Winner-Takes-All Games: Strategic Optimisation of Rank

In many competitive situations (including nearly all sports) a player’s aim is not simply to maximise his score but to maximise its rank among all scores. Examples include sales contests (where the salesman with highest monthly sales gets a bonus) and patent races (where lowest time is best). We assume the score Xi of player i is obtained costlessly, so that his utility is the probability of having the best score. This gives a constant-sum game. All that matters for player i is the distribution of his score, so we assume he chooses from a given convex set of distributions Fi . We call such games Distribution Ranking Games, and characterise their solution for various classes of the distribution sets Fi , such as distributions with given mean or moment, where we extend a result of Bell and Cover.