An auction model links seller revenue to who gets to learn

Before bidding for a business, a buyer may need access to financial records, management presentations or specialist reports. A seller controlling that access also shapes what bidders learn. A new auction-design preprint examines how that power interacts with two private facts: bidders’ initial beliefs about value and their costs of investigating it.

An auction’s design can begin before anyone submits a final bid. When a seller controls opportunities to investigate an asset, deciding who learns what becomes part of the selling mechanism.

In Auction Design with Private Priors and Costly Learning, Kemal Ozbek develops a theoretical framework for this problem. The preprint studies how a seller can jointly choose information acquisition, subsequent allocation and payments when bidders privately know both their initial beliefs about value and their learning costs. Its results concern that mathematical environment, rather than measured outcomes from real auctions.

Two kinds of private information

The model has a single indivisible object and at least two risk-neutral bidders. Each bidder’s underlying valuation takes one of two possible values. Before learning more, she privately knows the probability that her valuation is the higher one, along with a personal scale determining the cost of an assigned experiment.

The seller first collects reports about those private characteristics and assigns experiments. After receiving private signals, bidders report signal labels, and the mechanism determines allocation and payments.

The difficulty is that the two reporting problems interact. A bidder’s initial belief determines how useful information is to her and whether she would benefit from pretending to hold another belief. Such a misreport can change her experiment, the allocation rule and her payments. Screening learning costs therefore cannot generally be separated from eliciting initial beliefs.

Revenue is not the same as efficiency

Ozbek derives a revenue formula combining allocated posterior expected values, adjusted learning costs and prior-dependent information rents. These rents are utilities left to bidders under the mechanism’s incentive and participation requirements.

The learning-cost adjustment includes the informational rent associated with screening private costs. Revenue consequently evaluates experimentation using a “virtual” cost scale rather than the actual cost scale used in the welfare calculation.

Private beliefs introduce another constraint: the rents left to different belief types must discourage imitation between them. Changing the allocation rule after learning can alter those required rents. Giving the object to the bidder with the highest posterior expected value therefore need not maximize seller revenue.

For finitely many possible initial beliefs, the paper expresses the remaining cross-belief incentives as directed difference inequalities. These describe how much one belief type’s boundary utility must exceed another’s to deter a deviation.

A small example with a strict gain

The paper illustrates the distinction with two bidders whose valuations are either zero or one. Their private probabilities of having the higher valuation are equally likely to be one-fifth or two-fifths. Learning-cost scales are independently uniform between zero and one. The experiment menu contains a free uninformative option and a costly informative option.

When both bidders always receive the informative experiment, the paper derives optimal revenue of 19/49. By contrast, every dynamically incentive-compatible, individually rational mechanism with those fixed experiments and efficient continuation allocation earns at most 73/196.

The revenue-optimal fixed-information rule sometimes withholds the object even though posterior values are positive. Specifically, it withholds when both bidders have the lower prior and receive negative signals. This departure from efficiency reduces the rent burden associated with private beliefs.

Allowing experiment assignments to depend on reported beliefs, but not reported learning costs, does not improve optimal revenue in this example. Conditioning learning on reported costs does: an explicit mechanism earns 3043/7840, exceeding 19/49 by 3/7840. This establishes a strict improvement, not the optimal revenue of the unrestricted cost-contingent problem.

Constructive results, under substantive assumptions

Beyond the example, the paper establishes existence of an optimal mechanism for finite experiment menus, allowing arbitrary prior distributions. With both prior support and experiment menus finite, it constructs finite linear programs whose revenue error decreases proportionally to the largest learning-cost interval used in the approximation. Crucially, the resulting mechanisms preserve exact dynamic incentive compatibility and interim participation for every continuous cost type, not just selected grid points.

These conclusions rely on substantive assumptions. Initial beliefs and learning-cost scales are independent within bidders, and types are independent across bidders. Assigned experiments are enforceable and exclusive: bidders cannot secretly supplement them. Payments may include subsidies. The seller commits to the mechanism, and bidders accepting initially are bound to continue; participation after each signal is not required.

Within those boundaries, the preprint’s message is that information access and auction rules are a joint design problem. More learning and efficient allocation are not automatically the revenue-maximizing combination.