The Complexity of the Homotopy Method, Equilibrium Selection, and Lemke-Howson Solutions

We show that the widely used homotopy method for solving fixpoint problems, as well as the Harsanyi-Selten equilibrium selection process for games, are PSPACE-complete to implement. Extending our result for the Harsanyi-Selten process, we show that several other homotopy-based algorithms for finding equilibria of games are also PSPACE-complete to implement. A further application of our techniques yields the result that it is PSPACE-complete to compute any of the equilibria that could be found via the classical Lemke-Howson algorithm, a complexity-theoretic strengthening of the result in [Savani and von Stengel]. These results show that our techniques can be widely applied and suggest that the PSPACE-completeness of implementing homotopy methods is a general principle.Comment: 23 pages, 1 figure; to appear in FOCS 2011 conferenc

Structural Robustness of Large Games

This short survey discusses recent findings on the robustness of Nash equilibria of strategic games with many semianonymous players. It describes the notion of structural robustness and its general consequences, as well as implications to particular games, such as ones played on the web and market games.Nash Equilibrium,ex-post Nash, anonymous games, market games, rational expectations, structural robustness, information proofness, web games

Approximate Fixed Point Theorems in Banach Spaces with Applications in Game Theory

In this paper some new approximate fixed point theorems for multifunctions in Banach spaces are presented and a method is developed indicating how to use approximate fixed point theorems in proving the existence of approximate Nash equilibria for non-cooperative games.Nash equilibrium;noncooperative games

Approximate Equilibria for Bayesian Multi-Criteria Games

AMS Classification: 91A10non-cooperative games;Bayesian multi-criteria equilibria

Private Information in Large Games

Nash equilibrium, ex-post Nash,anonymous games, market games, rational expectations, extensive robustness, information proofness, web games

The Quality of Equilibria for Set Packing Games

We introduce set packing games as an abstraction of situations in which $n$ selfish players select subsets of a finite set of indivisible items, and analyze the quality of several equilibria for this class of games. Assuming that players are able to approximately play equilibrium strategies, we show that the total quality of the resulting equilibrium solutions is only moderately suboptimal. Our results are tight bounds on the price of anarchy for three equilibrium concepts, namely Nash equilibria, subgame perfect equilibria, and an equilibrium concept that we refer to as $k$-collusion Nash equilibrium