Risk Aversion Can Break a Standard Bidding Assumption in Multiunit Auctions

A limit to an intuitive bidding rule

In auctions selling several identical units, buyers choose both how much to bid and how many units to pursue. A common analytical assumption is monotonicity: if a bidder’s values for every successive unit increase, none of her bids decreases.

Wei He, Yeneng Sun and Hanping Xu challenge the availability of equilibria with this property in Universal Nonexistence of Monotone Equilibrium in Multiunit Auctions with Risk Aversion. The source is arXiv version 2611.06644v1, posted November 4, 2026, and is a theoretical preprint. Its findings concern mathematical auction models, not an empirical test of bidding behavior.

The crucial qualification is what “universal” means. The authors do not claim that all risk-averse multiunit auctions lack monotone equilibrium. Rather, for every permitted utility profile and bidder count, they construct distributions of private values that defeat the relevant existence guarantee.

What the model assumes

There are at least two bidders. Each privately observes a vector of marginal values: the value of the first unit, the second, and so on. These values are between zero and one and weakly decrease across successive units. Types are independent across bidders, although a bidder’s own values can be dependent.

Each bidder submits an ordered, nonincreasing bid vector. The highest unit bids win. Ties use a randomly drawn bidder priority, independent of values and applied consistently across tied bids.

Bidders maximize expected utility of their monetary surplus. All utility functions are strictly increasing and concave; at least one is strictly concave, representing strict risk aversion. The others may be risk neutral. The authors impose neither differentiability nor a minimum degree of risk aversion.

A pure strategy assigns a definite bid vector to each type rather than randomizing. In Bayesian equilibrium, almost every type is choosing a best response to opponents’ strategies. Monotonicity adds the requirement that coordinatewise higher values produce coordinatewise no-lower bids.

Different results for different payment rules

Under uniform pricing, every winning unit costs the same amount. The paper considers both the lowest accepted bid and the highest rejected bid as that common price.

For every supply of at least three units, the authors construct independent type distributions under which neither rule admits an admissible monotone pure-strategy equilibrium. Admissibility excludes bids weakly dominated by another bid or a lottery over bids. Their stronger result rules out no-overbidding pure equilibria even when only the designated risk-averse bidder’s final bid must be monotone.

The constructed distributions have densities bounded above and bounded away from zero across the entire ordered-value space. Thus, the result does not require leaving possible types out of the distribution. The same distributions work for both uniform-price rules—and, the appendix shows, every fixed weighted average of their two cutoff prices.

Under discriminatory, or pay-as-bid, pricing, winners pay their own accepted bids. Here the authors construct a finite supply, potentially dependent on the given utilities, and full-support distributions with no monotone pure-strategy equilibrium. This result needs no admissibility or no-overbidding restriction. It does not establish the claim for every supply of three or more units.

Why a higher value can mean a lower bid

The uniform-price example has two bidders and three units. One bidder faces either a weak rival seeking one unit or a strong rival seeking two.

Against the weak rival, she can obtain two units cheaply or compete for all three. Against the strong rival, she retains only one. Raising her third bid requires raising her second bid too; that losing second bid can then raise the price of her remaining unit.

Now increase her value for the second unit. This raises surplus when she wins two or three units, but not when she wins only one. Strict concavity reduces the utility gain from adding the third unit to the now-more-valuable two-unit purchase. Meanwhile, the utility cost of paying more in the one-unit outcome is unchanged.

The authors choose rival probabilities so these incentives reverse the preferred third bid. Small increases in the other values preserve the reversal: a bidder can value every unit more yet optimally bid less for the third.

What the findings do—and do not—establish

The paper also proves nonexistence results for all-pay auctions, where rejected bids cost money, and qualified extensions to finite bid grids. For discriminatory and all-pay auctions, suitable fixed counterexamples survive every sufficiently fine grid. Separate coarse-grid constructions do not imply robustness to arbitrary added bid levels.

An important boundary is the two-unit case. With constant absolute risk aversion, including heterogeneous coefficients and risk neutrality, the authors prove existence of monotone no-overbidding pure equilibria under both uniform-price rules and discriminatory pricing.

The broader conclusion is therefore specific: risk aversion can invalidate monotone pure-strategy equilibrium as a general modeling guarantee. The results do not rule out nonmonotone or mixed-strategy equilibria, nor do they establish which auction format delivers better revenue or allocation.