Coxeter tournaments

Abstract: In line with the trend of Coxeter combinatorics, and in response to a question of Stanley, we show that the Coxeter permutahedra, recently studied by Ardila, Castillo, Eur and Postnikov, can be described in terms of Coxeter tournaments. These polytopes have been described geometrically in terms of submodular functions. We show that they can also be viewed in terms of tournaments with cooperative and solitaire games, as well as the usual competitive games in classical graph tournaments. We establish a Coxeter analogue of Moon's classical theorem, regarding mean score sequences of random tournaments. We present a geometric proof by the Mirsky--Thompson generalized Birkhoff's theorem, a probabilistic proof by Strassen's coupling theorem, and an algorithmic proof by a Coxeter analogue of the Havel--Hakimi algorithm. These proofs have natural interpretations in terms of players seeking out potential competitors/collaborators with respect to their relative weakness/strength. We also observe that the Bradley--Terry model, from the statistical theory of paired comparisons, extends to the Coxeter setting. Finally, we show that an analogue of Landau's classical theorem, concerning deterministic tournaments, holds for balanced Coxeter tournaments.