Pure Strategy Equilibria in Symmetric Two-Player Zero-Sum Games

We observe that a symmetric two-player zero-sum game has a pure strategy equilibrium if and only if it is not a generalized rock-paper-scissors matrix. Moreover, we show that every finite symmetric quasiconcave two-player zero-sum game has a pure equilibrium. Further sufficient conditions for existence are provided. Our findings extend to general two-player zero-sum games using the symmetrization of zero-sum games due to von Neumann. We point out that the class of symmetric two-player zero-sum games coincides with the class of relative payoff games associated with symmetric two-player games. This allows us to derive results on the existence of finite population evolutionary stable strategies.Symmetric two-player games, zero-sum games, Rock-Paper-Scissors, single-peakedness, quasiconcavity, finite population evolutionary stable strategy, saddle point, exact potential games

Non-Zero Sum Games for Reactive Synthesis

In this invited contribution, we summarize new solution concepts useful for the synthesis of reactive systems that we have introduced in several recent publications. These solution concepts are developed in the context of non-zero sum games played on graphs. They are part of the contributions obtained in the inVEST project funded by the European Research Council.Comment: LATA'16 invited pape

Geometry, Correlated Equilibria and Zero-Sum Games

Ce papier porte à la fois sur la géométrie des équilibres de Nash et des équilibres corrélés et sur une généralisation des jeux à sommes nulles fondée sur les équilibres corrélés. L'ensemble des distributions d'équilibres corrélés de n'importe quel jeu fini est un polytope, qui contient les équilibres de Nash. Je caractérise la classe des jeux tels que ce polytope (s'il ne se réduit pas à un singleton) contienne un équilibre de Nash dans son intérieur relatif. Bien que cette classe de jeux ne soit pas définie par une propriété d'antagonisme entre les joueurs, je montre qu'elle inclut et qu'elle généralise la classe des jeux à deux joueurs et à somme nulle.Jeux à somme nulle;Equilibres corrélés;Géométrie

Continuous fictitious play in zero-sum games

Robinson (1951) showed that the learning process of Discrete Fictitious Play converges from any initial condition to the set of Nash equilibria in two-player zero-sum games. In several earlier works, Brown (1949, 1951) makes some heuristic arguments for a similar convergence result for the case of Continuous Fictitious Play (CFP). The standard reference for a formal proof is Harris (1998); his argument requires several technical lemmas, and moreover, involves the advanced machinery of Lyapunov functions. In this note we present a simple alternative proof. In particular, we show that Brown''s convergence result follows easily from a result obtained by Monderer et al. (1997).mathematical economics;

Survival of the Fittest and Zero Sum Games

Competition for available resources is natural amongst coexisting species, and the fittest contenders dominate over the rest in evolution. The dynamics of this selection is studied using a simple linear model. It has similarities to features of quantum computation, in particular conservation laws leading to destructive interference. Compared to an altruistic scenario, competition introduces instability and eliminates the weaker species in a finite time.Comment: 6 pages, formatted according to journal style. Special Issue on Game Theory and Evolutionary Processes. (v2) Published version. Some clarifications added. Topological interpretation pointed ou

Open-ended Learning in Symmetric Zero-sum Games

Zero-sum games such as chess and poker are, abstractly, functions that evaluate pairs of agents, for example labeling them `winner' and `loser'. If the game is approximately transitive, then self-play generates sequences of agents of increasing strength. However, nontransitive games, such as rock-paper-scissors, can exhibit strategic cycles, and there is no longer a clear objective -- we want agents to increase in strength, but against whom is unclear. In this paper, we introduce a geometric framework for formulating agent objectives in zero-sum games, in order to construct adaptive sequences of objectives that yield open-ended learning. The framework allows us to reason about population performance in nontransitive games, and enables the development of a new algorithm (rectified Nash response, PSRO_rN) that uses game-theoretic niching to construct diverse populations of effective agents, producing a stronger set of agents than existing algorithms. We apply PSRO_rN to two highly nontransitive resource allocation games and find that PSRO_rN consistently outperforms the existing alternatives.Comment: ICML 2019, final versio