CSC304 Lecture 17 Voting 4: Impartial selection CSC304 - Nisarg - PowerPoint PPT Presentation

CSC304 Lecture 17 Voting 4: Impartial selection CSC304 - Nisarg Shah 1

Recap • The Gibbard-Satterthwaite theorem says that we cannot design strategyproof voting rules that are also nondictatorial and onto. • Restricted settings (e.g., facility location on a line) ➢ There exist strategyproof, nondictatorial, and onto rules. ➢ They can be used to (perfectly or approximately) optimize the societal goal • Today, we will study another interesting setting called impartial selection CSC304 - Nisarg Shah 2

Impartial Selection • “How can we select 𝑙 people out of 𝑜 people?” ➢ Applications: electing a student representation committee, selecting 𝑙 out of 𝑜 grant applications to fund using peer review, … • Model ➢ Input: a directed graph 𝐻 = (𝑊, 𝐹) ➢ Nodes 𝑊 = {𝑤 1 , … , 𝑤 𝑜 } are the 𝑜 people ➢ Edge 𝑓 = 𝑤 𝑗 , 𝑤 𝑘 ∈ 𝐹 : 𝑤 𝑗 supports/approves of 𝑤 𝑘 o We do not allow or ignore self-edges (𝑤 𝑗 , 𝑤 𝑗 ) ➢ Output: a subset 𝑊 ′ ⊆ 𝑊 with 𝑊 ′ = 𝑙 ➢ 𝑙 ∈ {1, … , 𝑜 − 1} is given CSC304 - Nisarg Shah 3

Impartial Selection • Impartiality: A 𝑙 -selection rule 𝑔 is impartial if 𝑤 𝑗 ∈ 𝑔(𝐻) does not depend on the outgoing edges of 𝑤 𝑗 ➢ 𝑤 𝑗 cannot manipulate his outgoing edges to get selected ➢ Q: But the definition says 𝑤 𝑗 can neither go from 𝑤 𝑗 ∉ 𝑔(𝐻) to 𝑤 𝑗 ∈ 𝑔(𝐻) , nor from 𝑤 𝑗 ∈ 𝑔(𝐻) to 𝑤 𝑗 ∉ 𝑔(𝐻) . Why? • Societal goal: maximize the sum of in-degrees of selected agents σ 𝑤∈𝑔 𝐻 𝑗𝑜 𝑤 ➢ 𝑗𝑜(𝑤) = set of nodes that have an edge to 𝑤 ➢ 𝑝𝑣𝑢 𝑤 = set of nodes that 𝑤 has an edge to ➢ Note: OPT will pick the 𝑙 nodes with the highest indegrees CSC304 - Nisarg Shah 4

Optimal ≠ Impartial 𝑤 1 … 𝑤 3 𝑤 𝑜 𝑤 2 • An optimal 1-selecton rule must select 𝑤 1 or 𝑤 2 • The other node can remove his edge to the winner, and make sure the optimal rule selects him instead • This violates impartiality CSC304 - Nisarg Shah 5

Goal: Approximately Optimal • 𝛽 -approximation: We want a 𝑙 -selection system that always returns a set with total indegree at least 𝛽 times the total indegree of the optimal set • Q: For 𝑙 = 1 , what about the following rule? Rule: “Select the lowest index vertex in 𝑝𝑣𝑢 𝑤 1 . If 𝑝𝑣𝑢 𝑤 1 = ∅ , select 𝑤 2 .” ➢ A. Impartial + constant approximation ➢ B. Impartial + bad approximation ➢ C. Not impartial + constant approximation ➢ D. Not impartial + bad approximation CSC304 - Nisarg Shah 6

No Finite Approximation  • Theorem [Alon et al. 2011] For every 𝑙 ∈ {1, … , 𝑜 − 1} , there is no impartial 𝑙 - selection rule with a finite approximation ratio. • Proof: ➢ For small 𝑙 , this is trivial. E.g., consider 𝑙 = 1 . o What if 𝐻 has two nodes 𝑤 1 and 𝑤 2 that point to each other, and there are no other edges? o For finite approximation, the rule must choose either 𝑤 1 or 𝑤 2 o Say it chooses 𝑤 1 . If 𝑤 2 now removes his edge to 𝑤 1 , the rule must choose 𝑤 2 for any finite approximation. o Same argument as before. But applies to any “finite approximation rule”, and not just the optimal rule. CSC304 - Nisarg Shah 7

