Kuhn's Equivalence Theorem for Games in Intrinsic Form

TL;DR

This paper introduces the intrinsic form of games, a new representation based on sigma-fields, and proves Kuhn's equivalence theorem within this framework, simplifying analysis of games with information.

Contribution

It extends Kuhn's theorem to the intrinsic form, a more general game representation that does not rely on explicit temporality or tree structures.

Findings

Proves Kuhn's equivalence theorem for intrinsic games.

Shows intrinsic form simplifies information handling.

Demonstrates equivalence of strategies under perfect recall.

Abstract

We state and prove Kuhn's equivalence theorem for a new representation of games, the intrinsic form. First, we introduce games in intrinsic form where information is represented by $\sigma$-fields over a product set. For this purpose, we adapt to games the intrinsic representation that Witsenhausen introduced in control theory. Those intrinsic games do not require an explicit description of the play temporality, as opposed to extensive form games on trees. Second, we prove, for this new and more general representation of games, that behavioral and mixed strategies are equivalent under perfect recall (Kuhn's theorem). As the intrinsic form replaces the tree structure with a product structure, the handling of information is easier. This makes the intrinsic form a new valuable tool for the analysis of games with information.