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The Complexity of Pure Nash Equilibria
, 2004
"... We investigate from the computational viewpoint multiplayer games that are guaranteed to have pure Nash equilibria. We focus on congestion games, and show that a pure Nash equilibrium can be computed in polynomial time in the symmetric network case, while the problem is PLScomplete in general. ..."
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Cited by 150 (6 self)
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We investigate from the computational viewpoint multiplayer games that are guaranteed to have pure Nash equilibria. We focus on congestion games, and show that a pure Nash equilibrium can be computed in polynomial time in the symmetric network case, while the problem is PLScomplete in general. We discuss implications to nonatomic congestion games, and we explore the scope of the potential function method for proving existence of pure Nash equilibria.
The Structure and Complexity of Nash Equilibria for a Selfish Routing Game
, 2002
"... In this work, we study the combinatorial structure and the computational complexity of Nash equilibria for a certain game that models sel sh routing over a network consisting of m parallel links. We assume a collection of n users, each employing a mixed strategy, which is a probability distribu ..."
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Cited by 109 (26 self)
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In this work, we study the combinatorial structure and the computational complexity of Nash equilibria for a certain game that models sel sh routing over a network consisting of m parallel links. We assume a collection of n users, each employing a mixed strategy, which is a probability distribution over links, to control the routing of its own assigned trac. In a Nash equilibrium, each user sel shly routes its trac on those links that minimize its expected latency cost, given the network congestion caused by the other users. The social cost of a Nash equilibrium is the expectation, over all random choices of the users, of the maximum, over all links, latency through a link.
A.: Playing large games using simple strategies
 In: Proc. of the 4th ACM Conf. on El. Commerce (EC ’03). Assoc. of Comp. Mach
, 2003
"... We prove the existence of Nash equilibrium strategies with support logarithmic in the number of pure strategies. We also show that the payos to all players in any (exact) Nash equilibrium can be approximated by the payos to the players in some such logarithmic support Nash equilibrium. These str ..."
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Cited by 95 (3 self)
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We prove the existence of Nash equilibrium strategies with support logarithmic in the number of pure strategies. We also show that the payos to all players in any (exact) Nash equilibrium can be approximated by the payos to the players in some such logarithmic support Nash equilibrium. These strategies are also uniform on a multiset of logarithmic size and therefore this leads to a quasipolynomial algorithm for computing an Nash equilibrium. To our knowledge this is the rst subexponential algorithm for nding an Nash equilibrium. Our results hold for any multipleplayer game as long as the number of players is a constant (i.e., it is independent of the number of pure strategies). A similar argument also proves that for a xed number of players m, the payos to all players in any mtuple of mixed strategies can be approximated by the payos in some mtuple of constant support strategies. We also prove that if the payo matrices of a two person game have low rank then the game has an exact Nash equilibrium with small support. This implies that if the payo matrices can be well approximated by low rank matrices, the game has an equilibrium with small support. It also implies that if the payo matrices have constant rank we can compute an exact Nash equilibrium in polynomial time.
An application of boolean complexity to separation problems in bounded arithmetic
 Proc. London Math. Society
, 1994
"... We develop a method for establishing the independence of some Zf(a)formulas from S'2(a). In particular, we show that T'2(a) is not VZ*(a)conservative over S'2(a). We characterize the Z^definable functions of T2 as being precisely the functions definable as projections of polynomial ..."
