Auctions with Online Supply

We study the problem of selling identical goods to n unit-demand bidders in a setting in which the total supply of goods is unknown to the mechanism. Items arrive dynamically, and the seller must make the allocation and payment decisions online with the goal of maximizing social welfare. We consider two models of unknown supply: the adversarial supply model, in which the mechanism must produce a welfare guarantee for any arbitrary supply, and the stochastic supply model, in which supply is drawn from a distribution known to the mechanism, and the mechanism need only provide a welfare guarantee in expectation. Our main result is a separation between these two models. We show that all truthful mechanisms, even randomized, achieve a diminishing fraction of the optimal social welfare (namely, no better than a Omega(loglog n) approximation) in the adversarial setting. In sharp contrast, in the stochastic model, under a standard monotone hazard-rate condition, we present a truthful mechanism that achieves a constant approximation. We show that the monotone hazard rate condition is necessary, and also characterize a natural subclass of truthful mechanisms in our setting, the set of online-envy-free mechanisms. All of the mechanisms we present fall into this class, and we prove almost optimal lower bounds for such mechanisms. Since auctions with unknown supply are regularly run in many online-advertising settings, our main results emphasize the importance of considering distributional information in the design of auctions in such environments.

💡 Research Summary

The paper investigates online auctions for identical items when the total supply is unknown to the mechanism. A fixed set of n unit‑demand bidders submit their values before any items arrive, and items appear one by one. The mechanism must allocate each arriving item immediately and charge the winner at that moment (the “prompt” or “online‑envy‑free” requirement). The objective is to maximize social welfare (the sum of values of allocated bidders) while guaranteeing truthfulness: reporting true values must be a dominant strategy for every bidder, regardless of others’ reports, the random bits of the mechanism, or the realized supply.

Two models of supply uncertainty are considered. In the adversarial supply model, the mechanism must achieve a welfare guarantee for every possible supply ℓ. The authors prove strong impossibility results. Any deterministic truthful mechanism can do no better than an n‑approximation (trivially achieved by allocating the first item to the highest bidder and ignoring the rest). For randomized truthful mechanisms they derive a much tighter lower bound: no mechanism can guarantee better than an Ω(log log n) approximation. The proof proceeds by characterizing truthful mechanisms through a system of equations linking allocation probabilities and payments, then exhibiting a distribution over bidder values for which these equations have no feasible solution unless the approximation factor is at least Ω(log log n). When the additional natural fairness constraint of online‑envy‑freeness is imposed (all bidders face the same price rule and payments are collected instantly), the lower bound strengthens to Ω(log n / log log n). A matching upper bound of O(log n) is given by a simple randomized mechanism, leaving only a small gap for non‑envy‑free random mechanisms.

In the stochastic supply model, the supply ℓ is drawn from a known distribution D. The authors assume D satisfies a non‑decreasing hazard rate (MHR), a standard condition that holds for exponential, uniform, binomial, and many other natural distributions. Under this assumption they construct a deterministic, truthful, and computationally efficient mechanism that achieves a constant‑factor approximation to optimal expected welfare. The mechanism pre‑computes a “threshold” for each rank based on the hazard rate of D, allocates each arriving item to the highest‑valued unserved bidder, and charges that bidder the smallest value that would still have secured the item given the remaining supply distribution (a VCG‑style critical value). Truthfulness holds pointwise for every supply realization, not merely in expectation, which is crucial because bidders may not share the seller’s knowledge of D. The authors also prove that the MHR condition is necessary: for distributions with decreasing hazard rates, no deterministic truthful mechanism can achieve a constant approximation; the best possible factor degrades to Ω(log n / log log n). Thus, distributional information about supply dramatically changes what is achievable.

The paper further explores a subclass of bidders with knapsack (single‑minded) valuations, where each bidder desires a specific bundle of items. Even in this restricted setting, the authors derive strong lower bounds on the approximation ratio in both adversarial and stochastic models, and they present matching algorithms, showing that the hardness is not merely an artifact of unit‑demand preferences.

Related work is discussed in depth. Mahdian and Saberi (2010) studied unknown supply but allowed payments after the entire supply was exhausted, and they focused on revenue rather than welfare. Cole, Dobzinski, and Fleischer (2013) introduced the promptness requirement but assumed a known, fixed supply. Lavi and Nisan (2000) and Hajiaghaei et al. (2009) examined online allocation of expiring items, again with known supply. Recent Bayesian online mechanism design papers assume full priors over bidders’ values, which this work deliberately avoids. By contrast, the present paper isolates the effect of supply uncertainty while keeping bidders’ values completely unrestricted, thereby highlighting the intrinsic difficulty of achieving welfare‑optimal truthful online allocation without supply information.

In conclusion, the authors demonstrate a sharp separation between the adversarial and stochastic settings. Without any distributional knowledge of supply, truthful mechanisms cannot beat a polylogarithmic approximation, even with randomization. With modest distributional knowledge (MHR), a simple deterministic mechanism attains a constant factor. This underscores the practical importance of gathering reliable supply statistics (e.g., click‑through rates, impression volumes) in online advertising platforms, as such data enable the design of truthful, welfare‑maximizing auctions that are both computationally tractable and implementable in real‑time.