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Population uncertainty in contests.

Article Excerpt
Summary. We study contests where the set of players is a random variable. If it is known for certain that there will be at least one participant, then aggregate contest expenditure in equilibrium is strictly lower in a contest with population uncertainty than in a non-uncertain contest with the same expected number of players. This suggests an explanation of, for example, why empirical studies show rent-seeking expenditures to be much lower than predicted by other theories.

Keywords and Phrases: Contests, population uncertainty.

JEL Classification Numbers: C72, D44, D72, D82, K41.

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The probabilistic contest for control of an indivisible object is arguably a canonical model of non-market-based allocation of goods. It occurs in such seemingly diverse settings as the theory of anarchy and warfare (see, e.g., Hirshleifer [5, 6] and Skaperdas [15]), the theory of competition among political candidates under uncertainty (see, e.g., Coughlin [1]), the theory of rent-seeking (see, e.g., Tullock [20] and Nitzan [13]), corporate governance (see Muller and Warneryd [10]) and the theory of sports competitions (see, e.g., Szymanski [18]). More generally, a contest is an example of an imperfectly discriminating all-pay auction (see Hillman and Riley [4] and Krishna and Morgan [8]). Studies of the properties of abstract contests include Dixit [2] and Warneryd [21].

In this paper, we study contests where the standard assumption that the set of players is common knowledge is relaxed. Many of the applications mentioned above have in common that this assumption may be unreasonably demanding, given that we are usually interested in situations where the number of potential contestants may be very large.

The common knowledge assumption may be weakened in different ways. The particular approach we take here applies the population uncertainty notion of Myerson [11, 12]. Under population uncertainty, the number of players of a given type in a game is a random variable with a commonly known distribution. Since players cannot be distinguished by names in such a setting, the equilibrium concept cannot independently associate a strategy with each individual player. Instead, each type of player is associated with a strategy in equilibrium.

We shall go on to show that if the number of contestants is known to be at least one, then aggregate equilibrium expenditure in a contest where the number of contestants is a nontrivial random variable with expectation [mu] is strictly lower than in a nonuncertain contest with [mu]...

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