Undiscounted Equilibrium in Positive Recursive Absorbing Games with Non-Rectangular Absorption Structure

TL;DR

This paper proves the existence of undiscounted equilibrium payoffs in positive recursive absorbing games with non-rectangular nonabsorbing components, expanding understanding of equilibrium existence in complex stochastic game structures.

Contribution

It establishes the existence of equilibrium payoffs in a new class of positive recursive absorbing games with non-rectangular nonabsorbing components, a case not previously addressed.

Findings

Existence of undiscounted equilibrium payoffs proven

Applicable to positive recursive absorbing games with non-rectangular components

Extends equilibrium theory to more complex game structures

Abstract

An absorbing game is a stochastic game with a single nonabsorbing state. Such a game is called recursive if all players receive a payoff of 0 in the nonabsorbing state, and positive if all payoffs in absorbing states are positive. An action profile is nonabsorbing if, when it is played, the game remains in the nonabsorbing state with probability 1. The set of nonabsorbing action profiles can be partitioned into the connected components of an undirected graph, whose vertices are these profiles, with two vertices joined by an edge whenever the corresponding profiles differ in the action of a single player. A connected component is said to be rectangular if it is the Cartesian product of subsets of the players' action sets. We prove that every positive recursive absorbing game whose nonabsorbing components are all non-rectangular admits an undiscounted equilibrium payoff.