Paul Dütting

An expressive mechanism for auctions on the web

Auctions are widely used on the Web. Applications range from internet advertising to platforms such as eBay. In most of these applications the auctions in use are single/multi-item auctions with unit demand. The main drawback of standard mechanisms for this type of auctions, such as VCG and GSP, is the limited expressiveness that they offer to the bidders. The General Auction Mechanism (GAM) of [1] is taking a first step towards addressing the problem of limited expressiveness by computing a bidder optimal, envy free outcome for linear utility functions with identical slopes and a single discontinuity per bidder-item pair. We show that in many practical situations this does not suffice to adequately model the preferences of the bidders, and we overcome this problem by presenting the first mechanism for piece-wise linear utility functions with non-identical slopes and multiple discontinuities. Our mechanism runs in polynomial time. Like GAM it is incentive compatible for inputs that fulfill a certain non-degeneracy requirement, but our requirement is more general than the requirement of GAM. For discontinuous utility functions that are non-degenerate as well as for continuous utility functions the outcome of our mechanism is a competitive equilibrium. We also show how our mechanism can be used to compute approximately bidder optimal, envy free outcomes for a general class of continuous utility functions via piece-wise linear approximation. Finally, we prove hardness results for even more expressive settings.

The performance of deferred-acceptance auctions

Deferred-acceptance auctions are auctions for binary single-parameter mechanism design problems whose allocation rule can be implemented using an adaptive reverse greedy algorithm. Milgrom and Segal [2014] recently introduced these auctions and proved that they satisfy a remarkable list of incentive guarantees: in addition to being dominant-strategy incentive-compatible, they are weakly group-strategyproof, can be implemented by ascending-clock auctions, and admit outcome-equivalent full-information pay-as-bid versions. Neither forward greedy mechanisms nor the VCG mechanism generally possess any of these additional incentive properties. The goal of this paper is to initiate the study of deferred-acceptance auctions from an approximation standpoint. We study these auctions through the lens of two canonical welfare-maximization problems, in knapsack auctions and in combinatorial auctions with single-minded bidders. For knapsack auctions, we prove a separation between deferred-acceptance auctions and arbitrary dominant-strategy incentive-compatible mechanisms. While the more general class can achieve an arbitrarily good approximation in polynomial time, and a constant-factor approximation via forward greedy algorithms, the former class cannot obtain an approximation guarantee sub-logarithmic in the number of items m, even with unbounded computation. We also give a polynomial-time deferred-acceptance auction that achieves an approximation guarantee of O(log m) for knapsack auctions.

Sponsored search, market equilibria, and the Hungarian Method

Matching markets play a prominent role in economic theory. A prime example of such a market is the sponsored search market. Here, as in other markets of that kind, market equilibria correspond to feasible, envy free, and bidder optimal outcomes. For settings without budgets such an outcome always exists and can be computed in polynomial-time by the so-called Hungarian Method. Moreover, every mechanism that computes such an outcome is incentive compatible. We show that the Hungarian Method can be modified so that it finds a feasible, envy free, and bidder optimal outcome for settings with budgets. We also show that in settings with budgets no mechanism that computes such an outcome can be incentive compatible for all inputs. For inputs in general position, however, the presented mechanism-as any other mechanism that computes such an outcome for settings with budgets-is incentive compatible.

Simplicity-expressiveness tradeoffs in mechanism design

A fundamental result in mechanism design theory, the so-called revelation principle, asserts that for many questions concerning the existence of mechanisms with a given outcome one can restrict attention to truthful direct-revelation mechanisms. In practice, however, many mechanisms use a restricted message space. This motivates the study of the tradeoffs involved in choosing simplified mechanisms, which can sometimes bring benefits in precluding bad or promoting good equilibria, and other times impose costs on welfare and revenue. We study the simplicity-expressiveness tradeoff in two representative settings, sponsored search auctions and combinatorial auctions, each being a canonical example for complete information and incomplete information analysis, respectively. We observe that the amount of information available to the agents plays an important role for the tradeoff between simplicity and expressiveness.

