Abstract
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.
Highlights
IntroductionRandom utility models [15,16]. According to [23], in these models “a decision maker i faces a choice among N alternatives and will assign a certain level of utility to each alternative
Discrete choice models under the assumption of a utility-maximizing behavior by the decision maker and uncertainty over the estimation of the utility values are calledMontreal, Canada random utility models [15,16]
We have considered a multi-stage dynamic decision process in which decisions are nested each other
Summary
Random utility models [15,16]. According to [23], in these models “a decision maker i faces a choice among N alternatives and will assign a certain level of utility to each alternative. We consider a decision process evolving over multiple stages (e.g., over a discrete time horizon) in which a decision maker is asked to solve consecutively several random utility models, i.e. he needs to select, at each stage, an alternative among a finite set of mutually exclusive choices. When facing the special case in which only a single stage exists (i.e., when the decision maker has only a static set of alternatives to choose from), it is well-known that the choice probability reduces to a Multinomial Logit (MNL) model under the assumption that the random utilities are independent and identically distributed (i.i.d.) and the common distribution is a Gumbel function (see [1,2,5,12]).
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