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

3. 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 2

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

5. Ashton Kutcher vs. CNN 4

6. 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 5

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

8. 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+e i )=i –  i=1,...,n-1, |f -1 (i)| even  |dom(f)| even, but |dom(f)| = 2 n-1 -1 1 1 2 2 3 3 4 4 5 5 6 6 7 7 7

9. 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 8

10. 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 9

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

12. 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/k 1/3 ) for m~k 1/3 Theorem: there is no randomized SP k- selection mechanism with approx ratio < 1 + 1/(k 2 +k-1) 11

13. 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 12

14. 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 13

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

16. 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 15

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18. 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 17