Computational Extensive-Form Games

Abstract

We define solution concepts appropriate for computationally bounded players playing a fixed finite game. To do so, we need to define what it means for a computational game, which is a sequence of games that get larger in some appropriate sense, to represent a single finite underlying extensive-form game. Roughly speaking, we require all the games in the sequence to have essentially the same structure as the underlying game, except that two histories that are indistinguishable (i.e., in the same information set) in the underlying game may correspond to histories that are only computationally indistinguishable in the computational game. We define a computational version of both Nash equilibrium and sequential equilibrium for computational games, and show that every Nash (resp., sequential) equilibrium in the underlying game corresponds to a computational Nash (resp., sequential) equilibrium in the computational game. One advantage of our approach is that if a cryptographic protocol represents an abstract game, then we can analyze its strategic behavior in the abstract game, and thus separate the cryptographic analysis of the protocol from the strategic analysis.

On the Existence of Nash Equilibrium in Games with Resource-Bounded Players

“…of games where, for all n, G n = (J, B n , T n , P n , v n ) is a k-player game (k is fixed and does not depend on n). We adapt the definition of [8], which in turn is based on earlier definitions by Dodis, Halevy and Rabin [5] and Megiddo and Wigderson [13], and is applied to extensive-form games, to Bayesian games. For an integer s, recall that {0, 1} ≤s is the set of all bit strings of length at most s.…”

“…When dealing with games with polynomial-time players, we require slightly stronger properties summarized in the definition below. Following the definition of polynomial games for extensiveform games [8], we define polynomial games for a sequence of Bayesian games.…”

“…We model such settings using tools from game theory and complexity theory. Formally, consider computational games [8] {G 1 , G 2 , . .…”

“…The encryption at the first step involves a security parameter k, where larger security parameters provide more security (i.e.., more running time is required to break the scheme). This can be modeled as a sequence of games, where in the kth game the sender encrypts the bit using a security parameter k (see also [8]). Many cryptographic protocols, including secret sharing and multiparty computation, can be viewed as games in this way.…”

“….} [8]. We assume that every game G j is a k-player game and that for 1 ≤ i ≤ k, player i uses a TM M i that computes his actions in G j given j.…”