Computing pure strategy nash equilibria in compact symmetric games

Abstract

Abstract. We analyze the complexity of computing pure strategy Nash equilibria (PSNE) in symmetric games with a fixed number of actions. We restrict ourselves to "compact" representations, meaning that the number of players can be exponential in the representation size. We show that in the general case, where utility functions are represented as arbitrary circuits, the problem of deciding the existence of PSNE is NP-complete. For the special case of games with two actions, we show that there always exist a PSNE and give a polynomial-time algorithm for finding one. We then focus on a specific compact representation: piecewise-linear functions. We give polynomial-time algorithms for finding a sample PSNE and for counting the number of PSNE. Our approach makes use of Barvinok and Wood's rational generating function method [3], which enables us to encode the set of PSNE as a generating function of polynomial size.

Nash equilibria in symmetric graph games with partial observation

“…In the following definition we propose a fix to the above definition. The resulting definition is equivalent to the definitions in [37,38,39,40], but it is given in a form resembling Definition 35. Note that this formulation has similarities with our own definition of a symmetric game network (Definition 8).…”

“…In [10] and other sources where symmetric games are defined [37,38,39,29,40,41] the basic intuition of what a symmetric normal-form game should be is a game where all players have the same set of actions and the same utility functions in the sense that the utility of player should depend exactly on which action he chooses himself and on the number of other players choosing any particular action. However, the definition above does not seem to capture the scenario at hand and might even be erroneous.…”

“…For two players we agree with the definition, and indeed in [29,41] it is only defined for two players. In [37,38,39,40] however it is defined such that u i (a i , a −i ) = u j (a j , a −j ) whenever a i = a j and a −i = a −j . Here, a i means the action taken by player i and a −i means the set of actions taken by players other than player i.…”

The Positronic Economist: A Computational System for Analyzing Economic Mechanisms

“…Thus, games with large type or action spaces-such as combinatorial auctions-cannot be succinctly represented. There is very little work on representing games with implicitly specified action spaces- Koller and Milch (2003) and Ryan, Jiang, and Leyton-Brown (2010) are the two exceptions, unfortunately both without good implementations-but as this literature develops there may be opportunities for extending our encode-and-solve CMA approach to new game families.…”

Action-Graph Games

“…Another useful form of structure not generally captured by graphical games is dubbed anonymity; it holds when each agent's utility depends only on the number of agents who took each action, rather than on these agents' identities. 4 Recently, researchers such as Papadimitriou and Roughgarden (2008), Kalai (2005), Daskalakis and Papadimitriou (2007), Brandt et al (in press) and Ryan et al (2010) have explored the representational, computational and strategic benefits that can be derived from symmetry and anonymity assumptions.…”