2 research outputs found
Randomized Strategyproof Mechanisms for Facility Location and the Mini-Sum-of-Squares Objective
We consider the problem of locating a public facility on a line, where a set
of strategic agents report their \emph{locations} and a mechanism
determines, either deterministically or randomly, the location of the facility.
Game theoretic perspectives of the facility location problem advanced in two
main directions. The first direction is concerned with the characterization of
\emph{strategyproof} (SP) mechanisms; i.e., mechanisms that induce truthful
reporting as a dominant strategy; and the second direction quantifies how well
various objective functions can be approximated when restricted to SP
mechanisms. The current paper provides contributions in both directions. First,
we construct a parameterized randomized SP mechanism, and show that all of the
previously proposed deterministic and randomized SP mechanisms for the current
settings can be formalized as special cases of this mechanism. Second, we give
tight results for the approximation ratio of SP mechanisms with respect to the
objective of minimizing the sum of squares of distances to the agents
(\emph{miniSOS}). Holzman \cite{Holzman1990} provided an axiomatic foundation
for this function, showing that it is the unique function that satisfies
unanimity, continuity and invariance. We devise a randomized mechanism that
gives a 1.5-approximation for the miniSOS function, and show that no other
randomized SP mechanism can provide a better approximation. This mechanism
chooses the average location with probability 1/2 and a \emph{random dictator}
with probability 1/2. For deterministic mechanisms, we show that the median
mechanism provides a 2-approximation, and this is tight. Together, our study
provides fundamental understanding of the miniSOS objective function and makes
a step toward the characterization of randomized SP facility location
mechanisms
On the Power of Deterministic Mechanisms for Facility Location Games
We consider K-Facility Location games, where n strategic agents report their
locations in a metric space, and a mechanism maps them to K facilities. Our
main result is an elegant characterization of deterministic strategyproof
mechanisms with a bounded approximation ratio for 2-Facility Location on the
line. In particular, we show that for instances with n \geq 5 agents, any such
mechanism either admits a unique dictator, or always places the facilities at
the leftmost and the rightmost location of the instance. As a corollary, we
obtain that the best approximation ratio achievable by deterministic
strategyproof mechanisms for the problem of locating 2 facilities on the line
to minimize the total connection cost is precisely n-2. Another rather
surprising consequence is that the Two-Extremes mechanism of (Procaccia and
Tennenholtz, EC 2009) is the only deterministic anonymous strategyproof
mechanism with a bounded approximation ratio for 2-Facility Location on the
line.
The proof of the characterization employs several new ideas and technical
tools, which provide new insights into the behavior of deterministic
strategyproof mechanisms for K-Facility Location games, and may be of
independent interest. Employing one of these tools, we show that for every K
\geq 3, there do not exist any deterministic anonymous strategyproof mechanisms
with a bounded approximation ratio for K-Facility Location on the line, even
for simple instances with K+1 agents. Moreover, building on the
characterization for the line, we show that there do not exist any
deterministic strategyproof mechanisms with a bounded approximation ratio for
2-Facility Location on more general metric spaces, which is true even for
simple instances with 3 agents located in a star