Undiscounted Equilibrium in Positive Recursive Absorbing Games with Nonrectangular Absorption Structure

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 zero 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 one. 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 nonrectangular admits an undiscounted equilibrium payoff. Funding: E. Solan acknowledges the support of the Israel Science Foundation [Grant 211/22].