This paper studies equilibrium behavior in a class of games that models asymmetric competitions with unconditional and conditional investments. Such competitions include lobbying settings, labor-market tournaments, and R& races, among others. I provide an algorithm that constructs the unique equilibrium in these games and apply it to study competitions in which a fraction of each competitor's investment is sunk and the rest is paid only by the winners. Complete-information all-pay auctions are a special case. (JEL D44, D72, D82)

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We consider contests with many, possibly heterogeneous, players and prizes, and show that the equilibrium outcomes of such contests are approximated by the outcomes of mechanisms that implement the assortative allocation in an environment with a single agent that has a continuum of possible types. This makes it possible to easily approximate the equilibria of contests whose exact equilibrium characterization is complicated, as well as the equilibria of contests for which there is no existing equilibrium characterization.

This paper studies multiprize contests in which players' costs need not be strictly increasing in their performance. Such costs accommodate various types of asymmetries, including head starts. Head starts capture incumbency advantages, prior investments, and technological differences. I provide an algorithm that constructs the unique equilibrium in which players do not choose weakly- dominated strategies, and apply it to study multiprize all-pay auctions with head starts. A comparison to the standard all-pay auction shows that the strategic effects of head starts differ substantially from those of differing valuations. (JEL D11, D44)

Many sales, sports, and research contests are put in place to maximize contestants' performance. We investigate and provide a complete characterization of the prize structures that achieve this objective in settings with many contestants. The contestants may be ex ante asymmetric in their abilities and prize valuations, and there may be complete or incomplete information about these parameters. The prize valuations and performance costs may be linear, concave, or convex. A main novel takeaway is that awarding numerous different prizes whose values gradually decline with contestants' ranking is optimal in the typical case of contestants with convex performance costs and concave prize valuations. This suggests that many real-world contests can be improved by increasing the number of prizes and making them more heterogeneous. The techniques we develop can also be used to formulate and solve other contest design questions that have so far proven intractable.