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A Survey of PPADCompleteness for Computing Nash equilibria
, 2011
"... PPAD refers to a class of computational problems for which solutions are guaranteed to exist due to a specific combinatorial principle. The most wellknown such problem is that of computing a Nash equilibrium of a game. Other examples include the search for market equilibria, and envyfree allocatio ..."
Abstract

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PPAD refers to a class of computational problems for which solutions are guaranteed to exist due to a specific combinatorial principle. The most wellknown such problem is that of computing a Nash equilibrium of a game. Other examples include the search for market equilibria, and envyfree allocations in the context of cakecutting. A problem is said to be complete for PPAD if it belongs to PPAD and can be shown to constitute one of the hardest computational challenges within that class. In this paper, I give a relatively informal overview of the proofs used in the PPADcompleteness results. The focus is on the mixed Nash equilibria guaranteed to exist by Nashâ€™s theorem. I also give an overview of some recent work that uses these ideas to show PSPACEcompleteness for the computation of specific equilibria found by homotopy methods. I give a brief introduction to related problems of searching for market equilibria.
Linear Solvers for Nonlinear Games: Using Pivoting Algorithms to Find Nash Equilibria in nPlayer
"... Nash equilibria of twoplayer games are much easier to compute in practice than those of nplayer games, even though the two problems have the same asymptotic complexity. We used a recent constructive reduction to solve general games using a twoplayer algorithm. However, the reduction increases the ..."
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Nash equilibria of twoplayer games are much easier to compute in practice than those of nplayer games, even though the two problems have the same asymptotic complexity. We used a recent constructive reduction to solve general games using a twoplayer algorithm. However, the reduction increases the game size too much to be practically usable. An open problem is to find a more compact constructive reduction, which might make this approach feasible. Categories and Subject Descriptors: F.2.0 [Analysis of algorithms and problem complexity]: