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We study conditions for the existence of coalition games with the result invariant under cyclic shifts of players sequence numbers. Given a total number  n  of players, we estimate the number  k  of players of one coalition under which there exists a game in which this coalition wins under all cyclic shifts of players. We give procedures for construction of the so-called set-splitting digraphs on which risk-free nim-type games of a given coalition are defined.

Cyclic decomposition of sets, set-splitting digraphs and cyclic classes of risk-free games