Revenue-Maximizing Mechanism Design for Quasi-Proportional Auctions

TL;DR

This paper analyzes how the choice of weight functions in quasi-proportional auctions affects equilibrium outcomes and revenue, revealing conditions under which different functions optimize revenue based on bidder valuation and number.

Contribution

It provides a comprehensive analysis of the relationship between weight functions, bidder valuations, and revenue, including equilibrium existence, uniqueness, and revenue bounds.

Findings

Steeper weight functions maximize revenue as bidder valuations become similar.

Pure-strategy Nash equilibria exist for both flat and steep weight functions.

Optimal weight function selection depends on private value ratios and number of bidders.

Abstract

In quasi-proportional auctions, each bidder receives a fraction of the allocation equal to the weight of their bid divided by the sum of weights of all bids, where each bid's weight is determined by a weight function. We study the relationship between the weight function, bidders' private values, number of bidders, and the seller's revenue in equilibrium. It has been shown that if one bidder has a much higher private value than the others, then a nearly flat weight function maximizes revenue. Essentially, threatening the bidder who has the highest valuation with having to share the allocation maximizes the revenue. We show that as bidder private values approach parity, steeper weight functions maximize revenue by making the quasi-proportional auction more like a winner-take-all auction. We also show that steeper weight functions maximize revenue as the number of bidders increases. For flatter weight functions, there is known to be a unique pure-strategy Nash equilibrium. We show that a pure-strategy Nash equilibrium also exists for steeper weight functions, and we give lower bounds for bids at an equilibrium. For a special case that includes the two-bidder auction, we show that the pure-strategy Nash equilibrium is unique, and we show how to compute the revenue at equilibrium. We also show that selecting a weight function based on private value ratios and number of bidders is necessary for a quasi-proportional auction to produce more revenue than a second-price auction.