Optimal commitments in auctions with incomplete information

TL;DR

This paper introduces a novel approach to determine optimal commitments in rank-and-bid auctions, including first-price and all-pay types, revealing strategic bidding behaviors and offering closed-form solutions for various distributions.

Contribution

The paper develops a new method using equal-bid functions and equal-utility curves to solve for optimal commitments in auctions with continuous type distributions, applicable to multiple auction formats.

Findings

Optimal commitment strategies involve passive bidding at low types.

High types may bid above their own value as a threat.

Leader's strategies can be used to secure more utility in middle types.

Abstract

We are interested in the problem of optimal commitments in rank-and-bid based auctions, a general class of auctions that include first price and all-pay auctions as special cases. Our main contribution is a novel approach to solve for optimal commitment in this class of auctions, for any continuous type distributions. Applying our approach, we are able to solve optimal commitments for first-price and all-pay auctions in closed-form for fairly general distribution settings. The optimal commitments functions in these auctions reveal two surprisingly opposite insights: in the optimal commitment, the leader bids passively when he has a low type. We interpret this as a credible way to alleviate competition and to collude. In sharp contrast, when his type is high enough, the leader sometimes would go so far as to bid above his own value. We interpret this as a credible way to threat. Combing both insights, we show via concrete examples that the leader is indeed willing to do so to secure more utility when his type is in the middle. Our main approach consists of a series of nontrivial innovations. In particular we put forward a concept called equal-bid function that connects both players' strategies, as well as a concept called equal-utility curve that smooths any leader strategy into a continuous and differentiable strategy. We believe these techniques and insights are general and can be applied to similar problems.