How discontinuous is Computing Nash Equilibria?

arXiv:0907.1482v1 [cs.GT] 9 Jul 2009 How discontinuous is Computing Nash Equilibria? Arno Pauly University of Cambridge Computer Laboratory Cambridge, UK Arno.Pauly@cl.cam.ac.uk We investigate the degree of discontinuity of several solution concepts from non-cooperative game theory. While the consideration of Nash equilibria forms the core of our work, also pure and correlated equilibria are dealt with. Formally, we restrict the treatment to two player games, but results and proofs extend to the n-player case. As a side result, the degree of dis- continuity of solving systems of linear inequalities is settled. Keywords. Game Theory, Computable Analysis, Nash Equilibrium, Discontinuity 1. Introduction Both for applications and theoretical considerations, the algorithmic task of computing Nash equilibria from certain representations of games is of immense importance. A natural mathematical formulation of game theory uses the real numbers for payoffs and for mixed strategies, while classical models for algorithms require a restriction to countable sets. By imposing suitable restrictions and modifications to obtain countable problems, the complexity of computing a Nash equilibrium for a normal form game was proven to be PPAD-complete ([1], [2]). Here we will use another approach: Instead of limiting the problem, we will extend the theory of computation. While the TTE-framework ([3]) is perfectly capable of for- mulating the task of computing Nash equilibria from normal form games, we will see that even the most trivial cases are discontinuous, and hence not computable. To gain a deeper understanding of the problem, its degree of discontinuity will be studied. Mirroring an approach in the study of game theory using classical computational complexity, we will also examine other solution concepts such as correlated equilibria. While correlated equilibria seem to be computationally easier than Nash equilibria1, 1In [4] several decision problems regarding Nash equilibria and correlated equilibria were compared, most of them turned out to be NP-hard for Nash equilibria and to be in P for correlated equilibria. Discontinuity of Equilibria 2 we will prove that both concepts share a degree of discontinuity. Limitation to pure strategies yields a strictly less discontinuous problem, the classical problem can be solved by a cubic algorithm2. 2. Preliminaries 2.1. Game Theory An n × m bi-matrix game is simply given by two n × m real valued matrices A and B. Two players simultaneously pick an index, row player chooses an i ∈{1, 2, . . . , n} and column player chooses an j ∈{1, 2, . . . , m}. Row player gets Aij as a reward, column player gets Bij. We consider several solution concepts defined as equilibria, where no player has an incentive to change her strategy unilaterally. Definition 1. A pure equilibrium for a n × m bi-matrix game (A, B) is a pair (i, j) ∈ {1, . . . , n} × {1, . . . , m} satisfying Aij ≥Akj for all k ∈{1, . . . , n} and Bij ≥Bil for all l ∈{1, . . . , m}. As pure equilibria do not exist for all games, a more general notion is introduced. If both players can randomize independently over their actions, one is led to the definition of an m-mixed strategy as an m-dimensional real valued vector s with non-negative coefficients and m P j=1 sj = 1. The set of m-mixed strategies will be denoted by Sm. Definition 2. A Nash equilibrium for an n×m bi-matrix game (A, B) is a pair (ˆx, ˆy) ∈ Sn × Sm satisfying ˆxT Aˆy ≥xT Aˆy for all x ∈Sn and ˆxT Bˆy ≥ˆxT By for all y ∈Sm. If (ˆx, ˆy) is a Nash equilibrium, again neither of the players can improve her payoff by changing her mixed strategy unilaterally. A famous result by John Nash ([7]) es- tablished that Nash equilibria in bi-matrix games always exist. By identifying a pure strategy with the mixed strategy that puts weight 1 on it, pure equilibria can be con- sidered a special case of Nash equilibria. An even more general solution concept can be obtained by allowing the individual player’s randomization processes to be correlated ([8]). Definition 3. A correlated equilibrium for a n × m bi-matrix game is a real valued n × m matrix C with non-negative entries and nP i=1 m P j=1 Cij = 1 so that m X j=1 AijCij ≥ m X j=1 AljCij 2There are, however, several interesting hardness results for finding pure equilibria in games ([5], [6]), originating in other representations or requiring additional properties. Discontinuity of Equilibria 3 holds for all i, l ∈{1, 2, . . . , n} and n X i=1 BijCij ≥ n X i=1 BikCij holds for all j, k ∈{1, 2, . . . , m}. Given a Nash equilibrium (x, y), a correlated equilibrium can be constructed as Cij = xiyj, while each correlated equilibrium of this form can be obtained from a Nash equi- librium, allowing us to consider Nash equilibria as special cases of correlated equilibria. Thus, finding a correlated equilibrium has to be easier than finding a Nash equilibrium, as we just presented a reduction. Another way of creating an easier problem consists in

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