The Complexity of Optimal Mechanism Design

Abstract

Myerson's seminal work provides a computationally efficient revenue-optimal auction for selling one item to multiple bidders [18]. Generalizing this work to selling multiple items at once has been a central question in economics and algorithmic game theory, but its complexity has remained poorly understood. We answer this question by showing that a revenue-optimal auction in multi-item settings cannot be found and implemented computationally efficiently, unless ZPP ⊇ P #P . This is true even for a single additive bidder whose values for the items are independently distributed on two rational numbers with rational probabilities. Our result is very general: we show that it is hard to compute any encoding of an optimal auction of any format (direct or indirect, truthful or non-truthful) that can be implemented in expected polynomial time. In particular, under well-believed complexity-theoretic assumptions, revenue-optimization in very simple multi-item settings can only be tractably approximated. We note that our hardness result applies to randomized mechanisms in a very simple setting, and is not an artifact of introducing combinatorial structure to the problem by allowing correlation among item values, introducing combinatorial valuations, or requiring the mechanism to be deterministic (whose structure is readily combinatorial). Our proof is enabled by a flow-interpretation of the solutions of an exponential-size linear program for revenue maximization with an additional supermodularity constraint.

Mechanism design via optimal transport

“…They also show that in the single-buyer setting with identically distributed items, offering the grand bundle at some optimal price guarantees a O( 1 log n )-fraction of the optimal revenue. At the same time, exact polynomialtime solutions have been recently precluded by [8], where it is shown that computing optimal mechanisms is #P hard, even when there is a single additive bidder whose values for the items are independent of support 2.…”

“…It is also known that, unlike the single-item case, the optimal mechanism need not be deterministic [17,13,14,11]. Here is an example from [8]: Suppose there are two items and an additive bidder whose values are independent and uniformly distributed in {1, 2} and {1, 3} respectively. In this scenario, the optimal mechanism offers the bundle of both the items at price 4; it also offers at price 2.5 a lottery that, with probability 1/2, gives both items and, with probability 1/2, offers just the first item.…”

“…In the implicit setting, e.g. when the bidders have additive valuations with independent and/or continuous values for the items, these results do not apply, and it was recently shown that exact revenue optimization is intractable, even when there is only one bidder [8]. Even for item distributions with special structure, optimal mechanisms have been surprisingly rare [13] and the problem is challenging even in the two-item case [10].…”

Separation in Correlation-Robust Monopolist Problem with Budget

“…In particular, it may involve bundling and lottery pricing, the menu size could be very large or even infinite [32,29,24,18]. The task of finding the optimal mechanism could be computationally intractable [17,19].…”

“…In some special cases the optimal mechanism is relatively simple, e.g., in a natural case of values for different goods being independent and uniform [0, 1], the optimal mechanism offers a menu with separate prices for each of the items and a price for the bundle (despite a simple answer the proof of this fact is quite nontrivial [29].) For general distributions it has been shown that randomization might be necessary and even that the seller might have to offer an infinite menu of lotteries [24,19]. On the other note, the revenue of the optimal auction may be non-monotone [25] when the buyer's values in the prior distribution are moved upwards (in the stochastic dominance sense).…”