A Report on r-Nash Reduction

The problem of computing Nash equilibria of two-player zero-sum games may be reduced to a polynomialtime solvable linear programming problem. For r > 2, the equilibrium strategies of r-player zero-sum games may be irrational, and it is unclear whether non-zero-sum games, even for two players, may be solved quickly. In general, one gets the sense that the complexity of computing Nash equilibria in a general r-player game, henceforth r-Nash, may increase without bound with r. A series of recent papers has placed increasingly tighter upper bounds on this growth in complexity. Storage space for general games increases exponentially with the number of players, and so these and other works investigated reductions between r-Nash and d-Graphical-Nash, graphical games of degree ≤ d, in which each player is a vertex and only affects the payoffs of its neighbors in the graph. In [3], Goldberg and Papadimitriou first proved that d-Graphical-Nash and r-Nash for r ≥ 4 are reducible to 4-Nash, and in [2] followed up by proving that 4-Nash is complete in the complexity class PPAD by a reduction of the PPAD-complete problem 3-Dimensional Brouwer to 3-Graphical-Nash. Chen and Deng completed the series in [1] by proving, with a refinement of the previous reduction that bypasses graphical games, that 2-Nash is PPAD-complete. While PPAD is not a richly-described class at this time and therefore completeness within it gives no good general sense of the complexity of 2-Nash, this final result limits the sufficient scope of future work by establishing that r-Nash is equally hard for all r (up to a polynomial-time reduction), and therefore work may concentrate on 2-Nash without loss of generality. This paper proceeds by first presenting a shallow overview of the concepts and work leading up to Chen and Deng’s paper, then demonstrating a sketch of their reduction, and finally making explicit portions of the reductions in [2] and [1].