Proof-theoretic Analysis of Rationality for Strategic Games with Arbitrary Strategy Sets

TL;DR

This paper provides an axiomatic proof connecting common knowledge of rationality with the iterative elimination of strictly dominated strategies in strategic games, using formal languages and fixpoint logic.

Contribution

It introduces a formal proof framework using modal fixpoint logic to analyze rationality and strategy elimination in strategic games.

Findings

Common knowledge of rationality leads to elimination of non-optimal strategies.

Formal languages are used to model optimality and belief in strategic contexts.

The proof framework clarifies the logical foundations of strategic reasoning.

Abstract

In the context of strategic games, we provide an axiomatic proof of the statement Common knowledge of rationality implies that the players will choose only strategies that survive the iterated elimination of strictly dominated strategies. Rationality here means playing only strategies one believes to be best responses. This involves looking at two formal languages. One is first-order, and is used to formalise optimality conditions, like avoiding strictly dominated strategies, or playing a best response. The other is a modal fixpoint language with expressions for optimality, rationality and belief. Fixpoints are used to form expressions for common belief and for iterated elimination of non-optimal strategies.