Fast algorithms for finding randomized strategies in game trees

Abstract

Interactions among agents can be conveniently described by game trees. In order to analyze a game, it is important to derive optimal (or equilibrium) strategies for the different players. The standard approach to finding such strategies in games with imperfect information is, in general, computationally intractable. The approach is to generate the normal form of the game (the matrix containing the payoff for each strategy combination), and then solve a linear program (LP) or a linear complementarity problem (LCP). The size of the normal form, however, is typically exponential in the size of the game tree, thus making this method impractical in all but the simplest cases. This paper describes a new representation of strategies which results in a practical linear formulation of the problem of two-player games with perfect recall (i.e., games where players never forget anything, which is a standard assumption). Standard LP or LCP solvers can then be applied to find optimal randomized strategies. The resulting algorithms are, in general, exponentially better than the standard ones, both in terms of time and in terms of space.

Computing Proper Equilibria of Zero-Sum Games

“…However, the conversion from extensive to normal form incurs an exponential blowup in the size of the representation. Koller, Megiddo and von Stengel [8] showed how to use sequence form representation to efficiently compute minimax strategies for two-player extensive-form zero-sum games with imperfect information but perfect recall. The minimax strategies can be found from the sequence form by solving a linear program of size linear in the size of the game tree, avoiding the conversion to normal form altogether.…”

Bilateral bargaining with one-sided uncertain reserve prices

“…Computer science researchers have proposed a number of algorithms for computing Nash equilibria (e.g., [22,23,26]) or sequential equilibria (e.g., [20,29]). However, these algorithms are not applicable in solving bargaining games since they only consider finite strategy space rather than continuous strategy space considered in this paper.…”

Computing a quasi-perfect equilibrium of a two-player game

“…We mimic the derivation of Koller, Megiddo and von Stengel (1994) for the unperturbed case. If G is a zero-sum game, so is G(ǫ), and so the set of Nash equilibria of G(ǫ) is the Cartesian product of the sets of minimax strategies for each of the two players.…”

“…Instead, the linear complementarity program they devise has roughly the same number of variables and constraints as the number of information sets of the game considered and can therefore be used in practice to solve rather large games. For the case of zero-sum games, Koller, Megiddo and von Stengel (1994) gave a variant of their algorithm based on linear programming rather than linear complementarity programming. This version of the algorithm is motivated by the fact that linear programs are more efficiently solvable than linear complementarity programs, both in theory and in practice.…”