[Paper Review] A Note on Optimal Auctions for Two Uniformly Distributed Items.

Yiannis Giannakopoulos|arXiv (Cornell University)|Sep 24, 2014

Auction Theory and ApplicationsDecision Sciences2 references5 citations

TL;DR

This paper presents a simplified proof of Pavlov's (2011) result on revenue-maximizing auctions for two i.i.d. uniformly distributed items over [c, c+1], using explicit dual solutions to a relaxed problem. It shows that for c = 0 and c ≥ 0.092, the relaxed solution is feasible and optimal, but for 0 < c < 0.092, relaxing convexity incurs a loss, marking the first such example in a two-item i.i.d. setting.

ABSTRACT

We provide a new, much simplified and straightforward proof to a result of Pavlov [2011] regarding the revenue maximizing mechanism for selling two items with uniformly i.i.d. valuations over intervals [c, c + 1], to an additive bidder. This is done by explicitly defining optimal dual solutions to a relaxed version of the problem, where the convexity requirement for the bidder’s utility has been dropped. Their optimality comes directly from their structure, through the use of exact complementarity. For c = 0 and c ≥ 0.092 it turns out that the corresponding optimal primal solution is a feasible auction, thus the initial relaxation comes without a loss, and revenue maximality follows. However, for 0 &amp;lt; c &amp;lt; 0.092 that’s not the case, providing the first clear example where relaxing convexity provably does not come for free, even in a two-item regularly i.i.d. setting. 1

Motivation & Objective

• To simplify and clarify Pavlov's (2011) result on revenue-maximizing auctions for two i.i.d. uniformly distributed items.
• To analyze when relaxing the convexity constraint in the bidder's utility function affects optimality in two-item auction design.
• To identify the threshold c ≈ 0.092 where the relaxation ceases to be without loss, providing a counterexample in a regular i.i.d. setting.
• To demonstrate that dual solutions can directly imply primal optimality via exact complementarity, bypassing complex structural arguments.

Proposed method

• Construct explicit dual solutions for a relaxed auction design problem where the convexity requirement on the bidder's utility is dropped.
• Use exact complementarity between primal and dual solutions to verify optimality of the primal solution.
• Analyze the feasibility of the primal solution derived from the dual: if feasible, it is optimal due to strong duality.
• Compare the structure of the dual solutions across different values of c to determine when the primal solution remains feasible.
• Apply known results from duality theory in linear programming to establish optimality conditions without relying on convex analysis.
• Identify the critical threshold c = 0.092 where the primal solution transitions from feasible to infeasible under the relaxation.

Experimental results

Research questions

• RQ1 Under what conditions is the relaxation of the convexity constraint in the bidder’s utility function without loss in two-item auction design?
• RQ2 Can a simpler, more direct proof be constructed for Pavlov’s (2011) result on optimal auctions for two i.i.d. uniformly distributed items?
• RQ3 What is the precise threshold value of c for which the relaxed solution ceases to be feasible in the original problem?
• RQ4 Does the failure of convexity relaxation in the range 0 < c < 0.092 represent the first known example in a two-item i.i.d. setting where such relaxation incurs a loss?
• RQ5 How can exact complementarity between primal and dual solutions be leveraged to prove optimality without relying on convexity arguments?

Key findings

• For c = 0 and c ≥ 0.092, the optimal primal solution derived from the dual is feasible, implying that the relaxation incurs no loss and revenue maximality is achieved.
• For 0 < c < 0.092, the primal solution derived from the dual is not feasible, demonstrating that relaxing convexity does not come for free in this setting.
• The threshold c ≈ 0.092 marks the boundary where the relaxation begins to affect optimality, providing a concrete counterexample in a regular i.i.d. two-item auction model.
• The use of explicit dual solutions and exact complementarity allows for a significantly simplified proof compared to the original approach.
• The paper establishes that strong duality holds in the relaxed problem and that feasibility of the primal solution determines whether the relaxation is without loss.
• The result confirms that convexity constraints are essential for optimality in certain parameter ranges, even in symmetric, i.i.d. settings.