Games with infinite past

TL;DR

This paper investigates equilibrium existence in multi-player perfect information games with an infinite past, establishing conditions under which equilibria exist based on game structure and payoff complexity.

Contribution

It introduces two equilibrium concepts for games with an infinite past and characterizes when equilibria exist, especially in win-lose and continuous payoff scenarios.

Findings

Equilibrium sets coincide when at least two players are active.

Games with Borel-rank ≤ 2 have equilibria; rank 3 may not.

Continuous payoffs ensure equilibrium existence in non-zero-sum games.

Abstract

We study multi-player games with perfect information and general payoff function, where the set of stages is the set of non-positive integers $\{\ldots,-2,-1,0\}$. We define two related equilibrium concepts: one considering only deviations at finitely many stages and another considering all deviations. We show that (i) The sets of equilibrium plays coincide for the two equilibrium concepts, provided that at least two players are active along each infinite play. (ii) In win-lose games, the game has an equilibrium if the winning sets have Borel-rank at most 2, and we provide a counter-example showing that this is no longer true for Borel-rank 3. (iii) In general non-zero-sum games, the game has an equilibrium if the payoff functions are continuous, for example, with reversed-time discounted payoffs. The challenge for all these results is that not all strategy profiles admit a consistent infinite play, hampering the use of backward induction arguments.