On the existence of weak subgame perfect equilibria

TL;DR

This paper investigates the existence of weak subgame perfect equilibria in multi-player turn-based games on graphs, providing conditions for their existence and properties, including finite-memory and memoryless strategies.

Contribution

It introduces general conditions ensuring the existence of weak SPEs, including finite-memory and memoryless strategies, in broad classes of graph-based games.

Findings

Weak SPEs exist under certain structural and preference conditions.

Finite-range outcome functions guarantee weak SPEs.

Finite graph and prefix-independent outcomes lead to weak SPEs with memoryless strategies.

Abstract

We study multi-player turn-based games played on (potentially infinite) directed graphs. An outcome is assigned to every play of the game. Each player has a preference relation on the set of outcomes which allows him to compare plays. We focus on the recently introduced notion of weak subgame perfect equilibrium (weak SPE). This is a variant of the classical notion of SPE, where players who deviate can only use strategies deviating from their initial strategy in a finite number of histories. Having an SPE in a game implies having a weak SPE but the contrary is generally false. We propose general conditions on the structure of the game graph and on the preference relations of the players that guarantee the existence of a weak SPE, that additionally is finite-memory. From this general result, we derive two large classes of games for which there always exists a weak SPE: (i) the games with a finite-range outcome function, and (ii) the games with a finite underlying graph and a prefix-independent outcome function. For the second class, we identify conditions on the preference relations that guarantee memoryless strategies for the weak SPE.