Nash Equilibria In Finite Games Research Articles

A strategic learning algorithm for state-based games

Learning algorithm design and applications of state-based games are investigated. First, a heuristic uncoupled learning algorithm, which is a two memory better reply learning rule, is proposed. Under reachability conditions it is proved that for any initial state, if all agents in the state-based game follow the proposed learning algorithm, the action state pair converges almost surely to an action invariant set of recurrent state equilibria. The design of the learning algorithm relies on global and local searches with finite memory, inertia, and randomness. Then, existence of time-efficient universal learning algorithm is studied. Finally, applications of our proposed learning algorithm are discussed, including learning pure Nash equilibrium in finite games and cooperative control with time-varying communication structure.

Pure Nash Equilibria in Concurrent Deterministic Games

We study pure-strategy Nash equilibria in multi-player concurrent deterministic games, for a variety of preference relations. We provide a novel construction, called the suspect game, which transforms a multi-player concurrent game into a two-player turn-based game which turns Nash equilibria into winning strategies (for some objective that depends on the preference relations of the players in the original game). We use that transformation to design algorithms for computing Nash equilibria in finite games, which in most cases have optimal worst-case complexity, for large classes of preference relations. This includes the purely qualitative framework, where each player has a single omega-regular objective that she wants to satisfy, but also the larger class of semi-quantitative objectives, where each player has several omega-regular objectives equipped with a preorder (for instance, a player may want to satisfy all her objectives, or to maximise the number of objectives that she achieves.)

A survey on computation methods for Nash equilibrium

A dominant solution concept of non-cooperative game theory is the concept of Nash equilibrium. A Nash equilibrium is a strategy profile from where unilateral deviations do not pay. A nice property of this concept is the well known fact that every finite game has at least one Nash equilibrium. The proof given by Nash (1950) is based on Brouwer fixed point theorem which is very non-constructive. A natural question to ask is whether Nash equilibrium can be computed efficiently. This is still unknown in terms of complexity. Very recently (Daskalakis et al., 2006), it has been shown that the computation of Nash equilibrium is PPAD-complete; which is a new complexity class introduced by Papadimitriou (1994) mainly to capture the complexity of Nash equilibrium. This note tries to give a brief survey of the various algorithms that exist in the literature for computing the Nash equilibrium in finite games. Also, it will briefly touch upon the various existential conditions for Nash equilibrium for infinite games.

Axiomatic Equilibrium Selection for Generic Two-Player Games

We impose three conditions on refinements of the Nash equilibria of finite games with perfect recall that select closed connected subsets, called solutions. A. Each equilibrium in a solution uses undominated strategies; B. Each solution contains a quasi-perfect equilibrium; C. The solutions of a game map to the solutions of an embedded game, where a game is embedded if each player’s feasible strategies and payoffs are preserved by a multilinear map. We prove for games with two players and generic payoffs that these conditions characterize each solution as an essential component of equilibria in undominated strategies, and thus a stable set as defined by Mertens (1989).

Computation of Nash equilibria in finite games: introduction to the symposium

This is a collection of ten invited articles on the computation of Nash equilibria in finite games. Three (by Ruchira Datta, Jean-Jacques Herings and Ronald Peeters, and Tim Roughgarden) are longer surveys, the others present new research. All contributions attempt to make the material accessible and interesting to the non-specialist. The collection reflects in equal measure the state of research, as well as providing an exposition of the main ideas and questions of the field. The articles are also bridged by a unity of ideas that appear in many places. The topic of this collection is a computational problem: given a game in some finite description, find its Nash equilibria; or, a more modest task: find one Nash equilibrium of the game. These questions pose highly interesting challenges for economics, mathematics, and computer science. For the applied economist, noncooperative games are a modeling tool, with the Nash equilibrium as the central solution concept. Interpreting and discussing a Nash equilibrium is essential for the model, but the equilibrium has to be found first, which often takes a substantial amount of the analysis. A computer program for this analysis should save time and allow for more detailed modeling. Why should the applied economist, as a modeler and user of game theory, care about algorithms? It seems easy just to convert the equilibrium problem to a suitable optimization problem, run somemathematical software on it, and use the solution. The problem is that such a “black box algorithm” is unlikely to work on all but the smallest cases. Any algorithm that “scales” reasonably well in allowing for more detailed games as input, and still finishes in reasonable time, must exploit the mathematical structure of the problem.

A dynamic homotopy interpretation of the logistic quantal response equilibrium correspondence

This paper uses properties of the logistic quantal response equilibrium correspondence to compute Nash equilibria in finite games. It is shown that branches of the correspondence may be numerically traversed efficiently and securely. The method can be implemented on a multicomputer, allowing for application to large games. The path followed by the method has an interpretation analogous to that of Harsanyi and Selten's Tracing Proecdure. As an application, it is shown that the principal branch of any quantal response equilibrium correspondence satisfying a monotonicity property converges to the risk-dominant equilibrium in 2 × 2 games.

A global Newton method to compute Nash equilibria

A new algorithm is presented for computing Nash equilibria of finite games. Using Kohlberg and Mertens’ structure theorem we show that a homotopy method can be represented as a dynamical system and implemented by Smale's global Newton method. The algorithm is outlined and computational experience is reported.

Evolutionary Stability of Pure-Strategy Equilibria in Finite Games

Sufficient conditions for pure-strategy Nash equilibria of finite games to be (Lyapunov) stable under a large class of evolutionary dynamics, the regular monotonic selection dynamics, are discussed. In particular, it is shown that in almost all finite extensive-form games, all the pure-strategy equilibria are stable. In such games, all mixed-strategy equilibria close to pure-strategy equilibria are also stable.Journal of Economic LiteratureClassification Numbers: C70, C72.