Bigger markets cannot always fix restrictive bundle rules

Two different sources of waste

The researchers study a double auction, where buyers submit bids and sellers submit asking prices. Traders can offer individual goods or packages, but each participant trades at most one bundle with one counterparty.

Their central restriction, called No Recombination, prevents two activities: assembling a buyer’s package from several sellers’ components, and splitting one seller’s package among several buyers.

The auction selects trades that maximise the surplus implied by submitted reports, then chooses the highest prices supporting those trades. Surplus means buyers’ values minus sellers’ opportunity costs: the gains created by exchange, not platform revenue.

The authors compare three situations. An ideal planner knows everyone’s values and costs and can freely assemble or split bundles. Next comes the restricted auction with truthful reporting. Finally, buyers bid strategically within the restricted auction.

The first comparison measures the structural cost of bundle restrictions. The second isolates the additional cost of strategic bidding. The planner assumes recombination is free, so its advantage is an upper bound on what an intermediary could recover before integration or licensing costs.

When growth helps—and when it does not

The main structural results concern two goods. With normally distributed individual shocks to values and costs, the decisive question is whether demand for the two-good package matches the supply of whole packages at the prices that would prevail with free recombination.

If those quantities balance in expectation, the structural loss shrinks as a share of optimal gains from trade. Under the paper’s regularity and moment assumptions, that share falls approximately in inverse proportion to market size. Doubling participation eventually roughly halves it. The absolute amount lost need not disappear.

If package demand and whole-package supply remain mismatched, a positive percentage loss persists even as participation grows. More traders smooth out random shortages; they do not necessarily correct an imbalance built into preferences, production costs or the buyer–seller ratio.

In a mirror-image benchmark—equal numbers of buyers and sellers, Gaussian item shocks and the same distribution of package benefits on both sides—the structural share vanishes. Equal participation alone is insufficient.

For example, with equal numbers and unit-variance Gaussian item shocks, buyers receiving a fixed package benefit of one and sellers receiving no corresponding cost saving produce a limiting structural loss of 1.84% of optimal gains from trade. This is a model calculation, not a measured loss in an operating marketplace.

Strategic bidding has narrower guarantees

For sellers, truthful asking is weakly dominant in this mechanism: misreporting cannot improve their payoff, whatever others report.

The buyer results concern a restricted commitment game. Before observing their values, buyers choose fixed adjustments to their bids for each bundle. Every adjustment must lie within a fixed bounded interval, from minus Λ to plus Λ. This is a substantive assumption, not merely a computational convenience.

Within that bounded game, the authors prove that symmetric equilibria allowing randomised adjustments exist under their stated conditions. For two goods, additional distributional assumptions imply that equilibrium adjustments shrink at least as fast as inverse market size, while their welfare cost as a share shrinks at least as fast as inverse market size squared. These are not existence or efficiency guarantees for unrestricted bidding strategies. Existence of pure, non-randomised symmetric equilibria remains open in general.

What the simulations show

The numerical work mainly uses two goods, equal participation, Gaussian item shocks and exponentially distributed package benefits. Reinforcement learning searches for fixed bid adjustments; no neural network is involved. Welfare comparisons generally use one million simulated markets per main-grid setting, with fewer draws at sixteen traders per side.

Without package benefits, two traders per side produce a structural loss of about 2.7% of optimal welfare and a strategic loss of 8.3% of truthful restricted-auction welfare. At eight per side, those figures are about 1.2% and 0.5%. Their denominators differ, so the percentages should not simply be added.

Structural losses exceed strategic losses from around eight traders per side in this benchmark. That is not a universal participation threshold.

Limits and practical meaning

The learned strategies are equilibrium candidates, not certified equilibria. Searches can miss profitable deviations. With the common value known, separable valuation-dependent bidding rules improve a buyer’s payoff by up to about 4.2% in the reported searches; a separate cross-bundle search reaches 4.42%. These are individual payoff gains, not welfare-loss percentages.

The supplied source is arXiv preprint version 2608.06134v3, dated 8 November 2026; the evidence does not establish peer-reviewed publication. Its licensing and cloud-market examples motivate the restrictions, but the study does not measure their welfare costs in those markets.

The practical lesson is conditional: expanding participation can discipline bidding while leaving restrictive trading rules costly. Relaxing those rules helps only where recombination is feasible and its benefits exceed its costs.