19 research outputs found
Discrete Strategies in Keyword Auctions and Their Inefficiency for Locally Aware Bidders
We study formally discrete bidding strategies for the game induced by the Generalized Second Price keyword auction mechanism. Such strategies have seen experimental evaluation in the recent literature as parts of iterative best response procedures, which have been shown not to converge. We give a detailed definition of iterative best response under these strategies and, under appropriate discretization of the players' strategy spaces we find that the discretized configurations space {\em contains} socially optimal pure Nash equilibria. We cast the strategies under a new light, by studying their
performance for bidders that act based on local information; we prove bounds for the worst-case ratio of the social welfare of locally stable configurations, relative to the socially optimum welfare
Integrality gaps of semidefinite programs for Vertex Cover and relations to ell embeddability of negative type metrics
We study various SDP formulations for Vertex Cover by adding different constraints to the standard formulation. We rule out approximations better than
even when we add the so-called pentagonal inequality constraints to the standard SDP formulation, and thus almost meet the
best upper bound known due to Karakostas, of
. We further show the surprising fact that by strengthening the SDP with the (intractable) requirement that the metric interpretation
of the solution embeds into ℓ1 with no distortion, we get an exact relaxation (integrality gap is 1), and on the other hand if the solution is arbitrarily
close to being ℓ1 embeddable, the integrality gap is 2 − o(1). Finally, inspired by the above findings, we use ideas from the integrality gap construction of Charikar to provide a
family of simple examples for negative type metrics that cannot be embedded into ℓ1 with distortion better than 8/7 − ε. To this end we prove a new isoperimetric inequality for the hypercube.
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Some results on approximating the minimax solution in approval voting
Voting has been a very popular method for preference aggregation in multiagent environments. It is often the case that a set of agents with different preferences need to make a choice among a set of alternatives, where the alternatives could be various entities such as potential committee members, or joint plans of action. A standard methodology for this scenario is to have each agent express his preferences and then select an alternative according to some voting protocol. Several decision making applications in AI have followed this approach including problems in collaborative filtering [10] and planning [3, 4]
Cost sharing over combinatorial domains: Complement-free cost functions and beyond
We study mechanism design for combinatorial cost sharing models. Imagine that multiple items or services are available to be shared among a set of interested agents. The outcome of a mechanism in this setting consists of an assignment, determining for each item the set of players who are granted service, together with respective payments. Although there are several works studying specialized versions of such problems, there has been almost no progress for general combinatorial cost sharing domains until recently [7]. Still, many questions about the interplay between strategyproofness, cost recovery and economic efficiency remain unanswered. The main goal of our work is to further understand this interplay in terms of budget balance and social cost approximation. Towards this, we provide a refinement of cross-monotonicity (which we term trace-monotonicity) that is applicable to iterative mechanisms. The trace here refers to the order in which players become finalized. On top of this, we also provide two parameterizations (complementary to a certain extent) of cost functions which capture the behavior of their average cost-shares. Based on our trace-monotonicity property, we design a scheme of ascending cost sharing mechanisms which is applicable to the combinatorial cost sharing setting with symmetric submodular valuations. Using our first cost function parameterization, we identify conditions under which our mechanism is weakly group-strategyproof, O(1)-budget-balanced and O(Hn)-approximate with respect to the social cost. Further, we show that our mechanism is budget-balanced and Hn-approximate if both the valuations and the cost functions are symmetric submodular; given existing impossibility results, this is best possible. Finally, we consider general valuation functions and exploit our second parameterization to derive a more fine-grained analysis of the Sequential Mechanism introduced by Moulin. This mechanism is budget balanced by construction, but in general only guarantees a poor social cost approximation of n. We identify conditions under which the mechanism achieves improved social cost approximation guarantees. In particular, we derive improved mechanisms for fundamental cost sharing problems, including Vertex Cover and Set Cover
Comparing approximate relaxations of envy-freeness
In fair division problems with indivisible goods it is well known that one cannot have any guarantees for the classic fairness notions of envy-freeness and proportionality. As a result, several relaxations have been introduced, most of which in quite recent works. We focus on four such notions, namely envy-freeness up to one good (EF1), envy-freeness up to any good (EFX), maximin share fairness (MMS), and pairwise maximin share fairness (PMMS). Since obtaining these relaxations also turns out to be problematic in several scenarios, approximate versions of them have been considered. In this work, we investigate further the connections between the four notions mentioned above and their approximate versions. We establish several tight, or almost tight, results concerning the approximation quality that any of these notions guarantees for the others, providing an almost complete picture of this landscape. Some of our findings reveal interesting and surprising consequences regarding the power of these notions, e.g., PMMS and EFX provide the same worst-case guarantee for MMS, despite PMMS being a strictly stronger notion than EFX. We believe such implications provide further insight on the quality of approximately fair solutions
Undominated Groves Mechanisms
The family of Groves mechanisms, which includes the well-known VCG mechanism (also
known as the Clarke mechanism), is a family of efficient and strategy-proof mechanisms.
