When stronger complements make bundling the best sales strategy

A software platform can offer a basic package with optional modules, while refusing to sell those modules independently. That arrangement seems natural when the products work better together. But a seller might still benefit from offering an inferior standalone option: it can help separate buyers with different willingness to pay.

In Bundling Complements, Stanford researcher Weijie Zhong asks which force wins as products become more valuable together. The November 10, 2026 revision is an arXiv preprint. Its findings are mathematical results within a specified monopoly model, not an empirical demonstration that existing subscription packages maximize revenue.

What becomes more complementary?

The model has one seller offering several indivisible goods to a buyer whose private information has multiple dimensions. The seller knows the distribution of buyer types, but not the particular buyer’s values. Buyers choose the option giving them the greatest value minus payment, including the option of buying nothing.

Zhong designates some goods as a core. A parameter called the premium multiplies the sum of a buyer’s component values whenever a bundle contains the entire core. Bundles missing part of the core receive no multiplier. A premium of one gives additive values; above one, core-containing bundles gain extra value.

The question is whether increasing that premium eventually makes a core-peripheral menu optimal: the core is sold only as a block, and peripheral goods are available only alongside it. Importantly, the comparison allows competing mechanisms to offer lotteries over bundles, not just ordinary posted prices.

Turning pricing into a transport problem

Finding the best menu is difficult because changing one option can change which buyers choose several others. Zhong develops a mathematical alternative: bound the seller’s revenue through a minimum-cost flow problem across valuation vectors.

This is not a model of shipping products. The flow represents constraints on how much buyers’ utilities can differ. Its minimum cost equals the maximum revenue achievable by a menu, providing a way to certify optimality.

For linear valuation maps—including the premium model—the method separates two ingredients. The distribution of buyer types determines the utility comparisons that the certificate must accommodate. Preferences determine which bundles can generate the largest utility differences along those comparisons.

The paper describes these as “required” and “covered” directions. Raising the premium expands the directions covered by core-containing bundles without changing the underlying distribution-based accounting. With sufficient coverage, a core-peripheral menu can match the revenue bound and therefore beat every alternative mechanism.

Two thresholds, with different meanings

The first threshold concerns the grand bundle containing every good. Under the paper’s assumptions, once a positive-measure set of buyers values it at least as highly as every other bundle, every optimal menu must sell that deterministic grand bundle to a positive-measure set of buyers.

That does not mean selling only the grand bundle is optimal. The second, higher threshold concerns when a core-peripheral menu becomes optimal against unrestricted alternatives. Zhong proves a finite threshold exists when the candidate menu satisfies both inclusivity and a technical condition called strict core regularity. Above it, every optimal menu sells only core-containing bundles.

Core regularity compares the distribution-based sources and sinks within the regions where buyers select each option. It is an assumption with economic consequences, not something automatically guaranteed by stronger complementarity.

Why excluded buyers matter

An inclusive menu serves every buyer who values at least one good at its maximum possible level—not just buyers who value everything highly.

If a finite menu excludes some such buyers, no premium makes it optimal in the unrestricted problem under the theorem’s assumptions. Offering an individual good at a price slightly below its maximum value can improve revenue. Increasing the attractiveness of existing bundles does not eliminate that opportunity.

For pure bundling, where all goods form the core, the result is sharper. Under the stated regularity conditions, the revenue-maximizing bundle price must leave positive surplus for every buyer who values some component at its maximum. Otherwise, pure bundling remains suboptimal at every premium.

A benchmark, not a universal prescription

With two independent component values uniformly distributed between zero and one, and both goods in the core, the lower threshold is one-half. At the additive point, the optimum uses mixed bundling. Zhong certifies pure bundling for premiums of approximately 1.253 or more. That number is a sufficient bound, not the exact transition point.

The conclusions concern seller revenue, not consumer welfare. They also depend on the valuation structure and distribution assumptions. The broader lesson is conditional: stronger complementarity can simplify the optimal sales menu, but it cannot automatically repair a menu that excludes valuable customers.