# Revenue Loss in Shrinking Markets

## TL;DR

This paper investigates how even minimal market shrinkage, such as the retirement of a lower-value bidder, can cause a substantial decrease in revenue, addressing an open problem in auction theory.

## Contribution

It demonstrates that market shrinkage can lead to a multiplicative revenue loss of approximately 26.8%, solving an open question in auction revenue analysis.

## Key findings

- Market shrinkage causes significant revenue loss.
- Retirement of a lower-value bidder impacts revenue substantially.
- Provides a bound of 1/(e+1) for revenue decrease.

## Abstract

We analyze the revenue loss due to market shrinkage. Specifically, consider a simple market with one item for sale and $n$ bidders whose values are drawn from some joint distribution. Suppose that the market shrinks as a single bidder retires from the market. Suppose furthermore that the value of this retiring bidder is fixed and always strictly smaller than the values of the other players. We show that even this slight decrease in competition might cause a significant fall of a multiplicative factor of $\frac{1}{e+1}\approx0.268$ in the revenue that can be obtained by a dominant strategy ex-post individually rational mechanism. In particular, our results imply a solution to an open question that was posed by Dobzinski, Fu, and Kleinberg [STOC'11].