64 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
Nash Welfare and Facility Location
We consider the problem of locating a facility to serve a set of agents
located along a line. The Nash welfare objective function, defined as the
product of the agents' utilities, is known to provide a compromise between
fairness and efficiency in resource allocation problems. We apply this welfare
notion to the facility location problem, converting individual costs to
utilities and analyzing the facility placement that maximizes the Nash welfare.
We give a polynomial-time approximation algorithm to compute this facility
location, and prove results suggesting that it achieves a good balance of
fairness and efficiency. Finally, we take a mechanism design perspective and
propose a strategy-proof mechanism with a bounded approximation ratio for Nash
welfare
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
Characterization of Group-Strategyproof Mechanisms for Facility Location in Strictly Convex Space
We characterize the class of group-strategyproof mechanisms for the single
facility location game in any unconstrained strictly convex space. A mechanism
is \emph{group-strategyproof}, if no group of agents can misreport so that all
its members are \emph{strictly} better off. A strictly convex space is a normed
vector space where holds for any pair of different unit vectors , e.g., any space with .
We show that any deterministic, unanimous, group-strategyproof mechanism must
be dictatorial, and that any randomized, unanimous, translation-invariant,
group-strategyproof mechanism must be \emph{2-dictatorial}. Here a randomized
mechanism is 2-dictatorial if the lottery output of the mechanism must be
distributed on the line segment between two dictators' inputs. A mechanism is
translation-invariant if the output of the mechanism follows the same
translation of the input.
Our characterization directly implies that any (randomized)
translation-invariant approximation algorithm satisfying the
group-strategyproofness property has a lower bound of -approximation for
maximum cost (whenever ), and for social cost. We also find
an algorithm that -approximates the maximum cost and -approximates the
social cost, proving the bounds to be (almost) tight.Comment: Accepted to ACM Conference on Economics and Computation (EC) 202
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