Computational complexity of winning strategies in two-person polynomial games

Abstract

One considers two-person games, with players called I and II below. In order, they choose natural numbers, for example, for length 4, I chooses x1, II chooses x2. I chooses x3, II chooses x4. Then I wins if P(x1,x2,x3,x4)=0.Here P is a polynomial with integer coefficients. An old theorem of von Neumann and Zermelo shows that such a game is determined, i.e., there exists a winning strategy for one player or the other but not necessarily a computable winning strategy or one computable in polynomial time. It will be shown that there exists a game of polynomial type of length 4 for which there do not exist winning strategies for either player which are computable in polynomial time.