How Good Is a Strategy in a Game with Nature?

We consider games with two antagonistic players - Éloïse (modelling a program) and Abélard (modelling a Byzantine environment) - and a third, unpredictable and uncontrollable player, which 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 lead to decidability where the probabilistic one fails: one based on counting and one based on topology.