Undiscounted Equilibrium in Positive Recursive Absorbing Games with Non-Rectangular Absorption Structure
Eilon Solan, Nicolas Vieille
TL;DR
The paper resolves the existence of undiscounted equilibrium payoffs for positive recursive absorbing games under a non-rectangular absorption-structure condition, addressing a central open problem in stochastic games. It extends a dynamical-systems approach from quitting games by constructing an operator f on payoff vectors and concatenating finite-horizon equilibria along an approximate orbit of f to yield a long-run equilibrium strategy profile σ^*. The main contributions include identifying non-rectangular connected components as the critical barrier, providing a constructive proof that yields a uniform (and thus long-run) equilibrium payoff, and laying groundwork for extensions using public correlation devices. The results unify and simplify prior methods, offering a robust framework for analyzing undiscounted equilibria in multi-agent stochastic settings and guiding future work on broader classes and correlation-enabled equilibria.
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.