Three Problems of the Theory of Choice on Random Sets

Abstract

This paper discusses three problems which are united not only by the common topic of research stated in the title, but also by a somewhat surprising interlacing of the methods and techniques used. In the first problem, an attempt is made to resolve a very unpleasant metaproblem arising in general choice theory: why the conditions of rationality are not really necessary or, in other words, why in every-day life we are quite satisfied with choice methods which are far from being ideal. The answer, substantiated by a number of results, is as follows: situations in which the choice function "misbehaves" are very seldom met in large presentations. In the second problem, an overview of our studies is given on the problem of statistical properties of choice. One of the most astonishing phenomenon found when we deviate from scalar-extremal choice functions is in stable multiplicity of choice. If our presentation is random, then a random number of alternatives is chosen in it. But how many? The answer isn't trivial, and may be sought in many different directions. As we shall see below, usually a bottleneck case was considered in seeking the answer. It is interesting to note that statistical information effects the properties of the problem very much. The third problem is devoted to a model of a real life choice process. This process is typically spread in time, and we gradually (up to the time of making a final decision) accumulate experience, but once a decision is made we are not free to reject it. In the classical statement (i.e. when "optimality" is measured by some number) this model is referred to as a "secretary problem", and a great deal of literature is devoted to its study. We consider the case when the notions of optimality are most general. As will be seen below, the best strategy is practically determined by only the statistical properties of the corresponding choice function rather than its specific form.

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