Abstract
We investigate the problem of equilibrium computation for “large” n-player games. Large games have a Lipschitz-type property that no single player’s utility is greatly affected by any other individual player’s actions. In this paper, we mostly focus on the case where any change of strategy by a player causes other players’ payoffs to change by at most frac {1}{n}. We study algorithms having query access to the game’s payoff function, aiming to find ε-Nash equilibria. We seek algorithms that obtain ε as small as possible, in time polynomial in n. Our main result is a randomised algorithm that achieves ε approaching frac {1}{8} for 2-strategy games in a completely uncoupled setting, where each player observes her own payoff to a query, and adjusts her behaviour independently of other players’ payoffs/actions. O(log n) rounds/queries are required. We also show how to obtain a slight improvement over frac {1}{8}, by introducing a small amount of communication between the players. Finally, we give extension of our results to large games with more than two strategies per player, and alternative largeness parameters.
Highlights
In studying the computation of solutions of multi-player games, we encounter the well-known problem that a game’s payoff function has description length exponential in the number of players
We study the computation of approximate Nash equilibria of multiplayer games having the feature that if a player changes her behaviour, she only has a small effect on the payoffs that result to any other player
Our main result applies in the setting of completely uncoupled dynamics in equilibria computation
Summary
In studying the computation of solutions of multi-player games, we encounter the well-known problem that a game’s payoff function has description length exponential in the number of players. We study the computation of approximate Nash equilibria of multiplayer games having the feature that if a player changes her behaviour, she only has a small effect on the payoffs that result to any other player These games, sometimes called large games, or Lipschitz games, have recently been studied in the literature, since they model various real-world economic interactions; for example, an individual’s choice of what items to buy may have a small effect on prices, where other individuals are not strongly affected. Our main result applies in the setting of completely uncoupled dynamics in equilibria computation These dynamics have been studied extensively: Hart and Mas-Colell [13] show that there exist finite-memory uncoupled strategies that lead to pure Nash equilibria in every game where they exist. Both of these results motivate the study of specific subclasses of these games, such as the “large” games studied here
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