Lecture by Andrey Subochev on "Everything I know about stable sets: definitions, characterizations, correlations, generalizations, interpretations"
On Wednesday, October 14, the National Research University Higher School of Economics hosted a regular meeting of the All-Moscow seminar "Mathematical methods for analyzing optimal solutions in economics, business and politics".
Speaker: A. N. Subochev (HSE, ICS RAS)
This work presents a special approach to solving the problem of winner choice (best options) in a tournament. This approach is based on two ideas:
1) the group strength of the participant (candidate, variant) is measured, not the individual strength; a key member of a "strong" group is considered as a strong candidate,
2) one or another version of stability is considered as a strength of the group.
The report includes the collection and comparisons of different definitions of the stability of a subset of tournament alternatives (dominance, domination, external stability, self-defense ability, weak stability). Characterization of minimal stable sets is given for three versions of stability through a connection with versions of the uncovered set of tournament alternatives.
This paper demonstrates logical (set-theoretic) relations of unions of minimal stable sets with other tournament solutions.
There are also the review and a partial solution to the problem of generalizing the obtained results to the case of an infinite (uncountable) set of alternatives. A (new) interpretation of stable sets considered as winning coalitions in a simple non-cooperative game is proposed.