This paper shows that for distributions that have linear density, the bid function at any symmetric, increasing equilibrium of a kth price auction with k = 3 can be represented as a finite series of k-2 terms whose lth term involves the lth Catalan number. Using an integral representation of Catalan numbers together with some classical combinatorial identities, we derive the closed form of the unique symmetric, increasing equilibrium of a kth price auction for a non-uniform distribution. (C) 2018 Elsevier B.V. All rights reserved.
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