Equivalence classes for optimizing risk models in Markov decision processes

We consider eight problems in which we maximize or minimize threshold probabilities in discounted Markov decision processes with bounded reward set. We show that such problems are classified to two equivalence classes and give a relationship between optimal values and optimal policies of problems in each equivalence class. Literatures relative to such problems deal with only first equivalence class (cf. White(1993), Wu and Lin(1999) and Ohtsubo and Toyonaga(2002)). We consider a problem of the second equivalence class in the same situation as Ohtsubo and Toyonaga and characterize optimal values in finite and infinite horizon cases, by using an argument of a dual problem. We also give two sufficient conditions for the existence of an optimal policy. Finally we give a relationship of optimal values between first and second equivalence classes.