Winner-take-all games: existence of equilibria and three player dice

Abstract In many competitive situations, the aim of a player is not to maximize his ‘score’ but rather its rank among other scores. Think sales contests or mutual fund rankings. Following E.J. Anderson’s game of two players entering fair casinos trying to leave with the most money, Alpern and Howard abstracted to the following winner take all game: Each player chooses a distribution from a given set. The distributions are sampled independenly and the player with the highest sample (the winner) gets a unit prize. In case of a tie the winners split the prize. An example of such a two person game is by Bell and Cover in the context ot finance, where the set is all distributions on the unit interval with mean 1/2 and the solution is the uniform distribution. Alpern and Howard extended this and solved many other such games, including two person fair dice (mean 3.5), where the solution is the usual equiprobable die. This paper establishes the existence of equilibria for games where the sets of distributions are those supported on the unit interval with no interior atoms. In addition, we solve the three player game with fair dice, where the usual equiprobable die does not form an equilibrium (only faces 1, 2, 3 and 6 have positive probability at equilibrium). We also solve the limiting case of k sided dice, where players pick any point in $$\left[ 0,1\right]$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mfenced> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> </mml:mfenced> </mml:math> according to a distribution with mean 1/2.