An efficiency ordering of k-price auctions under complete information

TL;DR

This paper analyzes k-price auctions with complete information, characterizing equilibrium outcomes and showing how welfare varies with the auction type, highlighting the efficiency of first-price auctions.

Contribution

It provides a complete characterization of pure-strategy Nash equilibria in k-price auctions and compares their worst-case welfare outcomes.

Findings

Agents with higher valuations can win in equilibrium.

Welfare increases as k increases, from second-price to lowest-price auctions.

First-price auctions have the highest worst-case welfare.

Abstract

We study $k$-price auctions in a complete information environment and characterize all pure-strategy Nash equilibrium outcomes. In a setting with $n$ agents having ordered valuations, we show that any agent, except those with the lowest $k-2$ valuations, can win in equilibrium. As a consequence, worst-case welfare increases monotonically as we go from $k=2$ (second-price auction) to $k=n$ (lowest-price auction), with the first-price auction achieving the highest worst-case welfare.