[Paper Review] A Note on Selling Optimally Two Uniformly Distributed Goods

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

Auction Theory and Applications3 references3 citations

TL;DR

This paper presents a simplified, constructive duality-based proof for optimal revenue-maximizing mechanisms in a two-good monopoly with i.i.d. uniformly distributed valuations over [c, c+1]. By relaxing the convexity constraint on the buyer's utility function and explicitly constructing optimal dual solutions, the authors show that for c ≥ 0.092, the relaxed solution is feasible and optimal, but for 0 < c < 0.092, convexity is essential—demonstrating the first clear case where relaxing convexity incurs a revenue loss even in a two-item regular 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 goods with uniformly i.i.d. valuations over intervals $[c,c+1]$, to an additive buyer. 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 $c\geq 0.092$ it turns out that the corresponding optimal primal solution is a feasible selling mechanism, thus the initial relaxation comes without a loss, and revenue maximality follows. However, for $0

Motivation & Objective

• To provide a simplified, constructive proof of optimal revenue-maximizing mechanisms for two goods with i.i.d. uniform valuations on [c, c+1].
• To investigate the necessity of the convexity constraint in revenue maximization under duality frameworks.
• To demonstrate that relaxing convexity can lead to strictly higher objective values than achievable by any feasible truthful mechanism.
• To quantify the approximation gap between the relaxed primal solution and feasible mechanisms for c < 0.092.
• To present explicit, closed-form dual solutions that enable direct verification of optimality via complementarity.

Proposed method

• Relaxes the original revenue maximization problem by dropping the convexity requirement on the buyer's utility function and the non-negativity of derivatives.
• Constructs explicit, closed-form dual functions z₁ and z₂ over the valuation space [c, c+1]² with piecewise linear structure.
• Uses exact complementarity between primal and dual solutions to verify optimality without solving the primal directly.
• Applies the duality framework of Giannakopoulos and Koutsoupias [2] to verify that the dual objective matches the primal revenue.
• Analyzes the primal solution induced by the dual to determine whether it corresponds to a feasible truthful mechanism (i.e., convex and non-decreasing utility).
• Employs geometric and derivative analysis to verify non-negativity of dual variables and feasibility of the primal solution across different c ranges.

Experimental results

Research questions

• RQ1 Can a simpler, constructive duality framework be used to re-derive the optimal mechanism for two uniformly distributed goods?
• RQ2 For which values of c is the relaxed dual solution (without convexity) still feasible as a truthful selling mechanism?
• RQ3 What is the extent of the revenue loss when convexity is dropped in the two-item i.i.d. uniform setting?
• RQ4 How close is the relaxed primal solution to the best feasible mechanisms (randomized, deterministic, full-bundling) for small c?
• RQ5 Does the duality framework with explicit dual construction enable direct verification of optimality without solving the primal?

Key findings

• For c ≥ 0.092, the optimal solution to the relaxed primal-dual problem is convex and corresponds to a feasible truthful mechanism, specifically full bundling with price (4c + √(4c² + 6))/3.
• For 0 < c < 0.092, the optimal relaxed primal solution is not convex, and thus not a valid truthful mechanism, proving that convexity cannot be dropped without loss in this setting.
• The gap between the relaxed primal objective and the best feasible mechanism is at most 7.5‱ (0.075%) for c ∈ [0, 0.092], with deterministic and full-bundling mechanisms achieving 2‰ and 9‰ approximations, respectively.
• The primal solution for c = 0 corresponds to a deterministic mechanism with item prices 2/3 and bundle price (4 − √2)/3, matching Pavlov’s result.
• The dual solutions are explicitly constructed in closed form, enabling direct verification of optimality via complementarity, unlike prior existence-based duality proofs.
• The analysis shows that for c ≥ 0.092, the full-bundling mechanism is optimal, while for c < 0.092, a randomized mechanism with menu complexity 4 is optimal, as shown by Pavlov.