Abstract
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 the 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.
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