Abstract. We analyze the complexity of computing pure strategy Nash equilibria (PSNE) in symmetric games with a fixed number of actions, where the utilities are compactly represented. Such a representation is able to describe symmetric games whose number of players is 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, there always exist a PSNE and we give a polynomial algorithm for finding one. We then focus on a natural representation of utility as piecewise-linear functions, and show that such a representation has nice computational properties. In particular, we give polynomial-time algorithms to count the number of PSNE (thus deciding if such an equilibrium exists) and to find a sample PSNE, when one exists. Our approach makes use of Barvinok and Wood's rational generating function method [4], which enables us to encode the set of PSNE as a generating function of polynomial size.
In this paper we show that for a given set of pairwise comaximal ideals {X i } i∈I in a ring R with unity and any right R-module M with generating set Y and C(X i ) = k∈N ℓ M (X k i ), M = ⊕ i∈I C(X i ) if and only if for every y ∈ Y there exists a nonempty finite subset J ⊆ I and positive integers k j such that j∈J X k j i ⊆ r R (yR). We investigate this decomposition for a general class of modules. Our main theorem can be applied to a large class of rings including semilocal rings R with the Jacobson radical of R equal to the prime radical of R, left (or right) perfect rings, piecewise prime rings, and rings with ACC on ideals and satisfying the right AR property on ideals. This decomposition generalizes the decomposition of a torsion abelian group into a direct sum of its p-components. We also develop a torsion theory associated with sets of pairwise comaximal ideals.2010 Mathematics Subject Classification. 16D70.
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