48 research outputs found
Strategy-Proof Facility Location for Concave Cost Functions
We consider k-Facility Location games, where n strategic agents report their
locations on the real line, and a mechanism maps them to k facilities. Each
agent seeks to minimize his connection cost, given by a nonnegative increasing
function of his distance to the nearest facility. Departing from previous work,
that mostly considers the identity cost function, we are interested in
mechanisms without payments that are (group) strategyproof for any given cost
function, and achieve a good approximation ratio for the social cost and/or the
maximum cost of the agents.
We present a randomized mechanism, called Equal Cost, which is group
strategyproof and achieves a bounded approximation ratio for all k and n, for
any given concave cost function. The approximation ratio is at most 2 for Max
Cost and at most n for Social Cost. To the best of our knowledge, this is the
first mechanism with a bounded approximation ratio for instances with k > 2
facilities and any number of agents. Our result implies an interesting
separation between deterministic mechanisms, whose approximation ratio for Max
Cost jumps from 2 to unbounded when k increases from 2 to 3, and randomized
mechanisms, whose approximation ratio remains at most 2 for all k. On the
negative side, we exclude the possibility of a mechanism with the properties of
Equal Cost for strictly convex cost functions. We also present a randomized
mechanism, called Pick the Loser, which applies to instances with k facilities
and n = k+1 agents, and for any given concave cost function, is strongly group
strategyproof and achieves an approximation ratio of 2 for Social Cost
A Survey on Approximation Mechanism Design without Money for Facility Games
In a facility game one or more facilities are placed in a metric space to
serve a set of selfish agents whose addresses are their private information. In
a classical facility game, each agent wants to be as close to a facility as
possible, and the cost of an agent can be defined as the distance between her
location and the closest facility. In an obnoxious facility game, each agent
wants to be far away from all facilities, and her utility is the distance from
her location to the facility set. The objective of each agent is to minimize
her cost or maximize her utility. An agent may lie if, by doing so, more
benefit can be obtained. We are interested in social choice mechanisms that do
not utilize payments. The game designer aims at a mechanism that is
strategy-proof, in the sense that any agent cannot benefit by misreporting her
address, or, even better, group strategy-proof, in the sense that any coalition
of agents cannot all benefit by lying. Meanwhile, it is desirable to have the
mechanism to be approximately optimal with respect to a chosen objective
function. Several models for such approximation mechanism design without money
for facility games have been proposed. In this paper we briefly review these
models and related results for both deterministic and randomized mechanisms,
and meanwhile we present a general framework for approximation mechanism design
without money for facility games
Strategyproof Facility Location in Perturbation Stable Instances
We consider -Facility Location games, where strategic agents report
their locations on the real line, and a mechanism maps them to
facilities. Each agent seeks to minimize her distance to the nearest facility.
We are interested in (deterministic or randomized) strategyproof mechanisms
without payments that achieve a reasonable approximation ratio to the optimal
social cost of the agents. To circumvent the inapproximability of -Facility
Location by deterministic strategyproof mechanisms, we restrict our attention
to perturbation stable instances. An instance of -Facility Location on the
line is -perturbation stable (or simply, -stable), for some
, if the optimal agent clustering is not affected by moving any
subset of consecutive agent locations closer to each other by a factor at most
. We show that the optimal solution is strategyproof in
-stable instances whose optimal solution does not include any
singleton clusters, and that allocating the facility to the agent next to the
rightmost one in each optimal cluster (or to the unique agent, for singleton
clusters) is strategyproof and -approximate for -stable instances
(even if their optimal solution includes singleton clusters). On the negative
side, we show that for any and any , there is no
deterministic anonymous mechanism that achieves a bounded approximation ratio
and is strategyproof in -stable instances. We also prove
that allocating the facility to a random agent of each optimal cluster is
strategyproof and -approximate in -stable instances. To the best of our
knowledge, this is the first time that the existence of deterministic (resp.
randomized) strategyproof mechanisms with a bounded (resp. constant)
approximation ratio is shown for a large and natural class of -Facility
Location instances
Heterogeneous Facility Location with Limited Resources
We initiate the study of the heterogeneous facility location problem with limited resources. We mainly focus on the fundamental case where a set of agents are positioned in the line segment [0,1] and have approval preferences over two available facilities. A mechanism takes as input the positions and the preferences of the agents, and chooses to locate a single facility based on this information. We study mechanisms that aim to maximize the social welfare (the total utility the agents derive from facilities they approve), under the constraint of incentivizing the agents to truthfully report their positions and preferences. We consider three different settings depending on the level of agent-related information that is public or private. For each setting, we design deterministic and randomized strategyproof mechanisms that achieve a good approximation of the optimal social welfare, and complement these with nearly-tight impossibility results
Facility location with double-peaked preference
We 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 outperforms our
truthful-in-expectation mechanism
Mechanism Design for Facility Location Problems: A Survey
The study of approximate mechanism design for facility location problems has been in the center of research at the intersection of artificial intelligence and economics for the last decades, largely due to its practical importance in various domains, such as social planning and clustering. At a high level, the goal is to design mechanisms to select a set of locations on which to build a set of facilities, aiming to optimize some social objective and ensure desirable properties based on the preferences of strategic agents, who might have incentives to misreport their private information such as their locations. This paper presents a comprehensive survey of the significant progress that has been made since the introduction of the problem, highlighting the different variants and methodologies, as well as the most interesting directions for future research