Incentive Compatible Budget Elicitation in Multi-unit Auctions

In this paper, we consider the problem of designing incentive compatible auctions for multiple (homogeneous) units of a good, when bidders have private valuations and private budget constraints. When only the valuations are private and the budgets are public, Dobzinski {\em et al} show that the {\em adaptive clinching} auction is the unique incentive-compatible auction achieving Pareto-optimality. They further show thatthere is no deterministic Pareto-optimal auction with private budgets. Our main contribution is to show the following Budget Monotonicity property of this auction: When there is only one infinitely divisible good, a bidder cannot improve her utility by reporting a budget smaller than the truth. This implies that a randomized modification to the adaptive clinching auction is incentive compatible and Pareto-optimal with private budgets. The Budget Monotonicity property also implies other improved results in this context. For revenue maximization, the same auction improves the best-known competitive ratio due to Abrams by a factor of 4, and asymptotically approaches the performance of the optimal single-price auction. Finally, we consider the problem of revenue maximization (or social welfare) in a Bayesian setting. We allow the bidders have public size constraints (on the amount of good they are willing to buy) in addition to private budget constraints. We show a simple poly-time computable 5.83-approximation to the optimal Bayesian incentive compatible mechanism, that is implementable in dominant strategies. Our technique again crucially needs the ability to prevent bidders from over-reporting budgets via randomization.

💡 Research Summary

This paper tackles the design of incentive‑compatible auctions for selling multiple homogeneous units of a good when each bidder has a private per‑unit valuation and a private budget constraint. In the setting where budgets are public, Dobzinski et al. (2012) introduced the adaptive clinching auction, proved it to be the unique mechanism that is both incentive‑compatible and Pareto‑optimal, and showed that no deterministic Pareto‑optimal mechanism can be truthful when budgets are private. The authors of the present work make two major contributions that overcome this impossibility.

First, they identify a structural property of the adaptive clinching auction in the case of a single infinitely divisible good, which they call Budget Monotonicity: a bidder cannot increase her expected utility by reporting a budget lower than her true budget. The proof relies on a careful coupling of two executions of the auction—one with the true budget and one with a lower reported budget—tracking the evolution of demands, supplies, and remaining budgets over the continuous price‑rise process. A key lemma shows that at any price level the set of bidders receiving a positive allocation consists precisely of those with the highest remaining budgets, a property that holds only for divisible goods. The authors also demonstrate that Budget Monotonicity fails for the same auction when the good consists of a finite number of indivisible units, highlighting a sharp distinction between the divisible and indivisible cases.

Armed with Budget Monotonicity, the paper introduces a simple randomized extraction scheme to eliminate any incentive for over‑reporting a budget. In the deterministic adaptive clinching auction, a bidder i who is supposed to pay Pi (with Pi ≤ reported budget Bi) is instead charged her full reported budget Bi with probability Pi/Bi and charged zero otherwise. This randomization preserves the expected payment (hence voluntary participation) while making the expected utility of any over‑reporting strategy negative infinity, because the bidder would have to risk paying more than her true budget. Consequently, the randomized mechanism satisfies (VP) and (IC) in expectation, (NPT) always, and remains Pareto‑optimal ex‑post. Moreover, because the underlying deterministic auction already enjoys superior revenue guarantees in the public‑budget case, the randomized version inherits these guarantees: it improves the competitive ratio of Abrams (2011) by a factor of four and is asymptotically optimal, matching the performance of the optimal single‑price auction as the market grows large.

Second, the authors address the Bayesian setting where each bidder’s valuation, budget, and a public size constraint are drawn from independent discrete distributions known to the auctioneer. The optimal Bayesian incentive‑compatible mechanism can be expressed as an exponential‑size linear program, but solving it directly is computationally infeasible. By encoding only the “budget under‑reporting” constraints, the authors obtain a polynomial‑size LP relaxation that remains linear despite the presence of budgets. They then devise a novel rounding technique that converts the fractional LP solution into a feasible mechanism while preserving incentive compatibility via the same randomized extraction idea. The resulting mechanism achieves a 5.83‑approximation to the optimal expected revenue (or social welfare) and can be implemented in dominant strategies.

In summary, the paper makes three key advances: (1) it proves Budget Monotonicity for the adaptive clinching auction with a divisible good, (2) it shows how a minimal randomization of payments yields a truthful, Pareto‑optimal auction even with private budgets, and (3) it provides a polynomial‑time, constant‑factor approximation algorithm for Bayesian revenue (or welfare) maximization under private budgets and public size constraints. These results bridge a gap between theory and practice for markets where participants face hard budget limits, such as spectrum auctions or online advertising, and open avenues for extending budget‑monotone designs to other auction formats and to settings with indivisible goods.