The maximal number of totally mixed Nash equilibria in games of several
players equals the number of block derangements, as proved by McKelvey and
McLennan.On the other hand, counting the derangements is a well studied
problem. The numbers are identified as linearization coefficients for Laguerre
polynomials. MacMahon derived a generating function for them as an application
of his master theorem. This article relates the algebraic, combinatorial and
game-theoretic problems that were not connected before. New recurrence
relations, hypergeometric formulas and asymptotics for the derangement counts
are derived. An upper bound for the total number of all Nash equilibria is
given.Comment: 22 pages, 1 table; Theorem 3.3 adde