66 research outputs found
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 [S. Dobzinski and S. Ovadia, 2017]. 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(H_n)-approximate with respect to the social cost. Further, we show that our mechanism is budget-balanced and H_n-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
On the Complexity of Winner Determination and Strategic Control in Conditional Approval Voting
We focus on a generalization of the classic Minisum approval voting rule,
introduced by Barrot and Lang (2016), and referred to as Conditional Minisum
(CMS), for multi-issue elections with preferential dependencies. Under this
rule, voters are allowed to declare dependencies between different issues, but
the price we have to pay for this higher level of expressiveness is that we end
up with a computationally hard rule. Motivated by this, we first focus on
finding special cases that admit efficient algorithms for CMS. Our main result
in this direction is that we identify the condition of bounded treewidth (of an
appropriate graph, emerging from the provided ballots) as the necessary and
sufficient condition for exact polynomial algorithms, under common complexity
assumptions. We then move to the design of approximation algorithms. For the
(still hard) case of binary issues, we identify natural restrictions on the
voters' ballots, under which we provide the first multiplicative approximation
algorithms for the problem. The restrictions involve upper bounds on the number
of dependencies an issue can have on the others and on the number of
alternatives per issue that a voter can approve. Finally, we also investigate
the complexity of problems related to the strategic control of conditional
approval elections by adding or deleting either voters or alternatives and we
show that in most variants of these problems, CMS is computationally resistant
against control. Overall, we conclude that CMS can be viewed as a solution that
achieves a satisfactory tradeoff between expressiveness and computational
efficiency, when we have a limited number of dependencies among issues, while
at the same time exhibiting sufficient resistance to control
As Time Goes By: Adding a Temporal Dimension Towards Resolving Delegations in Liquid Democracy
In recent years, the study of various models and questions related to Liquid
Democracy has been of growing interest among the community of Computational
Social Choice. A concern that has been raised, is that current academic
literature focuses solely on static inputs, concealing a key characteristic of
Liquid Democracy: the right for a voter to change her mind as time goes by,
regarding her options of whether to vote herself or delegate her vote to other
participants, till the final voting deadline. In real life, a period of
extended deliberation preceding the election-day motivates voters to adapt
their behaviour over time, either based on observations of the remaining
electorate or on information acquired for the topic at hand. By adding a
temporal dimension to Liquid Democracy, such adaptations can increase the
number of possible delegation paths and reduce the loss of votes due to
delegation cycles or delegating paths towards abstaining agents, ultimately
enhancing participation. Our work takes a first step to integrate a time
horizon into decision-making problems in Liquid Democracy systems. Our
approach, via a computational complexity analysis, exploits concepts and tools
from temporal graph theory which turn out to be convenient for our framework
Cooperative Games with Overlapping Coalitions
In the usual models of cooperative game theory, the outcome of a coalition
formation process is either the grand coalition or a coalition structure that
consists of disjoint coalitions. However, in many domains where coalitions are
associated with tasks, an agent may be involved in executing more than one
task, and thus may distribute his resources among several coalitions. To tackle
such scenarios, we introduce a model for cooperative games with overlapping
coalitions--or overlapping coalition formation (OCF) games. We then explore the
issue of stability in this setting. In particular, we introduce a notion of the
core, which generalizes the corresponding notion in the traditional
(non-overlapping) scenario. Then, under some quite general conditions, we
characterize the elements of the core, and show that any element of the core
maximizes the social welfare. We also introduce a concept of balancedness for
overlapping coalitional games, and use it to characterize coalition structures
that can be extended to elements of the core. Finally, we generalize the notion
of convexity to our setting, and show that under some natural assumptions
convex games have a non-empty core. Moreover, we introduce two alternative
notions of stability in OCF that allow a wider range of deviations, and explore
the relationships among the corresponding definitions of the core, as well as
the classic (non-overlapping) core and the Aubin core. We illustrate the
general properties of the three cores, and also study them from a computational
perspective, thus obtaining additional insights into their fundamental
structure
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
Inequity Aversion Pricing over Social Networks: Approximation Algorithms and Hardness Results
We study a revenue maximization problem in the context of social networks. Namely, we consider a model introduced by Alon, Mansour, and Tennenholtz (EC 2013) that captures inequity aversion, i.e., prices offered to neighboring vertices should not be significantly different. We first provide approximation algorithms for a natural class of instances, referred to as the class of single-value revenue functions. Our results improve on the current state of the art, especially when the number of distinct prices is small. This applies, for example, to settings where the seller will only consider a fixed number of discount types or special offers. We then resolve one of the open questions posed in Alon et al., by establishing APX-hardness for the problem. Surprisingly, we further show that the problem is NP-complete even when the price differences are allowed to be relatively large. Finally, we also provide some extensions of the model of Alon et al., regarding the allowed set of prices
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