 
Summary: Sum of Us: Strategyproof Selection from the Selectors
Noga Alon Felix Fischer Ariel D. Procaccia Moshe Tennenholtz §
Abstract
We consider directed graphs over a set of n agents, where an edge (i, j) is taken to mean that agent i
supports or trusts agent j. Given such a graph and an integer k n, we wish to select a subset of k agents
that maximizes the sum of indegrees, i.e., a subset of k most popular or most trusted agents. At the same
time we assume that each individual agent is only interested in being selected, and may misreport its
outgoing edges to this end. This problem formulation captures realistic scenarios where agents choose
among themselves, which can be found in the context of Internet search, social networks like Twitter, or
reputation systems like Epinions.
Our goal is to design mechanisms without payments that map each graph to a ksubset of agents
to be selected and satisfy the following two constraints: strategyproofness, i.e., agents cannot benefit
from misreporting their outgoing edges, and approximate optimality, i.e., the sum of indegrees of the
selected subset of agents is always close to optimal. Our first main result is a surprising impossibility:
for k {1,...,n  1}, no deterministic strategyproof mechanism can provide a finite approximation
ratio. Our second main result is a randomized strategyproof mechanism with an approximation ratio that
is bounded from above by four for any value of k, and approaches one as k grows.
Microsoft Israel R&D Center, 13 Shenkar Street, Herzeliya 46725, Israel, and Schools of Mathematics and Computer Science,
Tel Aviv University, Tel Aviv, 69978, Israel, Email: nogaa@tau.ac.il. Research supported in part by a USA Israeli BSF grant,
by a grant from the Israel Science Foundation, by an ERC advanced grant and by the Hermann Minkowski Minerva Center for