Unfortunately, the Groves mechanisms are generally not budget balanced. That is, under
such mechanisms, payments may flow into or out of the system of the agents, resulting
in deficits or reduced utilities for the agents. We consider the following problem: within
the family of Groves mechanisms, we want to identify mechanisms that give the agents the
highest utilities, under the constraint that these mechanisms must never incur deficits.
We adopt a prior-free approach. We introduce two general measures for comparing
mechanisms in prior-free settings. We say that a non-deficit Groves mechanism M in-
dividually dominates another non-deficit Groves mechanism M′ if for every type profile,
every agent’s utility under M is no less than that under M′, and this holds with strict
inequality for at least one type profile and one agent. We say that a non-deficit Groves
mechanism M collectively dominates another non-deficit Groves mechanism M′ if for every
type profile, the agents’ total utility under M is no less than that under M′, and this holds
with strict inequality for at least one type profile. The above definitions induce two partial
orders on non-deficit Groves mechanisms. We study the maximal elements corresponding
to these two partial orders, which we call the individually undominated mechanisms and
the collectively undominated mechanisms, respectively
Cost sharing over combinatorial domains
We study the problem of designing cost-sharing mechanisms for combinatorial domains. Suppose that multiple items or services are available to be shared among a set of interested agents. The outcome of a mechanism in this setting consists of an assignment, determining for each item the set of players who are granted service, together with respective payments. Although there are several works studying specialized versions of such problems, there has been almost no progress for general combinatorial cost-sharing domains until recently [9]. Still, many questions about the interplay between strategyproofness, cost recovery, and economic efficiency remain unanswered.The main goal of our work is to further understand this interplay in terms of budget balance and social cost approximation. Towards this, we provide a refinement of cross-monotonicity (which we term trace-monotonicity) that is applicable to iterative mechanisms. The trace here refers to the order in which players become finalized. On top of this, we also provide two parameterizations (complementary to a certain extent) of cost functions, which capture the behavior of their average cost-shares.Based on our trace-monotonicity property, we design an Iterative Ascending Cost-Sharing Mechanism, which is applicable to the combinatorial cost-sharing setting with symmetric submodular valuations. Using our first cost function parameterization, we identify conditions under which our mechanism is weakly group-strategyproof, O(1)-budget-balanced, and O(Hn)-approximate with respect to the social cost. Furthermore, we show that our mechanism is budget-balanced and Hn-approximate if both the valuations and the cost functions are symmetric submodular; given existing impossibility results, this is best possible.Finally, we consider general valuation functions and exploit our second parameterization to derive a more fine-grained analysis of the Sequential Mechanism introduced by Moulin. This mechanism is budget balanced by construction, but in general, only guarantees a poor social cost approximation of n. We identify conditions under which the mechanism achieves improved social cost approximation guarantees. In particular, we derive improved mechanisms for fundamental cost-sharing problems, including Vertex Cover and Set Cover