Shaddin Dughmi

On the Power of Randomization in Algorithmic Mechanism Design

In many settings the power of truthful mechanisms is severely bounded. In this paper we use randomization to overcome this problem. In particular, we construct an FPTAS for multi-unit auctions that is truthful in expectation, whereas there is evidence that no polynomial-time truthful deterministic mechanism provides an approximation ratio better than 2. We also show for the first time that truthful in expectation polynomial-time mechanisms are provably stronger than polynomial-time universally truthful mechanisms. Specifically, we show that there is a setting in which: (1) there is a non-polynomial time truthful mechanism that always outputs the optimal solution, and that (2) no universally truthful randomized mechanism can provide an approximation ratio better than 2 in polynomial time, but (3) an FPTAS that is truthful in expectation exists.

Truthful Approximation Schemes for Single-Parameter Agents

We present the first monotone randomized polynomial-time approximation scheme (PTAS) for minimizing the makespan of parallel related machines (Q||Cmax), the paradigmatic problem in single-parameter algorithmic mechanism design. This result immediately gives a polynomial-time, truthful (in expectation) mechanism whose approximation guarantee attains the best-possible one for all polynomial-time algorithms (assuming P not equal to NP). Our algorithmic techniques are flexible and also yield, among other results, a monotone deterministic quasi-PTAS for Q||Cmax and a monotone randomized PTAS for max-min scheduling on related machines.

From convex optimization to randomized mechanisms: toward optimal combinatorial auctions

We design an expected polynomial time, truthful in expectation, (1-1/e)-approximation mechanism for welfare maximization in a fundamental class of combinatorial auctions. Our results apply to bidders with valuations that are matroid rank sums (MRS), which encompass most concrete examples of submodular functions studied in this context, including coverage functions and matroid weighted-rank functions. Our approximation factor is the best possible, even for known and explicitly given coverage valuations, assuming P ≠ NP. Ours is the first truthful-in-expectation and polynomial-time mechanism to achieve a constant-factor approximation for an NP-hard welfare maximization problem in combinatorial auctions with heterogeneous goods and restricted valuations. Our mechanism is an instantiation of a new framework for designing approximation mechanisms based on randomized rounding algorithms. A typical such algorithm first optimizes over a fractional relaxation of the original problem, and then randomly rounds the fractional solution to an integral one. With rare exceptions, such algorithms cannot be converted into truthful mechanisms. The high-level idea of our mechanism design framework is to optimize directly over the (random) output of the rounding algorithm, rather than over the input to the rounding algorithm. This approach leads to truthful-in-expectation mechanisms, and these mechanisms can be implemented efficiently when the corresponding objective function is concave. For bidders with MRS valuations, we give a novel randomized rounding algorithm that leads to both a concave objective function and a (1-1/e)-approximation of the optimal welfare.

Truthful assignment without money

We study the design of truthful mechanisms that do not use payments for the generalized assignment problem (GAP) and its variants. An instance of the GAP consists of a bipartite graph with jobs on one side and machines on the other. Machines have capacities and edges have values and sizes; the goal is to construct a welfare maximizing feasible assignment. In our model of private valuations, motivated by impossibility results, the value and sizes on all job-machine pairs are public information; however, whether an edge exists or not in the bipartite graph is a jobs private information. That is, the selfish agents in our model are the jobs, and their private information is their edge set. We want to design mechanisms that are truthful without money (henceforth strategyproof), and produce assignments whose welfare is a good approximation to the optimal omniscient welfare. We study several variants of the GAP starting with matching. For the unweighted version, we give an optimal strategyproof mechanism. For maximum weight bipartite matching, we show that no strategyproof mechanism, deterministic or randomized, can be optimal, and present a 2-approximate strategyproof mechanism along with a matching lowerbound. Next we study knapsack-like problems, which, unlike matching, are NP-hard. For these problems, we develop a general LP-based technique that extends the ideas of Lavi and Swamy [14] to reduce designing a truthful approximate mechanism without money to designing such a mechanism for the fractional version of the problem. We design strategyproof approximate mechanisms for the fractional relaxations of multiple knapsack, size-invariant GAP, and value-invariant GAP, and use this technique to obtain, respectively, 2, 4 and 4-approximate strategyproof mechanisms for these problems. We then design an O(log n)-approximate strategyproof mechanism for the GAP by reducing, with logarithmic loss in the approximation, to our solution for the value-invariant GAP. Our technique may be of independent interest for designing truthful mechanisms without money for other LP-based problems.

