Kuhn's Equivalence Theorem for Games in Product Form

TL;DR

This paper introduces a novel representation for extensive form games called 'games in product form', which simplifies modeling complex games with continuous actions and uncertain play order, and extends Kuhn's theorem to this framework.

Contribution

It proposes an alternative to tree-based game representations using sigma-fields over product sets and proves Kuhn's theorem for these generalized games with continuous actions.

Findings

Introduces 'games in product form' as a new representation.

Proves Kuhn's theorem for games with continuous action sets.

Encompasses games with continuum actions, randomness, and uncertain play order.

Abstract

We propose an alternative to the tree representation of extensive form games. Games in product form represent information with $\sigma$-fields over a product set, and do not require an explicit description of the play temporality, as opposed to extensive form games on trees. This representation encompasses games with a continuum of actions, randomness and players, as well as games for which the play order cannot be determined in advance. We adapt and prove Kuhn's theorem-regarding equivalence between mixed and behavioral strategies under perfect recall-for games in product form with continuous action sets.