  A family of models of individual discrete choice are constructed by means of
statistical averaging of choices made by a subject in a reinforcement learning
process, where the subject has short, k-term memory span. The choice
probabilities in these models combine in a non-trivial, non-linear way the
initial learning bias and the experience gained through learning. The
properties of such models are discussed and, in particular, it is shown that
probabilities deviate from Luce's Choice Axiom, even if the initial bias
adheres to it. Moreover, we shown that the latter property is recovered as the
memory span becomes large.
  Two applications in utility theory are considered. In the first, we use the
discrete choice model to generate binary preference relation on simple
lotteries. We show that the preferences violate transitivity and independence
axioms of expected utility theory. Furthermore, we establish the dependence of
the preferences on frames, with risk aversion for gains, and risk seeking for
losses. Based on these findings we propose next a parametric model of choice
based on the probability maximization principle, as a model for deviations from
expected utility principle. To illustrate the approach we apply it to the
classical problem of demand for insurance.
