Efficiency Loss in Revenue Optimal Auctions

TL;DR

This paper analyzes the efficiency loss in Bayesian revenue optimal auctions with single-parameter buyers, providing bounds on the worst-case social welfare loss in various valuation scenarios.

Contribution

It quantifies the worst-case efficiency loss ratio in revenue optimal auctions with finite valuation sets and derives bounds for different valuation and buyer configurations.

Findings

Worst case efficiency loss ratio is at most 1/2 for binary valuations.

Efficiency loss bounds depend on number of buyers and valuation set size.

Results extend to i.i.d. valuations with explicit bounds.

Abstract

We study efficiency loss in Bayesian revenue optimal auctions. We quantify this as the worst case ratio of loss in the realized social welfare to the social welfare that can be realized by an efficient auction. Our focus is on auctions with single-parameter buyers and where buyers' valuation sets are finite. For binary valued single-parameter buyers with independent (not necessarily identically distributed) private valuations, we show that the worst case efficiency loss ratio (ELR) is no worse than it is with only one buyer; moreover, it is at most 1/2. Moving beyond the case of binary valuations but restricting to single item auctions, where buyers' private valuations are independent and identically distributed, we obtain bounds on the worst case ELR as a function of number of buyers, cardinality of buyers' valuation set, and ratio of maximum to minimum possible values that buyers can have for the item.