A game-theoretic classification of interactive complexity classes

Game-theoretic characterisations of complexity classes have often proved useful in understanding the power and limitations of these classes. One well-known example tells us that PSPACE can be characterized by two-person, perfect-information games in which the length of a played game is polynomial in the length of the description of the initial position [by Chandra et al., see Journal of the ACM, vol. 28, p. 114-33 (1981)]. In this paper, we investigate the connection between game theory and interactive computation. We formalize the notion of a polynomially definable game system for the language L, which, informally, consists of two arbitrarily powerful players P₁ and P₂ and a polynomial-time referee V with a common input w. Player P₁ claims that w∈L, and player P₂ claims that w∈L; the referee´s job is to decide which of these two claims is true. In general, we wish to study the following question: What is the effect of varying the system´s game-theoretic properties on the class of languages recognizable by polynomially definable game systems? There are many possible game-theoretic properties that we could investigate in this context. The focus of this paper is the question of what happens when one or both of the players P₁ and P₂ have imperfect information or imperfect recall. We use polynomially definable game systems to derive new characterizations of the complexity classes NEXP and coNEXP