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