When selling items separately earns the most: a finite-value auction result

Selling products individually is simple, but a seller might earn more by offering bundles or lotteries. In Optimality of Separate Selling with Finite-Support Values, Sho Mikami of the University of Tokyo presents a mathematical characterization of when those more elaborate options cannot improve expected revenue. Submitted to arXiv on November 4, 2026, the supplied version is a preprint; the landing metadata does not establish peer-reviewed publication.

What the model assumes

The paper considers a seller with at least two different items and buyers whose values are additive: a buyer’s value for a bundle equals the sum of their values for its items. Buyers are risk-neutral, and utility is the value received minus payment.

Every buyer–item value is drawn independently from a known distribution with finitely many possible, nonnegative values. For a given item, all buyers face the same distribution, although distributions can differ between items. Each item has at least two possible values, all with positive probability.

“Separate selling” means running an optimal single-item auction for each item independently—not choosing arbitrary prices or using an arbitrary familiar auction format. “Top selling” is more restrictive: an item sells only when someone reports its highest possible value, and the winner pays that value. Otherwise, it remains unsold.

The main equivalence requires at least two items to have a strictly positive value below their respective highest values. That technical restriction is essential, not incidental.

A one-buyer test for multiple-buyer auctions

Mikami’s central theorem states that, under these assumptions, separate selling maximizes expected revenue for any fixed number of buyers of at least two if and only if top selling is optimal for one buyer.

Consequently, for fixed item distributions, the answer does not change as the number of buyers increases beyond two. Either separate selling is exactly optimal throughout that range, or it is not exactly optimal for any finite buyer count in that range.

The theorem covers two different incentive standards. Dominant-strategy incentive compatibility means truthful reporting is optimal regardless of other buyers’ reports, paired here with nonnegative truthful utility at every profile. Bayesian incentive compatibility requires truthfulness only in expectation over other buyers’ values, paired with nonnegative expected utility for each buyer type. The paper establishes the equivalence under both standards.

This does not say competition is irrelevant. The paper explains that separate selling can approach optimal revenue as buyer numbers grow. Exact equality is a stronger requirement than a shrinking revenue gap.

Why values near the top matter

The author calls the common economic feature behind the distributional restrictions “top-value isolation.” Values below the maximum must be sufficiently low, or sufficiently uncommon relative to the maximum, to limit profitable alternatives to top selling.

One necessary test, the top–tail revenue bound, examines every possible below-maximum threshold for each item. It compares the probability of values between that threshold and the maximum, multiplied by the threshold, with the maximum’s probability multiplied by the remaining value gap. The largest ratio for each item is selected, and their sum must not exceed one.

Passing this test alone is not enough. The paper gives two independent, identically distributed items valued at 0.2, 0.6 or 1, with probabilities 0.55, 0.10 and 0.35. The bound holds, and separate top-price sales earn expected revenue of 0.70 from one buyer. Yet a bundle priced at 1.20 earns 0.705 when the buyer purchases at indifference. Even this small improvement defeats exact optimality.

Stronger conditions and proof certificates

Under Myerson regularity—a condition requiring the paper’s discrete virtual values to be nondecreasing—the characterization becomes simpler. For each item, multiply its second-highest value by that value’s probability, then divide by the top value’s probability times the gap between the two values. Top selling is optimal exactly when these ratios sum to at most one.

For general finite supports, the paper gives an exact test using “top-preserving flows”: mathematical certificates from linear-programming duality that connect buyer types while preserving coordinates already at their maximum. Checking them can require a rapidly growing linear program. A stronger, easier-to-state slice-tail condition guarantees optimality through upward-only flows, but is not necessary. Another sufficient route combines the top–tail bound with a boundary-dominance condition.

These are theoretical results, not empirical estimates or field-tested auction recommendations. They do not establish the same conclusions for correlated values, continuous distributions, nonadditive preferences or buyer-specific distributions. The supplied extraction includes proofs and appendices, although some mathematical formatting is disrupted; it supports reporting the stated results, not claiming an independent verification of every proof.