647 research outputs found
Hardness Results for Consensus-Halving
No full textThe Consensus-halving problem is the problem of dividing an object into two portions, such that each of n agents has equal valuation for the two portions. We study the epsilon-approximate version, which allows each agent to have an epsilon discrepancy on the values of the portions. It was recently proven in [Filos-Ratsikas and Goldberg, 2018] that the problem of computing an epsilon-approximate Consensus-halving solution (for n agents and n cuts) is PPA-complete when epsilon is inverse-exponential. In this paper, we prove that when epsilon is constant, the problem is PPAD-hard and the problem remains PPAD-hard when we allow a constant number of additional cuts. Additionally, we prove that deciding whether a solution with n-1 cuts exists for the problem is NP-hard
Facility location with double-peaked preferences
No full textWe study the problem of locating a single facility on a real line based on the reports of self-interested agents, when agents have double-peaked preferences, with the peaks being on opposite sides of their locations.We observe that double-peaked preferences capture real-life scenarios and thus complement the well-studied notion of single-peaked preferences. We mainly focus on the case where peaks are equidistant from the agents’ locations and discuss how our results extend to more general settings. We show that most of the results for single-peaked preferences do not directly apply to this setting; this makes the problem essentially more challenging. As our main contribution, we present a simple truthful-in-expectation mechanism that achieves an approximation ratio of 1+b/c for both the social and the maximum cost, where b is the distance of the agent from the peak and c is the minimum cost of an agent. For the latter case, we provide a 3/2 lower bound on the approximation ratio of any truthful-in-expectation mechanism. We also study deterministic mechanisms under some natural conditions, proving lower bounds and approximation guarantees. We prove that among a large class of reasonable mechanisms, there is no deterministic mechanism that outpeforms our truthful-in-expectation mechanism
Consensus-halving: does it ever get easier?
No full textIn the ε-Consensus-Halving problem, a fundamental problem in fair division, there are n agents with valuations over the interval [0,1], and the goal is to divide the interval into pieces and assign a label “+” or “−” to each piece, such that every agent values the total amount of “+” and the total amount of “−” almost equally. The problem was recently proven by Filos-Ratsikas and Goldberg [Proceedings of the 50th Annual ACM Symposium on Theory of Computing, 2018, pp. 51–64; Proceedings of the 51st Annual ACM Symposium on Theory of Computing, 2019, pp. 638–649] to be the first “natural” complete problem for the computational class PPA, answering a decade-old open question. In this paper, we examine the extent to which the problem becomes easy to solve if one restricts the class of valuation functions. To this end, we provide the following contributions. First, we obtain a strengthening of the PPA-hardness result of Filos-Ratsikas and Goldberg [Proceedings of the 51st Annual ACM Symposium on Theory of Computing, 2019, pp. 638–649] to the case when agents have piecewise uniform valuations with only two blocks. We obtain this result via a new reduction, which is in fact conceptually much simpler than the corresponding one in Filos-Ratsikas and Goldberg [Proceedings of the 51st Annual ACM Symposium on Theory of Computing, 2019, pp. 638–649]. Then, we consider the case of single-block (uniform) valuations and provide a parameterized polynomial-time algorithm for solving ε-Consensus-Halving for any ε, as well as a polynomial-time algorithm for ε=1/2. Finally, an important application of our new techniques is the first hardness result for a generalization of Consensus-Halving, the Consensus-1/k1-Division problem [F. W. Simmons and F. E. Su, Math. Social Sci., 45 (2003), pp. 15–25]. In particular, we prove that ε-Consensus-1/3-Division is PPAD-hard
Walrasian Pricing in Multi-Unit Auctions
No full textMulti-unit auctions are a paradigmatic model, where a seller brings multiple units of a good, while several buyers bring monetary endowments. It is well known that Walrasian equilibria do not always exist in this model, however compelling relaxations such as Walrasian envy-free pricing do. In this paper we design an optimal envy-free mechanism for multi-unit auctions with budgets. When the market is even mildly competitive, the approximation ratios of this mechanism are small constants for both the revenue and welfare objectives, and in fact for welfare the approximation converges to 1 as the market becomes fully competitive. We also give an impossibility theorem, showing that truthfulness requires discarding resources, and in particular, is incompatible with (Pareto) efficiency
The Pareto Frontier of Inefficiency in Mechanism Design
