How Good Is a Strategy in a Game With Nature?

Abstract

We consider games with two antagonistic players --- Éloïse (modelling a program) and Abélard (modelling a byzantine environment) --- and a third, unpredictable and uncontrollable player, that we call Nature. Motivated by the fact that the usual probabilistic semantics very quickly leads to undecidability when considering either infinite game graphs or imperfect-information, we propose two alternative semantics that leads to decidability where the probabilistic one fails: one based on counting and one based on topology.