Sampling and Representation Complexity of Revenue Maximization

We consider (approximate) revenue maximization in mechanisms where the distribution on input valuations is given via “black box” access to samples from the distribution. We analyze the following model: a single agent, m outcomes, and valuations represented as m-dimensional vectors indexed by the outcomes and drawn from an arbitrary distribution presented as a black box. We observe that the number of samples required – the sample complexity – is tightly related to the representation complexity of an approximately revenue-maximizing auction. Our main results are upper bounds and an exponential lower bound on these complexities. We also observe that the computational task of “learning” a good mechanism from a sample is nontrivial, requiring careful use of regularization in order to avoid over-fitting the mechanism to the sample. We establish preliminary positive and negative results pertaining to the computational complexity of learning a good mechanism for the original distribution by operating on a sample from said distribution.

Limitations of randomized mechanisms for combinatorial auctions

We address the following fundamental question in the area of incentive-compatible mechanism design: Are truthful-in-expectation mechanisms compatible with polynomial-time approximation? In particular, can polynomial-time truthful-in-expectation mechanisms achieve a near-optimal approximation ratio for combinatorial auctions with submodular valuations?

Limitations of Randomized Mechanisms for Combinatorial Auctions

The design of computationally efficient and incentive compatible mechanisms that solve or approximate fundamental resource allocation problems is the main goal of algorithmic mechanism design. A central example in both theory and practice is welfare-maximization in combinatorial auctions. Recently, a randomized mechanism has been discovered for combinatorial auctions that is truthful in expectation and guarantees a (1-1/e)-approximation to the optimal social welfare when players have coverage valuations [DRY11]. This approximation ratio is the best possible even for non-truthful algorithms, assuming P does not equal NP. Given the recent sequence of negative results for combinatorial auctions under more restrictive notions of incentive compatibility, this development raises a natural question: Are truthful-in-expectation mechanisms compatible with polynomial-time approximation in a way that deterministic or universally truthful mechanisms are not? In particular, can polynomial-time truthful-in-expectation mechanisms guarantee a near-optimal approximation ratio for more general variants of combinatorial auctions? We prove that this is not the case. Specifically, the result of [DRY11] cannot be extended to combinatorial auctions with sub modular valuations in the value oracle model. (Absent strategic considerations, a (1-1/e)-approximation is still achievable in this setting.) More precisely, we prove that there is a constant \gamma>0 such that there is no randomized mechanism that is truthful-in-expectation -- or even approximately truthful-in-expectation -- and guarantees an m^{-\gamma}-approximation to the optimal social welfare for combinatorial auctions with sub modular valuations in the value oracle model. We also prove an analogous result for the flexible combinatorial public projects (CPP) problem, where a truthful-in-expectation

A truthful randomized mechanism for combinatorial public projects via convex optimization

(1-1/e)

Mixture Selection, Mechanism Design, and Signaling

-approximation for coverage valuations has been recently developed [Dughmi11]. We show that there is no truthful-in-expectation -- or even approximately truthful-in-expectation -- mechanism that achieves an m^{-\gamma}-approximation to the optimal social welfare for combinatorial public projects with sub modular valuations in the value oracle model. Both our results present an unexpected separation between coverage functions and sub modular functions, which does not occur for these problems without strategic considerations.

Posting Prices with Unknown Distributions

In Combinatorial Public Projects, there is a set of projects that may be undertaken, and a set of self-interested players with a stake in the set of projects chosen. A public planner must choose a subset of these projects, subject to a resource constraint, with the goal of maximizing social welfare. Combinatorial Public Projects has emerged as one of the paradigmatic problems in Algorithmic Mechanism Design, a field concerned with solving fundamental resource allocation problems in the presence of both selfish behavior and the computational constraint of polynomial time. We design a polynomial-time, truthful-in-expectation, (1-1/e)-approximation mechanism for welfare maximization in a fundamental variant of combinatorial public projects. Our results apply to combinatorial public projects when players have valuations that are matroid rank sums (MRS), which encompass most concrete examples of submodular functions studied in this context, including coverage functions and matroid weighted-rank functions. Our approximation factor is the best possible, assuming P ≠ NP. Ours is the first mechanism that achieves a constant factor approximation for a natural NP-hard variant of combinatorial public projects.