Title: Modal Logic of Games

Author: Eric Werner, Ph.D.


Contact: eric.werner@cellnomica.com

Abstract

It is shown that modal reasoning in games has its own logic. The logic of games is a temporal modal logic that also depends on information. For, the axioms that are valid in the temporal modal logic depend on the information conditions that hold in the game. It is shown that von Neumann games with perfect information have a particular modal logic. Other logics of games hold given imperfect information. The limitations of the classical mathematical definition of games are described, and the way to possible extensions of game theory for games with non-von Neumann modal logics is indicated. The results obtained apply to temporal modal reasoning within any open, indeterministic domain where the information about the domain may be imperfect. \end{abstract}

Introduction

Modal reasoning in games has its own logic. The logic of games is information and time dependent. The axioms that are valid in the temporal modal logic depend on the information conditions that obtain in the game. It is shown that von Neumann games with perfect information have a particular modal logic. The logic of games with imperfect information depends on the information conditions that obtain in the game. The limits of the classical mathematical definition of a game are described, and the way to possible extensions of game theory for games with non-von Neumann modal logics is indicated. The results obtained are quite general, and apply to the temporal modal reasoning of any agent with possibly imperfect information about an open, indeterministic domain.

The semantics developed has some novel features not found in traditional semantics of modal logic (e.g., [Kripke 63], [Montague 76]). The semantics is an information based semantics that integrates the concept of an angent's information state. This step brings the sematnics of modal logic closer to the work in communication theory [Werner 88a] and economic theory [von Neumann \& Morgenstern 47], [Myerson 88]. Furthermore, the logics and semantics investigated have a close relationship to the logics used to investigate distributed systems [Halpern 87].

Also of interest is that the accessiblity relation is relativized not just to information but to time. The general properties of temporal accessibility relations are investigated. This makes the semantics adaptible. With minor modifications it can serve as a semantics for tensed modal logics, i.e., logics that contain both tense and modal operators. This allows one to investigate the meaning and logic of expressions like $P \Box \alpha$ = " It was necessary that $\alpha$" or $\diamond F\alpha$ = "It is possible that it will be the case that $\alpha$" [Werner 89a].

We start with a description of the temporal modal language. Then we define information states and the information based semantics. We then look at temporal accessibility relations in general. Such relations are then used to generalize the information-based semantics to what we call the abstract semantics. Next the axioms of information-based temporal modal logic are presented. Completeness and consistency theorems are proven. Finally, the results are applied to game theory.