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

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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

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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

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Ashton Kutcher vs. CNN 4

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The Model Continued Agents 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

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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

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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

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A Mathematicians 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

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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

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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

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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

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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

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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

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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

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Variety of domains Approval Alon+Fischer+P+Tennenholtz Regression and classification Dekel+Fischer+P SODA08 Meir+P+Rosenschein AAAI08, IJCAI09, AAMAS10 Facility location P+Tennenholtz EC09, Alon+Feldman+P+Tennenholtz Lu+Wang+Zhou WINE09, Lu+Sun+Wang+Zhu EC10 Allocation of items Guo+Conitzer, AAMAS10 Generalized assignment Dughmi+Ghosh, EC10 Matching / kidney exchange Ashlagi+Kash+Fischer+P, EC10 15

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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