Vasilis Gkatzelis

Mechanism design for fair division: allocating divisible items without payments

We revisit the classic problem of fair division from a mechanism design perspective and provide an elegant truthful mechanism that yields surprisingly good approximation guarantees for the widely used solution of Proportional Fairness. This solution, which is closely related to Nash bargaining and the competitive equilibrium, is known to be not implementable in a truthful fashion, which has been its main drawback. To alleviate this issue, we propose a new mechanism, which we call the Partial Allocation mechanism, that discards a carefully chosen fraction of the allocated resources in order to incentivize the agents to be truthful in reporting their valuations. This mechanism introduces a way to implement interesting truthful outcomes in settings where monetary payments are not an option. For a multi-dimensional domain with an arbitrary number of agents and items, and for the very large class of homogeneous valuation functions, we prove that our mechanism provides every agent with at least a 1/e ≈ 0.368 fraction of her Proportionally Fair valuation. To the best of our knowledge, this is the first result that gives a constant factor approximation to every agent for the Proportionally Fair solution. To complement this result, we show that no truthful mechanism can guarantee more than 0.5 approximation, even for the restricted class of additive linear valuations. In addition to this, we uncover a connection between the Partial Allocation mechanism and VCG-based mechanism design. We also ask whether better approximation ratios are possible in more restricted settings. In particular, motivated by the massive privatization auction in the Czech republic in the early 90s we provide another mechanism for additive linear valuations that works really well when all the items are highly demanded.

Inner product spaces for MinSum coordination mechanisms

We study coordination mechanisms aiming to minimize the weighted sum of completion times of jobs in the context of selfish scheduling problems. Our goal is to design local policies that achieve a good price of anarchy in the resulting equilibria for unrelated machine scheduling. To obtain these approximation bounds, we introduce a new technique that while conceptually simple, seems to be quite powerful. The method entails mapping strategy vectors into a carefully chosen inner product space; costs are shown to correspond to the norm in this space, and the Nash condition also has a simple description. With this structure in place, we are able to prove a number of results, as follows. First, we consider Smiths Rule, which orders the jobs on a machine in ascending processing time to weight ratio, and show that it achieves an approximation ratio of 4. We also demonstrate that this is the best possible for deterministic non-preemptive strongly local policies. Since Smiths Rule is always optimal for a given fixed assignment, this may seem unsurprising, but we then show that better approximation ratios can be obtained if either preemption or randomization is allowed. We prove that ProportionalSharing, a preemptive strongly local policy, achieves an approximation ratio of 2.618 for the weighted sum of completion times, and an approximation ratio of 2.5 in the unweighted case. We also observe that these bounds are tight. Next, we consider Rand, a natural non-preemptive but randomized policy. We show that it achieves an approximation ratio of at most 2.13; moreover, if the sum of the weighted completion times is negligible compared to the cost of the optimal solution, this improves to π/2. Finally, we show that both ProportionalSharing and Rand induce potential games, and thus always have a pure Nash equilibrium (unlike Smiths Rule). This allows us to design the first combinatorial constant-factor approximation algorithm minimizing weighted completion time for unrelated machine scheduling. It achieves a factor of 2+ε for any ε > 0, and involves imitating best response dynamics using a variant of ProportionalSharing as the policy.

Decentralized utilitarian mechanisms for scheduling games

Game Theory and Mechanism Design are by now standard tools for studying and designing massive decentralized systems. Unfortunately, designing mechanisms that induce socially efficient outcomes often requires full information and prohibitively large computational resources. In this work we study simple mechanisms that require only local information. Specifically, in the setting of a classic scheduling problem, we demonstrate local mechanisms that induce outcomes with social cost close to that of the socially optimal solution. Somewhat counter-intuitively, we find that mechanisms yielding Pareto dominated outcomes may in fact enhance the overall performance of the system, and we provide a justification of these results by interpreting these inefficiencies as externalities being internalized. We also show how to employ randomization to obtain yet further improvements. Lastly, we use the game-theoretic insights gained to obtain a new combinatorial approximation algorithm for the underlying optimization problem.

Pricing private data

We consider a market where buyers can access unbiased samples of private data by appropriately compensating the individuals to whom the data corresponds (the sellers) according to their privacy attitudes. We show how bundling the buyers’ demand can decrease the price that buyers have to pay per data point, while ensuring that sellers are willing to participate. Our approach leverages the inherently randomized nature of sampling, along with the risk-averse attitude of sellers in order to discover the minimum price at which buyers can obtain unbiased samples. We take a prior-free approach and introduce a mechanism that incentivizes each individual to truthfully report his preferences in terms of different payment schemes. We then show that our mechanism provides optimal price guarantees in several 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.

