On the existence of weak subgame perfect equilibria

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 set outcomes which allows him compare plays. focus recently introduced notion weak subgame perfect equilibrium (weak SPE). This variant classical SPE, where players who deviate can only use strategies deviating from their initial strategy in finite number histories. Having an SPE game implies having but contrary generally false. propose general conditions structure graph and relations that guarantee existence additionally finite-memory. From this result, we derive two large classes for there always exists SPE: (i) with finite-range function, (ii) underlying prefix-independent function. For second class, identify memoryless SPE.