Analytical solution of $k$th price auction

TL;DR

This paper derives an exact analytical solution for the Nash equilibrium in the $k$th price auction using inverse distribution functions, covering various valuation distributions.

Contribution

It introduces a method to explicitly solve for Nash equilibria in $k$th price auctions across multiple valuation distributions.

Findings

Explicit solutions for polynomial, fat tail, and exponential valuation distributions.

Unique symmetric equilibrium identified for each distribution type.

Method enhances understanding of strategic bidding in auction theory.

Abstract

We provide an exact analytical solution of the Nash equilibrium for the $k$th price auction by using inverse of distribution functions. As applications, we identify the unique symmetric equilibrium where the valuations have polynomial distribution, fat tail distribution and exponential distributions.