Optimal Cost-Sharing in Weighted Congestion Games

We identify how to share costs locally in weighted congestion games with polynomial cost functions in order to minimize the worst-case price of anarchy (PoA). First, we prove that among all cost-sharing methods that guarantee the existence of pure Nash equilibria, the Shapley value minimizes the worst-case PoA. Second, if the guaranteed existence condition is dropped, then the proportional cost-sharing method minimizes the worst-case PoA over all cost-sharing methods. As a byproduct of our results, we obtain the first PoA analysis of the simple marginal contribution cost-sharing rule, and prove that marginal cost taxes are ineffective for improving equilibria in (atomic) congestion games.

Optimal Cost-Sharing in General Resource Selection Games

Resource selection games provide a model for a diverse collection of applications where a set of resources is matched to a set of demands. Examples include routing in traffic and in telecommunication networks, service of requests on multiple parallel queues, and acquisition of services or goods with demand-dependent prices. In reality, demands are often submitted by selfish entities (players) and congestion on the resources results in negative externalities for their users. We consider a policy maker that can set a priori rules to minimize the inefficiency induced by selfish players. For example, these rules may assume the form of scheduling policies or pricing decisions. We explore the space of such rules abstracted as cost-sharing methods. We prescribe desirable properties that the cost-sharing method should possess and prove that, in this natural design space, the cost-sharing method induced by the Shapley value minimizes the worst-case inefficiency of equilibria.

The Performance of Deferred-Acceptance Auctions

Deferred-acceptance auctions are mechanisms whose allocation rule can be implemented using an adaptive reverse greedy algorithm. Milgrom and Segal recently introduced these auctions and proved that they satisfy remarkable incentive guarantees: in addition to being dominant strategy and incentive compatible, they are weakly group-strategyproof and can be implemented by ascending-clock auctions. 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 what fraction of the optimal social welfare can be guaranteed by these auctions in two canonical problems, knapsack auctions and combinatorial auctions with single-minded bidders. For knapsack auctions, we prove a separation between deferred-acceptance auctions and arbitrary dominant-strategy incentive-compatible mechanisms. For combinatorial auctions with single-minded bidders...

The Impact of Social Ignorance on Weighted Congestion Games

We consider weighted linear congestion games, and investigate how social ignorance, namely lack of information about the presence of some players, affects the inefficiency of pure Nash equilibria (PNE) and the convergence rate of the ε-Nash dynamics. To this end, we adopt the model of graphical linear congestion games with weighted players, where the individual cost and the strategy selection of each player only depends on his neighboring players in the social graph. We show that such games admit a potential function, and thus a PNE. Next, we investigate the Price of Anarchy (PoA) and the Price of Stability (PoS) of graphical linear congestion games with respect to the players’ total actual cost. Our main result is that the impact of social ignorance on the PoA and on the PoS is naturally quantified by the independence numberα(G) of the social graph G. In particular, we show that the PoA grows roughly as α(G)(α(G)+2), which is essentially tight as long as α(G) does not exceed half the number of players, and that the PoS lies between α(G) and 2α(G). Moreover, we show that the ε-Nash dynamics reaches an α(G)(α(G)+2)-approximate configuration in polynomial time that does not directly depend on the social graph. For unweighted graphical linear games with symmetric strategies, we show that the ε-Nash dynamics reaches an ε-approximate PNE in polynomial time that exceeds the corresponding time for symmetric linear games by a factor at most as large as the number of players.

Coordination Mechanisms, Cost-Sharing, and Approximation Algorithms for Scheduling

We reveal a connection between coordination mechanisms for unrelated machine scheduling and cost-sharing protocols. Using this connection, we interpret three coordination mechanisms from the recent literature as Shapley-value-based cost-sharing protocols, thus providing a unifying justification regarding why these mechanisms induce potential games. More importantly, this connection provides a template for designing novel coordination mechanisms, as well as approximation algorithms for the underlying optimization problem. The designer need only decide the total cost to be suffered on each machine, and then the Shapley value can be used to induce games guaranteed to possess a potential function; these games can, in turn, be used to design algorithms. To verify the power of this approach, we design a combinatorial algorithm that achieves an approximation guarantee of 1.81 for the problem of minimizing the total weighted completion time for unrelated machines. To the best of our knowledge, this is the best approximation guarantee among combinatorial polynomial-time algorithms for this problem.