Specifying a Game-Theoretic Extensive Form as an Abstract 5-ary Relation

TL;DR

This paper introduces a new formalism called pentaform for representing extensive form games as sets of quintuples satisfying specific axioms, establishing a rigorous link to traditional game representations.

Contribution

It defines pentaforms as a novel abstract structure for extensive games and proves a bijection with traditional game models, enhancing formal understanding and analysis.

Findings

Pentaforms are characterized by eight abstract axioms.

A bijection between pentaform games and traditional games is established.

Applications include subgames and perfect-recall in extensive form games.

Abstract

This paper specifies an extensive form as a 5-ary relation (that is, as a set of quintuples) which satisfies eight abstract axioms. Each quintuple is understood to list a player, a situation (that is, a name for an information set), a decision node, an action, and a successor node. Accordingly, the axioms are understood to specify abstract relationships between players, situations, nodes, and actions. Such an extensive form is called a "pentaform". Finally, a "pentaform game" is defined to be a pentaform together with utility functions. To ground this new specification in the literature, the paper defines the concept of a "traditional game" to represent the literature's many specifications of finite-horizon and infinite-horizon games. The paper's main result is to construct an intuitive bijection between pentaform games and traditional games. Secondary results concern disaggregating pentaforms by subsets, constructing pentaforms by unions, and initial pentaform applications to Selten subgames and perfect-recall (an extensive application to dynamic programming is in Streufert 2023, arXiv:2302.03855).