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Tournament relation-based choice. (English. Russian original) Zbl 0832.90004
Phys.-Dokl. 39, No. 5, 322-325 (1994); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 336, No. 3, 324-327 (1994).
Summary: An important problem in tournament relation analysis is establishing preferences among the elements of $$X$$ according to a relation $$R$$ or, more generally, defining an ordering of $$X$$ according to $$R$$. The problem is nontrivial because usually the relation $$R$$ is cyclic and does not define any ordering over the entire set $$X$$. The methods developed for this case either are heuristic, giving no idea of the ordering principle, or select extremely large sets of the “best” elements.
Another problem consists in weakening the conditions (1) $$\forall a\in X R(a, a)= 0$$; (2) $$(a\neq b)\Rightarrow R(a, b)= \overline{R(b, a)}$$, and constructing an appropriate ordering for connected but not necessarily asymmetric relations.
To solve these problems, we use the methods of self-consistent choice.
##### MSC:
 91B08 Individual preferences