On the Inefficiency of the Uniform Price Auction

TL;DR

This paper analyzes the strategic behavior and efficiency of Uniform Price Auctions with multiple identical units and submodular bidders, providing bounds on social welfare in equilibrium states.

Contribution

It characterizes pure Nash equilibria in undominated strategies and establishes tight bounds on social welfare efficiency for both complete and incomplete information settings.

Findings

A fraction 1-1/e of the optimal welfare is always achieved in pure Nash equilibria.

The inefficiency bound for Bayes Nash equilibria is 4-2/k.

The results are tight bounds on social welfare in strategic uniform price auctions.

Abstract

We present our results on Uniform Price Auctions, one of the standard sealed-bid multi-unit auction formats, for selling multiple identical units of a single good to multi-demand bidders. Contrary to the truthful and economically efficient multi-unit Vickrey auction, the Uniform Price Auction encourages strategic bidding and is socially inefficient in general. The uniform pricing rule is, however, widely popular by its appeal to the natural anticipation, that identical items should be identically priced. In this work we study equilibria of the Uniform Price Auction for bidders with (symmetric) submodular valuation functions, over the number of units that they win. We investigate pure Nash equilibria of the auction in undominated strategies; we produce a characterization of these equilibria that allows us to prove that a fraction 1-1/e of the optimum social welfare is always recovered in undominated pure Nash equilibrium -- and this bound is essentially tight. Subsequently, we study the auction under the incomplete information setting and prove a bound of 4-2/k on the economic inefficiency of (mixed) Bayes Nash equilibria that are supported by undominated strategies.