Polymatroid Prophet Inequalities

Prophet inequalities bound the reward of an online algorithm—or gambler—relative to the optimum offline algorithm—the prophet—in settings that involve making selections from a sequence of elements whose order is chosen adversarially but whose weights are random. The goal is to maximize total weight.

Valuation Compressions in VCG-Based Combinatorial Auctions

The focus of classic mechanism design has been on truthful direct-revelation mechanisms. In the context of combinatorial auctions the truthful direct-revelation mechanism that maximizes social welfare is the VCG mechanism. For many valuation spaces computing the allocation and payments of the VCG mechanism, however, is a computationally hard problem. We thus study the performance of the VCG mechanism when bidders are forced to choose bids from a subspace of the valuation space for which the VCG outcome can be computed efficiently. We prove improved upper bounds on the welfare loss for restrictions to additive bids and upper and lower bounds for restrictions to non-additive bids. These bounds show that the welfare loss increases in expressiveness. All our bounds apply to equilibrium concepts that can be computed in polynomial time as well as to learning outcomes.

Algorithms against Anarchy: Understanding Non-Truthful Mechanisms

The algorithmic requirements for dominant strategy incentive compatibility, or truthfulness, are well understood. Is there a similar characterization of algorithms that when combined with a suitable payment rule yield near-optimal welfare in all equilibria? We address this question by providing a tight characterization of a (possibly randomized) mechanisms Price of Anarchy provable via smoothness, for single-parameter settings. The characterization assigns a unique value to each allocation algorithm; this value provides an upper and a matching lower bound on the Price of Anarchy of a derived mechanism provable via smoothness. The characterization also applies to the sequential or simultaneous composition of single-parameter mechanisms. Importantly, the factor that we identify is typically not in one-to-one correspondence to the approximation guarantee of the algorithm. Rather, it is usually the product of the approximation guarantee and the degree to which the mechanism is loser independent. We apply our characterization to show the optimality of greedy mechanisms for single-minded combinatorial auctions, whether these mechanisms are polynomial-time computable or not. We also use it to establish the optimality of a non-greedy, randomized mechanism for independent set in interval graphs and show that it is strictly better than any other deterministic mechanism.

Auctions for Heterogeneous Items and Budget Limits

We study individual rational, Pareto-optimal, and incentive compatible mechanisms for auctions with heterogeneous items and budget limits. We consider settings with multiunit demand and additive valuations. For single-dimensional valuations we prove a positive result for randomized mechanisms, and a negative result for deterministic mechanisms. While the positive result allows for private budgets, the negative result is for public budgets. For multidimensional valuations and public budgets we prove an impossibility result that applies to deterministic and randomized mechanisms. Taken together this shows the power of randomization in certain settings with heterogeneous items, but it also shows its limitations.

Auctions with heterogeneous items and budget limits

We study individual rational, Pareto optimal, and incentive compatible mechanisms for auctions with heterogeneous items and budget limits. For multi-dimensional valuations we show that there can be no deterministic mechanism with these properties for divisible items. We use this to show that there can also be no randomized mechanism that achieves this for either divisible or indivisible items. For single-dimensional valuations we show that there can be no deterministic mechanism with these properties for indivisible items, but that there is a randomized mechanism that achieves this for either divisible or indivisible items. The impossibility results hold for public budgets, while the mechanism allows private budgets, which is in both cases the harder variant to show. While all positive results are polynomial-time algorithms, all negative results hold independent of complexity considerations.

Bidder Optimal Assignments for General Utilities

We study the problem of matching bidders to items where each bidder i has general, strictly monotonic utility functions u i,j (p j ) expressing her utility of being matched to item j at price p j . For this setting we prove that a bidder optimal outcome always exists, even when the utility functions are non-linear and non-continuous. Furthermore, we give an algorithm to find such a solution. Although the running time of this algorithm is exponential in the number of items, it is polynomial in the number of bidders.