Pricing to Maximize Revenue and Welfare Simultaneously in Large Markets

Abstract

We study large markets with a single seller which can produce many types of goods, and many multi-minded buyers. The seller chooses posted prices for its many items, and the buyers purchase bundles to maximize their utility. For this setting, we consider the following questions: What fraction of the optimum social welfare does a revenue maximizing solution achieve? Are there pricing mechanisms which achieve both good revenue and good welfare simultaneously? To address these questions, we give efficient pricing schemes which are guaranteed to result in both good revenue and welfare, as long as the buyer valuations for the goods they desire have a nice (although reasonable) structure, e.g., that the aggregate buyer demand has a monotone hazard rate or is not too convex. We also show that our pricing schemes have implications for any pricing which achieves high revenue: specifically that even if the seller cares only about revenue, they can still ensure that their prices result in good social welfare without sacrificing profit. Our results holds for general multi-minded buyers in large markets; we also provide improved guarantees for the important special case of unit-demand buyers.1 For convenience, we will use revenue and profit interchangeably in this work

Price Doubling and Item Halving

“…Discussion of Bicriteria Results Simultaneous bicriteria (revenue,welfare) approximations have featured previously in single-item auctions [15], and item pricing for special cases of unit-demand and multi-minded buyers [2]. Our result, on the other hand, has a far greater scope since it is not tied to any class of valuation (see Table 1).…”

“…A noteworthy aspect of this work is that, as in much of the previous literature, all of our approximation factors are derived by comparing the revenue obtained by our algorithm to the social welfare of the optimum allocation. Measuring revenue in terms of social welfare allows for our results to have some useful implications [2]. For instance, since the optimum welfare is an upper bound on the revenue obtained by any 'individually rational mechanism', our main result implies that no pricing mechanism can improve upon item pricing (specifically, our algorithm's revenue) by a factor larger than O (log 2 m) for unit-supply settings.…”

Price Doubling and Item Halving: Robust Revenue Guarantees for Item Pricing

“…Revenue, Welfare, and Bicriteria Approximations A noteworthy aspect of this work is that, as in much of the previous literature, all of our approximation factors are derived by comparing the revenue obtained by our algorithm to the social welfare of the optimum allocation, SW (OP T ). While one could speculate that the choice of a better benchmark might result in improved approximations, measuring revenue in terms of social welfare allows for our results to have some useful implications [3]. For instance, since the optimum welfare is an upper bound on the revenue obtained by any individually rational mechanism, our main result provides a O(log 2 m)-approximation (for unit supply) to the performance of any reasonable algorithm, for e.g., even a centralized non-item pricing mechanism that arbitrarily allocates goods and charges each buyer her exact valuation for the allocated bundle.…”

“…The proof proceeds by showing that the sequence of solutions ( p (t) , S (t) ) γ t=1 satisfies the simple charging property, and therefore, the bicriteria result follows from Claim 2.2 with c = 3 2 . Details can be found in the Appendix.…”

“…The charging property follows. Now, we can directly apply Claim 2.2 with c = 3 2 to get the following lower bounds on the social welfare of the solution returned by our black-box algorithm, Rev( p (ℓ) , S (ℓ) ) ≥ SW ( S (1) ) 6γ and SW ( S (ℓ) ) ≥ 2SW ( S (1) ) 3 .…”