Recovering Preferences From Finite Data

We study preferences estimated from finite choice experiments and provide sufficient conditions for convergence to a unique underlying “true” preference. Our conditions are weak and, therefore, valid in a wide range of economic environments. We develop applications to expected utility theory, choice over consumption bundles, and menu choice. Our framework unifies the revealed preference tradition with models that allow for errors.

On winning shifts of marked uniform substitutions

The second author introduced with I. Törmä a two-player word-building game [Fund. Inform. 132 (2014) 131–152]. The game has a predetermined (possibly finite) choice sequence α1, α2, … of integers such that on round n the player A chooses a subset Sn of size αn of some fixed finite alphabet and the player B picks a letter from the set Sn. The outcome is determined by whether the word obtained by concatenating the letters B picked lies in a prescribed target set X (a win for player A) or not (a win for player B). Typically, we consider X to be a subshift. The winning shift W(X) of a subshift X is defined as the set of choice sequences for which A has a winning strategy when the target set is the language of X. The winning shift W(X) mirrors some properties of X. For instance, W(X) and X have the same entropy. Virtually nothing is known about the structure of the winning shifts of subshifts common in combinatorics on words. In this paper, we study the winning shifts of subshifts generated by marked uniform substitutions, and show that these winning shifts, viewed as subshifts, also have a substitutive structure. Particularly, we give an explicit description of the winning shift for the generalized Thue–Morse substitutions. It is known that W(X) and X have the same factor complexity. As an example application, we exploit this connection to give a simple derivation of the first difference and factor complexity functions of subshifts generated by marked substitutions. We describe these functions in particular detail for the generalized Thue–Morse substitutions.

Distances of Fuzzy Choice Functions

In this paper, we introduce four distances on the set of fuzzy choice functions defined on a finite choice space. They are studied along with four distances on the set of fuzzy relations. The two types of distance allow to investigate the way the changes in fuzzy preferences are reflected in the changes of fuzzy choice associated with them. Also the way the changes in fuzzy choices manifest themselves in changes in fuzzy preferences are studied. The coefficient of normality of a fuzzy choice function is defined as a measure of normality and its variation is evaluated with respect to the variation of fuzzy choices. Finally, the variation of some congruence indicators is evaluated as effect of the changes in fuzzy choices.