parallel queries. This principle of choice implies that co-NP has the separation property.

Computational complexity of winning strategies in two-person polynomial games

One considers two-person games, with players called I and II below. In order, they choose natural numbers, for example, for length 4, I chooses x1, II chooses x2. I chooses x3, II chooses x4. Then I wins if P(x1,x2,x3,x4)=0.Here P is a polynomial with integer coefficients. An old theorem of von Neumann and Zermelo shows that such a game is determined, i.e., there exists a winning strategy for one player or the other but not necessarily a computable winning strategy or one computable in polynomial time. It will be shown that there exists a game of polynomial type of length 4 for which there do not exist winning strategies for either player which are computable in polynomial time.