Seminar "Expert opinion and data analysis"

On Wednesday, January 11, the National Research University Higher School of Economics hosted a seminar "Expert opinion and data analysis".

Theme: S-Resilient Tournament Solutions: Overview and New Results
Speaker: Subochev A. (HSE), Yudina A. (HSE)

Annotation:
The optimal choice problem consists in choosing in some sense the best alternatives from the set presented to the chooser. An important special case is the choice based on the results of a pairwise comparison of options. A similar problem can be thought of as the problem of choosing a winner(s) in a sports tournament.Several different solutions to this problem (tournament solutions) have been proposed in the literature. These solutions differ in their properties, and particular importance is attached to the stability of the choice, understood as the independence of the results of the choice from the presence or absence of non-optimal alternatives in the presentation (the set of available choices). This understanding of stability generalizes Nash's notion of independence from extraneous alternatives [Nash, 1950].However, it was also found that stability can be understood differently, namely as a choice from the minimum subsets of presentation that are stable with respect to a given tournament decision S. This understanding generalizes the concepts of external and internal stability of a set of alternatives proposed by von Neumann and Morgenstern [von Neumann & Morgenstern, 1944]. A consequence of the idea of choosing from S-stable sets is that each concept of a tournament solution S generates a new concept of a tournament solution, the union of minimal S-stable sets.It was shown that solutions that are stable in the first sense must also be stable in the second sense, but the converse is not true. Thus, the search for stable solutions among S-stable ones is a non-trivial and interesting theoretical problem. At the same time, researchers did not pay attention to the fact that in order to determine new tournament decisions using the S-stability principle, one can use not only tournament decisions proper, but also decision concepts that allow for emptiness of choice. The presented report contains an overview of stable tournament solutions represented as unions of minimal S-stable sets. The report presents new results related both to the use of solutions that allow emptiness of choice and to generalizations of such solutions to the case of the presence of pairs of equivalent alternatives (draw tournaments).