Sum of Us: Truthful Self-Selection

We consider directed graphs over a set of agents, where an edge 〈i, j〉 is taken to mean that agent i trusts or supports agent j. Given such a graph, our goal is to select a subset of agents of fixed size that maximizes the sum of indegrees, that is, a subset of most popular or most trusted agents. On the other hand, each 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, in the context of, e.g., social networks such as Twitter, reputation systems such as Epinions, and Internet search. We wish to design mechanisms—functions that map graphs to selected subsets (without making payments)—which satisfy two constraints: strategyproofness, i.e., agents cannot benefit from misreporting their outgoing edges; and approximation, that is, the mechanism must always select a subset of agents that is close to optimal in terms of the sum of indegrees. Our first major result is a surprising impossibility: no deterministic strategyproof mechanism can yield a finite approximation ratio for any k ∈ {1, . . . , n−1}, where k is the size of the selected subset and n is the number of agents. Our second major result is a randomized strategyproof mechanism that yields an approximation ratio of four for any value of k, and provides a ratio that 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 Geometry at Tel Aviv University. †Institut fur Informatik, Ludwig-Maximilians-Universitat Munchen, 80538 Munchen, Germany, email: fischerf@tcs.ifi.lmu.de. Part of the work was done while the author was visiting The Hebrew University of Jerusalem. This visit was supported by a Minerva Short-Term Research Grant. ‡Microsoft Israel R&D Center, 13 Shenkar Street, Herzeliya 46725, Israel. Email: arielpro@gmail.com. §Microsoft Israel R&D Center, 13 Shenkar Street, Herzeliya 46725, Israel, and Technion, IIT, Haifa 32000, Israel. Email: moshet@microsoft.com