114 research outputs found
Truthful Interval Covering
We initiate the study of a novel problem in mechanism design without money,
which we term Truthful Interval Covering (TIC). An instance of TIC consists of
a set of agents each associated with an individual interval on a line, and the
objective is to decide where to place a covering interval to minimize the total
social cost of the agents, which is determined by the intersection of this
interval with their individual ones. This fundamental problem can model
situations of provisioning a public good, such as the use of power generators
to prevent or mitigate load shedding in developing countries. In the strategic
version of the problem, the agents wish to minimize their individual costs, and
might misreport the position and/or length of their intervals to achieve that.
Our goal is to design truthful mechanisms to prevent such strategic misreports
and achieve good approximations to the best possible social cost. We consider
the fundamental setting of known intervals with equal lengths and provide tight
bounds on the approximation ratios achieved by truthful deterministic
mechanisms. We also design a randomized truthful mechanism that outperforms all
possible deterministic ones. Finally, we highlight a plethora of natural
extensions of our model for future work, as well as some natural limitations of
those settings
Truthful Mechanisms for Matching and Clustering in an Ordinal World
We study truthful mechanisms for matching and related problems in a partial
information setting, where the agents' true utilities are hidden, and the
algorithm only has access to ordinal preference information. Our model is
motivated by the fact that in many settings, agents cannot express the
numerical values of their utility for different outcomes, but are still able to
rank the outcomes in their order of preference. Specifically, we study problems
where the ground truth exists in the form of a weighted graph of agent
utilities, but the algorithm can only elicit the agents' private information in
the form of a preference ordering for each agent induced by the underlying
weights. Against this backdrop, we design truthful algorithms to approximate
the true optimum solution with respect to the hidden weights. Our techniques
yield universally truthful algorithms for a number of graph problems: a
1.76-approximation algorithm for Max-Weight Matching, 2-approximation algorithm
for Max k-matching, a 6-approximation algorithm for Densest k-subgraph, and a
2-approximation algorithm for Max Traveling Salesman as long as the hidden
weights constitute a metric. We also provide improved approximation algorithms
for such problems when the agents are not able to lie about their preferences.
Our results are the first non-trivial truthful approximation algorithms for
these problems, and indicate that in many situations, we can design robust
algorithms even when the agents may lie and only provide ordinal information
instead of precise utilities.Comment: To appear in the Proceedings of WINE 201
Truthful ownership transfer with expert advice: Blending mechanism design with and without money
When a company undergoes a merger or transfers its ownership, the existing governing body has an opinion on which buyer should take over as the new owner. Similar situations occur while assigning the host of big sports tournaments, like the World Cup or the Olympics. In all these settings, the values of the external bidders are as important as the opinions of the internal experts. Motivated by such scenarios, we consider a social welfare maximizing approach to design and analyze truthful mechanisms in {\em hybrid social choice} settings, where payments can be imposed to the bidders, but not to the experts. Since this problem is a combination of mechanism design with and without monetary transfers, classical solutions like VCG cannot be applied, making this a novel mechanism design problem. We consider the simple but fundamental scenario with one expert and two bidders, and provide tight approximation guarantees of the optimal social welfare. We distinguish between mechanisms that use ordinal and cardinal information, as well as between mechanisms that base their decisions on one of the two sides (either the bidders or the expert) or both. Our analysis shows that the cardinal setting is quite rich and admits several non-trivial randomized truthful mechanisms, and also allows for closer-to-optimal welfare guarantees
PPAD-membership for problems with exact rational solutions: a general approach via convex optimization
We introduce a general technique for proving membership of search problems with exact rational solutions in PPAD, one of the most well-known classes containing total search problems with polynomial-time verifiable solutions. In particular, we construct a "pseudogate", coined the linear-OPT-gate, which can be used as a "plug-and-play" component in a piecewise-linear (PL) arithmetic circuit, as an integral component of the "Linear-FIXP" equivalent definition of the class. The linear-OPT-gate can solve several convex optimization programs, including quadratic programs, which often appear organically in the simplest existence proofs for these problems. This effectively transforms existence proofs to PPAD-membership proofs, and consequently establishes the existence of solutions described by rational numbers. Using the linear-OPT-gate, we are able to significantly simplify and generalize almost all known PPAD-membership proofs for finding exact solutions in the application domains of game theory, competitive markets, auto-bidding auctions, and fair division, as well as to obtain new PPAD-membership results for problems in these domains.Using the linear-OPT-gate, we are able to significantly simplify and generalize almost all known PPADmembership proofs for finding exact solutions in the application domains of game theory, competitive markets, auto-bidding auctions, and fair division, as well as to obtain new PPAD-membership results for problems in these domains
Donât Roll the Dice, Ask Twice: The Two-Query Distortion of Matching Problems and Beyond
In most social choice settings, the participating agents are typically
required to express their preferences over the different alternatives in the
form of linear orderings. While this simplifies preference elicitation, it
inevitably leads to high distortion when aiming to optimize a cardinal
objective such as the social welfare, since the values of the agents remain
virtually unknown. A recent array of works put forward the agenda of designing
mechanisms that can learn the values of the agents for a small number of
alternatives via queries, and use this extra information to make a
better-informed decision, thus improving distortion. Following this agenda, in
this work we focus on a class of combinatorial problems that includes most
well-known matching problems and several of their generalizations, such as
One-Sided Matching, Two-Sided Matching, General Graph Matching, and
-Constrained Resource Allocation. We design two-query mechanisms that
achieve the best-possible worst-case distortion in terms of social welfare, and
outperform the best-possible expected distortion that can be achieved by
randomized ordinal mechanisms
Truthful Facility Assignment with Resource Augmentation: An Exact Analysis of Serial Dictatorship
We study the truthful facility assignment problem, where a set of agents with
private most-preferred points on a metric space are assigned to facilities that
lie on the metric space, under capacity constraints on the facilities. The goal
is to produce such an assignment that minimizes the social cost, i.e., the
total distance between the most-preferred points of the agents and their
corresponding facilities in the assignment, under the constraint of
truthfulness, which ensures that agents do not misreport their most-preferred
points.
We propose a resource augmentation framework, where a truthful mechanism is
evaluated by its worst-case performance on an instance with enhanced facility
capacities against the optimal mechanism on the same instance with the original
capacities. We study a very well-known mechanism, Serial Dictatorship, and
provide an exact analysis of its performance. Although Serial Dictatorship is a
purely combinatorial mechanism, our analysis uses linear programming; a linear
program expresses its greedy nature as well as the structure of the input, and
finds the input instance that enforces the mechanism have its worst-case
performance. Bounding the objective of the linear program using duality
arguments allows us to compute tight bounds on the approximation ratio. Among
other results, we prove that Serial Dictatorship has approximation ratio
when the capacities are multiplied by any integer . Our
results suggest that even a limited augmentation of the resources can have
wondrous effects on the performance of the mechanism and in particular, the
approximation ratio goes to 1 as the augmentation factor becomes large. We
complement our results with bounds on the approximation ratio of Random Serial
Dictatorship, the randomized version of Serial Dictatorship, when there is no
resource augmentation
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