The multi-stage dynamic stochastic decision process with unknown distribution of the random utilities

We consider a decision maker who performs a stochastic decision process over a multiple number of stages, where the choice alternatives are characterized by random utilities with unknown probability distribution. The decisions are nested each other, i.e. the decision taken at each stage is affected by the subsequent stage decisions. The problem consists in maximizing the total expected utility of the overall multi-stage stochastic dynamic decision process. By means of some results of the extreme values theory, the probability distribution of the total maximum utility is derived and its expected value is found. This value is proportional to the logarithm of the accessibility of the decision maker to the overall set of alternatives in the different stages at the start of the decision process. It is also shown that the choice probability to select alternatives becomes a Nested Multinomial Logit model.