We study a contest with multiple (not necessarily equal) prizes. Contestants have private information about an ability parameter that affects their costs of bidding. The contestant with the highest bid wins the first prize, the contestant with the second-highest bid wins the second prize, and so on until all the prizes are allocated. All contestants incur their respective costs of bidding. The contest's designer maximizes the expected sum of bids. Our main results are: 1) We display bidding equlibria for any number of contestants having linear, convex or concave cost functions, and for any distribution of abilities. 2) If the cost functions are linear or concave, then, no matter what the distribution of abilities is, it is optimal for the designer to allocate the entire prize sum to a single ''first'' prize. 3) We give a necessary and sufficient conditions ensuring that several prizes are optimal if contestants have a convex cost function.
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