We review the current state of the art of methods for numerical computation of Nash equilibria for finite n-person games. Classical path following methods, such as the Lemke-Howson algorithm for two person games, and Scarf-type fixed point algorithms for n-person games provide globally convergent methods for finding a sample equilibrium. For large problems, methods which are not globally convergent, such as sequential linear complementarity methods may be preferred on the grounds of speed. None of these methods are capable of characterizing the entire set of Nash equilibria. More computationally intensive methods, which derive from the theory of semi-algebraic sets are required for finding all equilibria. These methods can also be applied to compute various equilibrium refinements.

According to conventional wisdom, Nash equilibrium in a game “involves” common knowl-edge of the payoff functions, of the rationality of the players, and of the strategies played. The basis for this wisdom is explored, and it turns out that considerably weaker conditions suffice. First, note that if each player is rational and knows his own payoff function, and the strategy choices of the players are mutually known, then these choices form a Nash equilibrium. The other two results treat the mixed strategies of a player not as conscious randomization of that player, but as conjectures of the other players about what he will do. When n = 2, mutual knowledge of the payoff functions, of rationality, and of the conjectures yields Nash equilibrium. When n ≥ 3, mutual knowledge of the payoff functions and of rationality, and common knowl-edge of the conjectures yield Nash equilibrium when there is a common prior. Examples are provided showing these results to be sharp.

We introduce the concept of a sink equilibrium. A sink equilibrium is a strongly connected component with no outgoing arcs in the strategy profile graph associated with a game. The strategy profile graph has a vertex set induced by the set of pure strategy profiles; its arc set corresponds to transitions between strategy profiles that occur with nonzero probability. (Here our focus will just be on the special case in which the strategy profile graph is actually a best response graph; that is, its arc set corresponds exactly to best response moves that result from myopic or greedy behaviour.) We argue that there is a natural convergence process to sink equilibria in games where agents use pure strategies. This leads to an alternative measure of the social cost of a lack of coordination, the price of sinking, which

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We present GAMUT 1, a suite of game generators designed for testing game-theoretic algorithms. We explain why such a generator is necessary, offer a way of visualizing relationships between the sets of games supported by GAMUT, and give an overview of GAMUT’s architecture. We highlight the importance of using comprehensive test data by benchmarking existing algorithms. We show surprisingly large variation in algorithm performance across different sets of games for two widely-studied problems: computing Nash equilibria and multiagent learning in repeated games. 2 1.

This paper is a survey and exposition of linear methods for finding Nash equilibria. Above all, these apply to games with two players. In an equilibrium of a twoperson game, the mixed strategy probabilities of one player equalize the expected payoffs for the pure strategies used by the other player. This defines an optimization problem with linear constraints. We do not consider nonlinear methods like simplicial subdivision for approximating fixed points, or systems of inequalities for higher-degree polynomials as they arise for noncooperative games with more than two players. These are surveyed in McKelvey and McLennan (1996)

Action-graph games (AGGs) are a fully expressive game representation which can compactly express both strict and context-specific independence between players' utility functions. Actions are represented as nodes in a graph G, and the payoff to an agent who chose the action s depends only on the numbers of other agents who chose actions connected to s. We present algorithms for computing both symmetric and arbitrary equilibria of AGGs using a continuation method. We analyze the worst-case cost of computing the Jacobian of the payoff function, the exponential-time bottleneck step, and in all cases achieve exponential speedup. When the indegree of G is bounded by a constant and the game is symmetric, the Jacobian can be computed in polynomial time.

In this paper we consider online auction mechanisms for the allocation of M items that are identical to each other except for the fact that the items have dierent expiration times, and each item must be allocated before it expires. A computational application is the allocation of time slots in a scheduling problem, and an economic application is the allocation of transportation tickets.

. We propose the sequence form as a new strategic description for an extensive game with perfect recall. It is similar to the normal form but has linear instead of exponential complexity, and allows a direct representation and efficient computation of behavior strategies. Pure strategies and their mixed strategy probabilities are replaced by sequences of consecutive choices and their realization probabilities. A zero-sum game is solved by a corresponding linear program that has linear size in the size of the game tree. General two-person games are studied in the paper by Koller, Megiddo, and von Stengel in this journal issue. Journal of Economic Literature Classification Number: C72 Keywords. Behavior strategy, equilibrium, extensive game, linear programming, normal form, reduced normal form. 1. Introduction In applications, it is often convenient to describe a game in extensive form. The game tree, with its information sets, possible moves, chance probabilities and payoffs, gives a...

We describe algorithms for computing Nash equilibria in structured game representations, including both graphical games and multi-agent influence diagrams (MAIDs). The algorithms are derived from a continuation method for normal-form and extensive-form games due to Govindan and Wilson; they follow a trajectory through the space of perturbed games and their equilibria. Our algorithms exploit game structure through fast computation of the Jacobian of the game's payoff function. They are guaranteed to find at least one equilibrium of the game and may find more. Our approach provides the first exact algorithm for computing an exact equilibrium in graphical games with arbitrary topology, and the first algorithm to exploit fine-grain structural properties of MAIDs. We present experimental results for our algorithms. The running time for our graphical game algorithm is similar to, and often better than, the running time of previous approximate algorithms. Our algorithm for MAIDs can effectively solve games that arc much larger than those that could be solved using previous methods. 1