Optimal Auction Design Under Asymmetric Information

Abstract

We study the problem of a seller who wishes to maximize expected revenue when allocating a single indivisible good to $n$ bidders. Each bidder $i$ privately observes a valuation $v_i \sim F_i$ drawn from a distribution that may differ across bidders. Following Myerson (1981), we characterize the optimal mechanism via virtual valuations and extend the analysis to settings with correlated types.

1. Introduction

The design of revenue-maximizing auctions is a central problem in mechanism design. Myerson's (1981) celebrated result shows that under independent private values, the optimal direct revelation mechanism takes a clean virtual-value form: allocate to the bidder with the highest virtual valuation, provided it is non-negative.

This paper revisits that framework under two generalizations:

1. Correlated types — bidder valuations are drawn from a joint distribution with non-trivial dependence structure.
2. Interdependent values — bidder $i$'s willingness to pay depends not only on her own signal $s_i$ but on the vector $s = (s_1, \ldots, s_n)$.

2. Model

Let $\mathcal{N} = \{1, \ldots, n\}$ be the set of bidders. A direct revelation mechanism is a pair $(q, t)$ where:

• $q : \mathbb{R}^n \to \Delta(\mathcal{N} \cup \{0\})$ maps reported type profiles to allocation probabilities;
• $t : \mathbb{R}^n \to \mathbb{R}^n$ maps reported profiles to transfer payments.

A mechanism is incentive compatible (IC) if truthful reporting is a Bayes-Nash equilibrium, and individually rational (IR) if each bidder weakly prefers participation over opting out.

3. Main Result

Theorem 1. Under independent private values with regular distributions $F_1, \ldots, F_n$, the revenue-maximizing IC and IR mechanism allocates the good to the bidder $i^* = \arg\max_i \psi_i(v_i)$ where

$\psi_i(v) = v - \frac{1 - F_i(v)}{f_i(v)}$

is the virtual valuation of bidder $i$, provided $\psi_{i^*}(v_{i^*}) \geq 0$.

The proof follows from the standard envelope argument and the observation that IC pins down transfers up to a constant once allocation rules are fixed.

References

• Myerson, R. B. (1981). Optimal auction design. Mathematics of Operations Research, 6(1), 58–73.
• Bulow, J., & Roberts, J. (1989). The simple economics of optimal auctions. Journal of Political Economy, 97(5), 1060–1090.