[Paper Review] On Welfare Approximation and Stable Pricing

Michal Feldman, Nick Gravin+1 more|arXiv (Cornell University)|Nov 7, 2015

Auction Theory and Applications18 references4 citations

TL;DR

This paper investigates the limits of item pricing in approximating social welfare in combinatorial markets with indivisible items and strategic buyers. It shows that even for two submodular buyers, stable item prices cannot guarantee better than an Ω(√m) approximation to optimal welfare, implying that revenue approximation via welfare benchmarks fails for submodular and single-minded valuations.

ABSTRACT

We study the power of item-pricing as a tool for approximately optimizing social welfare in a combinatorial market. We consider markets with $m$ indivisible items and $n$ buyers. The goal is to set prices to the items so that, when agents purchase their most demanded sets simultaneously, no conflicts arise and the obtained allocation has nearly optimal welfare. For gross substitutes valuations, it is well known that it is possible to achieve optimal welfare in this manner. We ask: can one achieve approximately efficient outcomes for valuations beyond gross substitutes? We show that even for submodular valuations, and even with only two buyers, one cannot guarantee an approximation better than $Ω(\sqrt{m})$. The same lower bound holds for the class of single-minded buyers as well. Beyond the negative results on welfare approximation, our results have daunting implications on revenue approximation for these valuation classes: in order to obtain good approximation to the collected revenue, one would necessarily need to abandon the common approach of comparing the revenue to the optimal welfare; a fundamentally new approach would be required.

Motivation & Objective

• To determine whether stable item pricing can achieve constant-factor welfare approximation beyond gross substitutes valuations.
• To investigate whether submodular or single-minded valuations allow for efficient welfare approximation using item prices.
• To examine the implications of poor welfare approximation for revenue approximation in combinatorial markets.
• To explore whether structural subclasses of submodular valuations (e.g., budget-additive) admit constant approximation guarantees.
• To understand the role of item indivisibility and complementarity in limiting the power of pricing mechanisms.

Proposed method

• Constructs a combinatorial auction instance with n single-minded buyers and m = n(n−1)/2 items, where each buyer desires a specific set of size n−1.
• Uses a counting argument on item prices to show that if the optimal allocation (to buyer n) is supported, then some other buyer must also demand a set, violating stability.
• Demonstrates that any stable pricing equilibrium must allocate at most one of the high-value sets, leading to welfare at most O(√m).
• Analyzes the configuration LP and its dual to show that an O(√m) approximation is always achievable for single-minded bidders via a greedy allocation of fractional solutions.
• Applies existential arguments independent of computation, proving that no stable pricing can achieve better than Ω(√m) welfare approximation.
• Uses the fact that no buyer can have strictly positive utility in the stable outcome to ensure stability of the greedy allocation.

Experimental results

Research questions

• RQ1 Can stable item pricing achieve a constant-factor approximation to optimal social welfare for submodular valuations?
• RQ2 Is there a fundamental limit to welfare approximation via stable pricing for complement-free valuations beyond gross substitutes?
• RQ3 To what extent can revenue approximation be achieved using stable pricing when welfare approximation is inherently poor?
• RQ4 Do subclasses of submodular valuations (e.g., budget-additive) admit constant-factor welfare approximation via stable pricing?
• RQ5 Can the Ω(√m) lower bound be improved or matched by a positive construction for single-minded bidders?

Key findings

• For two submodular buyers, no stable pricing can achieve better than an Ω(√m) approximation to optimal social welfare.
• The same Ω(√m) lower bound holds for single-minded valuations, even with only two buyers.
• The lower bound is existential and not due to computational complexity, meaning no stable pricing can achieve better than Ω(√m) welfare regardless of algorithmic design.
• For single-minded bidders, there always exists a stable outcome achieving an O(√m) approximation to optimal welfare, matching the lower bound.
• The result implies that any revenue approximation method relying on optimal welfare as a benchmark cannot generalize to submodular or single-minded valuations.
• For two budget-additive buyers, a constant-factor welfare approximation via stable pricing is always possible, suggesting a potential dichotomy between valuation classes.