$k$th price auctions and Catalan numbers

TL;DR

This paper reveals a novel mathematical connection between $k$th price auctions and Catalan numbers, providing a closed-form solution for the equilibrium bid function for certain distributions.

Contribution

It introduces a new link between auction theory and combinatorics, deriving explicit equilibrium bid functions using Catalan numbers for non-uniform distributions.

Findings

Bid functions can be expressed as finite series involving Catalan numbers.

Derived closed-form equilibrium bid functions for non-uniform distributions.

Established a mathematical link between auction theory and combinatorial identities.

Abstract

This paper establishes an interesting link between $k$th price auctions and Catalan numbers by showing that for distributions that have linear density, the bid function at any symmetric, increasing equilibrium of a $k$th price auction with $k\geq 3$ can be represented as a finite series of $k-2$ terms whose $\ell$th term involves the $\ell$th 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 $k$th price auction for a non-uniform distribution.