Sum of Us: Strategyproof Selection From the Selectors

1. Sum of Us: Strategyproof Selection From the Selectors NogaAlon, Felix Fischer, Ariel Procaccia, Moshe Tennenholtz

2. Approval Voting • A set of agents vote over a set of alternatives • Must choose k alternatives • Agents designate approved alternatives • Most popular alternatives win • Used by AMS, IEEE, GTS, IFAAMAS

3. The Model • Agents and alternatives coincide • Directed graph • n vertices = agents • Edge from ito j means that i approves of, trusts, or supports j • Internet-based examples: • Web search • “Directed” social networks (Twitter, Epinions)

4. Ashton Kutcher vs. CNN

5. The Model Continued • Agent’s outgoing edges are private info • k-selection mechanism maps graphs to k-subset of agents • Utility of an agent = 1 if selected, 0 otherwise • Mechanism is strategyproof (SP) if agents cannot gain by misreporting edges • Optimization target: sum of indegrees of selected agents • Optimal solution not SP  Looking for SP approx

6. Deterministic k-Selection Mechanisms • k = n: no problem • k = 1: no finite SP approx • k = n-1: no finite SP approx! 1 2

7. An Impossibility Result • Theorem: For all k n-1, there is no deterministic SP k-selection mechanism w. finite approx ratio • Proof (k = n-1): • Assume for contradiction • WLOG n eliminated given empty graph • Consider stars with n as center, n cannot be eliminated • Function f: {0,1}n-1\{0}{1,...,n-1} satisfies: f(x)=i f(x+ei)=i • i=1,...,n-1, |f-1(i)| even  |dom(f)| even, but |dom(f)| = 2n-1-1 1 2 6 7 3 5 4

8. A Mathematician’s “Survivor” • Each tribe member votes for at most one member • One member must be eliminated • Any SP rule cannot have property: if unique member received votes he is not eliminated

9. Randomized Mechanisms • The randomized m-Partition Mechanism (roughly) • Assign agents uniformly i.i.d. to m subsets • For each subset, select ~k/m agents with highest indegrees based on edges from other subsets

10. Example (k=2,m=2) 1 3 2 2 3 4 1 4 5 5 6 6

11. Randomized bounds • A randomized mechanism is universally SP if it is a distribution over SP mechanisms • Theorem:n,k,m, the mechanism is universally SP. Furthermore: • The approx ratio is 4 with m=2 • The approx ratio is 1+O(1/k1/3) for m~k1/3 • Theorem: there is no randomized SP k-selection mechanism with approx ratio < 1 + 1/(k2+k-1)

12. Discussion • Randomized m-Partition is practical when k is not very small! • Very general model • Application to conference reviews • More results about group strategyproofness • Payments

13. Approximate MD Without Money • You are all familiar with Algorithmic Mechanism Design • All the work in the field considers mechanisms with payments • Money unavailable in many settings

14. Some cool animations Opt SP mech with money + tractable Class 1 Opt SP mechanism with money Problem intractable Class 3 No opt SP mech w/o money Class 2 No opt SP mech with money

15. Variety of domains • Approval • Alon+Fischer+P+Tennenholtz • Regression and classification • Dekel+Fischer+P SODA’08 • Meir+P+Rosenschein AAAI’08, IJCAI’09, AAMAS’10 • Facility location • P+Tennenholtz EC’09, Alon+Feldman+P+Tennenholtz • Lu+Wang+Zhou WINE’09, Lu+Sun+Wang+Zhu EC’10 • Allocation of items • Guo+Conitzer, AAMAS’10 • Generalized assignment • Dughmi+Ghosh, EC’10 • Matching / kidney exchange • Ashlagi+Kash+Fischer+P, EC’10

16. Thank Y u!

17. Group Strategyproofness • k-selection mechanism is group strategyproof (GSP) if a coalition of deviators cannot all gain by lying • Selecting a random k-subset is GSP and gives a n/k-approx • Theorem: no randomized GSP k-selection mechanism has approx ratio < (n-1)/k