No full textWe study the trade-off between the price of anarchy (PoA) and the price of stability (PoS) in mechanism design in the prototypical problem of unrelated machine scheduling. We give bounds on the space of feasible mechanisms with respect to these metrics and observe that two fundamental mechanisms, namely the first price (FP) and the second price (SP), lie on the two opposite extrema of this boundary. Furthermore, for the natural class of anonymous task-independent mechanisms, we completely characterize the PoA/PoS Pareto frontier; we design a class of optimal mechanisms SPα that lie exactly on this frontier. In particular, these mechanisms range smoothly with respect to parameter α ≥ 1 across the frontier, between the first price (SP1) and second price (SP∞) mechanisms. En route to these results, we also provide a definitive answer to an important question related to the scheduling problem, namely whether nontruthful mechanisms can provide better makespan guarantees in the equilibrium compared with truthful ones. We answer this question in the negative by proving that the price of anarchy of all scheduling mechanisms is at least n, where n is the number of machines
Hardness results for consensus-halving
No full textWe study the consensus-halving problem of dividing an object into two portions, such that each of n agents has equal valuation for the two portions. The ϵ-approximate consensus-halving problem allows each agent to have an ϵ discrepancy on the values of the portions. We prove that computing ϵ-approximate consensus-halving solution using n cuts is in PPA, and is PPAD-hard, where ϵ is some positive constant; the problem remains PPAD-hard when we allow a constant number of additional cuts. It is NP-hard to decide whether a solution with n−1 cuts exists for the problem. As a corollary of our results, we obtain that the approximate computational version of the Continuous Necklace Splitting Problem is PPAD-hard when the number of portions t is two
Welfare ratios in one-sided matching mechanisms
No full textWe study the Price of Anarchy of mechanisms for the well-known problem of one-sided matching, or house allocation, with respect to the social welfare objective. We consider both ordinal mechanisms, where agents submit preference lists over the items, and cardinal mechanisms, where agents may submit numerical values for the items being allocated. We present a general lower bound of ?(?n) on the Price of Anarchy, which applies to all mechanisms and we show that a very well-known mechanisms, Probabilistic Serial achieves a matching upper bound. We extend our lower bound to the Price of Stability of a large class of mechanisms that satisfy a common proportionality propert
Social welfare in one-sided matchings: Random priority and beyond.
No full textWe study the problem of approximate social welfare maximization (without money) in onesided matching problems when agents have unrestricted cardinal preferences over a finite set of items. Random priority is a very well-known truthful-in-expectation mechanism for the problem. We prove that the approximation ratio of random priority is Θ(n1=2) while no truthful-in-expectation mechanism can achieve an approximation ratio better than O(n1=2), where n is the number of agents and items. Furthermore, we prove that the approximation ratio of all ordinal (not necessarily truthful-in-expectation)mechanisms is upper bounded by O(n1=2), indicating that random priority is asymptotically the besttruthful-in-expectation mechanism and the best ordinal mechanism for the problem
The Pareto Frontier of Inefficiency in Mechanism Design
No full textWe study the trade-off between the Price of Anarchy (PoA) and the Price of Stability (PoS) in mechanism design, in the prototypical problem of unrelated machine scheduling. We give bounds on the space of feasible mechanisms with respect to the above metrics, and observe that two fundamental mechanisms, namely the First-Price (FP) and the Second-Price (SP), lie on the two opposite extrema of this boundary. Furthermore, for the natural class of anonymous task-independent mechanisms, we completely characterize the PoA/PoS Pareto frontier; we design a class of optimal mechanisms (formula presented) that lie exactly on this frontier. In particular, these mechanisms range smoothly, with respect to parameter (formula presented) across the frontier, between the First-Price (formula presented) and Second-Price (formula presented) mechanisms. En route to these results, we also provide a definitive answer to an important question related to the scheduling problem, namely whether non-truthful mechanisms can provide better makespan guarantees in the equilibrium, compared to truthful ones. We answer this question in the negative, by proving that the Price of Anarchy of all scheduling mechanisms is at least n, where n is the number of machines
On the Complexity of Equilibrium Computation in First-Price Auctions
No full textWe consider the problem of computing a (pure) Bayes-Nash equilibrium in the
first-price auction with continuous value distributions and discrete bidding
space. We prove that when bidders have independent subjective prior beliefs
about the value distributions of the other bidders, computing an
-equilibrium of the auction is PPAD-complete, and computing an
exact equilibrium is FIXP-complete. We also provide an efficient algorithm for
solving a special case of the problem, for a fixed number of bidders and
available bids.Comment: Journal version. Preliminary version appeared at EC '2
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