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Cited by 58 (15 self)
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We develop a method for establishing the independence of some Zf(a)formulas from S'2(a). In particular, we show that T'2(a) is not VZ*(a)conservative over S'2(a). We characterize the Z^definable functions of T2 as being precisely the functions definable as projections of polynomial local search (PLS) problems. Although it is still an open problem whether bounded arithmetic S2 is finitely axiomatizable, considerable progress on this question has been made: S2 +1 is V2f+1conservative over T'2 [3], but it is not V2!f+2conservative unless £f+2 = Ylf+2 [10], and in addition, T2 is not VZf+1conservative over S'2 unless LogSpace s? = Af+1 [8]. In particular, S2 is not finitely axiomatizable provided that the polynomialtime hierarchy does not collapse [10]. For the theory S2(a) these results imply (with some additional arguments) absolute results: S'2 + (a) is V2f+,(a)conservative but not VZf+2(a)conservative over T'2(a), and T'2(a) is not VZf+i(c*)conservative over S'2(a). Here a represents a new uninterpreted predicate symbol adjoined to the language of arithmetic which may be used in induction formulas; from a computer science perspective, a represents an oracle. In this paper we pursue this line of investigation further by showing that T'2(a) is also not V2f(a)conservative over S'2(a). This was known for / = 1, 2 by [9,17] (see also [2]), and our present proof uses a version of the pigeonhole principle similar to the arguments in [2,9]. Perhaps more importantly, we formulate a general method (Theorem 2.6) which can be used to show the unprovability of other 2f(a)formulas from S'2(a). Our methods are analogous in spirit to the proof strategy of [8]: prove a witnessing theorem to show that provability of a Zf+1(a)formula A in S'2(a) implies that it is witnessed by a function of certain complexity and then employ techniques of boolean complexity to construct an oracle a such that the formula A cannot be witnessed by a function of the prescribed complexity. Our formula A shall be 2f(a) and thus we can use the original witnessing theorem of [2]. The boolean complexity used is the same as in [8], namely Hastad's switching lemmas [6].
Inverting Onto Functions
, 1996
"... We look at the hypothesis that all honest onto polynomialtime computable functions have a polynomialtime computable inverse. We show this hypothesis equivalent to several other complexity conjectures including ffl In polynomial time, one can find accepting paths of nondeterministic polynomialtim ..."
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Cited by 36 (6 self)
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We look at the hypothesis that all honest onto polynomialtime computable functions have a polynomialtime computable inverse. We show this hypothesis equivalent to several other complexity conjectures including ffl In polynomial time, one can find accepting paths of nondeterministic polynomialtime Turing machines that accept \Sigma . ffl Every total multivalued nondeterministic function has a polynomialtime computable refinement. ffl In polynomial time, one can compute satisfying assignments for any polynomialtime computable set of satisfiable formulae. ffl In polynomial time, one can convert the accepting computations of any nondeterministic Turing machine that accepts SAT to satisfying assignments. We compare these hypotheses with several other important complexity statements. We also examine the complexity of these statements where we only require a single bit instead of the entire inverse. 1 Introduction Understanding the power of nondeterminism has been one of the pri...
Classification of Search Problems and Their Definability in Bounded Arithmetic
, 2001
"... We present a new framework for the study of search problems and their definability in bounded arithmetic. We identify two notions of complexity of search problems: verification complexity and computational complexity. Notions of exact solvability and exact reducibility are developed, and exact b ..."
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Cited by 3 (2 self)
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We present a new framework for the study of search problems and their definability in bounded arithmetic. We identify two notions of complexity of search problems: verification complexity and computational complexity. Notions of exact solvability and exact reducibility are developed, and exact b i definability of search problems in bounded arithmetic is introduced. We specify a new machine model called the oblivious witnessoracle Turing machines. Based on
Integer Polyhedra: Combinatorial Properties and Complexity
, 2001
"... A polyhedron having vertices is called integer if all of its vertices are integer. This property is coNPcomplete in general. Recognizing integral setpacking polyhedra is one of the biggest challenges of graph theory (perfectness test). Various other special cases are major problems of discrete ..."
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A polyhedron having vertices is called integer if all of its vertices are integer. This property is coNPcomplete in general. Recognizing integral setpacking polyhedra is one of the biggest challenges of graph theory (perfectness test). Various other special cases are major problems of discrete mathematics.
Ecient Computation of Nash Equilibria for Very Sparse
"... It is known that nding a Nash equilibrium for winlose bimatrix games, i.e., twoplayer games where the players ' payos are zero and one, is complete for the class PPAD. We describe a linear time algorithm which computes a Nash equilibrium for winlose bimatrix games where the number of winning ..."
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It is known that nding a Nash equilibrium for winlose bimatrix games, i.e., twoplayer games where the players ' payos are zero and one, is complete for the class PPAD. We describe a linear time algorithm which computes a Nash equilibrium for winlose bimatrix games where the number of winning positions per strategy of each of the players is at most two. The algorithm acts on the directed graph that represents the zeroone pattern of the payo matrices describing the game. It is based upon the ecient detection of certain subgraphs which enable us to determine the support of a Nash equilibrium. 1