No Finite Approximation  • Theorem [Alon et al. 2011] For every 𝑙 ∈ {1, … , 𝑜 − 1} , there is no impartial 𝑙 - selection rule with a finite approximation ratio. • Proof: ➢ Proof is more intricate for larger 𝑙 . Let’s do 𝑙 = 𝑜 − 1 . o 𝑙 = 𝑜 − 1 : given a graph, “eliminate” a node. ➢ Suppose for contradiction that there is such a rule 𝑔 . ➢ W.l.o.g., say 𝑤 𝑜 is eliminated in the empty graph. ➢ Consider a family of graphs in which a subset of {𝑤 1 , … , 𝑤 𝑜−1 } have edges to 𝑤 𝑜 . CSC304 - Nisarg Shah 8

No Finite Approximation  • Proof ( 𝑙 = 𝑜 − 1 continued): 𝑤 2 𝑤 1 𝑤 3 ➢ Consider star graphs in which a non-empty subset of {𝑤 1 , … , 𝑤 𝑜−1 } have edge to 𝑤 𝑜 , and 𝑤 𝑜 there are no other edges 𝑤 𝑜−1 𝑤 4 o Represented by bit strings 0,1 𝑜−1 \{0} ➢ 𝑤 𝑜 cannot be eliminated in any star graph o Otherwise we have infinite approximation 𝑤 2 ➢ 𝑔 maps 0,1 𝑜−1 \{0} to {1, … , 𝑜 − 1} 𝑤 1 𝑤 3 o “Who will be eliminated?” 𝑤 𝑜 ➢ Impartiality: 𝑔 Ԧ 𝑦 = 𝑗 ⇔ 𝑔 Ԧ 𝑦 + Ԧ 𝑓 𝑗 = 𝑗 𝑓 𝑗 has 1 at 𝑗 𝑢ℎ coordinate, 0 elsewhere 𝑤 𝑜−1 𝑤 4 o Ԧ o In words, 𝑗 cannot prevent elimination by adding or removing his edge to 𝑤 𝑜 CSC304 - Nisarg Shah 9

No Finite Approximation  • Proof ( 𝑙 = 𝑜 − 1 continued): 𝑤 2 𝑤 1 𝑤 3 ➢ 𝑔: 0,1 𝑜−1 \{0} → {1, … , 𝑜 − 1} 𝑤 𝑜 ➢ 𝑔 Ԧ 𝑦 = 𝑗 ⇔ 𝑔 Ԧ 𝑦 + Ԧ 𝑓 𝑗 = 𝑗 𝑤 𝑜−1 𝑤 4 𝑓 𝑗 has 1 only in 𝑗 𝑢ℎ coordinate o Ԧ ➢ Pairing implies… o The number of strings on which 𝑔 outputs 𝑗 is 𝑤 2 even, for every 𝑗 . o Thus, total number of strings in the domain 𝑤 1 𝑤 3 must be even too. 𝑤 𝑜 o But total number of strings is 2 𝑜−1 − 1 (odd) 𝑤 𝑜−1 𝑤 4 ➢ So impartiality must be violated for some pair of Ԧ 𝑦 and Ԧ 𝑦 + Ԧ 𝑓 𝑗 CSC304 - Nisarg Shah 10

Back to Impartial Selection • Question: So what can we do to select impartially? • Answer: Randomization! ➢ Impartiality now requires that the probability of an agent being selected be independent of his outgoing edges. • Examples: Randomized Impartial Mechanisms ➢ Choose 𝑙 nodes uniformly at random o Sadly, this still has arbitrarily bad approximation. o Imagine having 𝑙 special nodes with indegree 𝑜 − 1 , and all other nodes having indegree 0 . o Mechanism achieves Τ 𝑙 𝑜 ∗ 𝑃𝑄𝑈 ⇒ approximation = 𝑜/𝑙 o Good when 𝑙 is comparable to 𝑜 , but bad when 𝑙 is small. CSC304 - Nisarg Shah 11

Random Partition • Idea: ➢ What if we partition 𝑊 into 𝑊 1 and 𝑊 2 , and select 𝑙 nodes from 𝑊 1 based only on edges coming to them from 𝑊 2 ? • Mechanism: ➢ Assign each node to 𝑊 1 or 𝑊 2 i.i.d. with probability ½ ➢ Choose 𝑊 𝑗 ∈ 𝑊 1 , 𝑊 2 at random ➢ Choose 𝑙 nodes from 𝑊 𝑗 that have most incoming edges from nodes in 𝑊 3−𝑗 CSC304 - Nisarg Shah 12

Random Partition • Analysis: ➢ We want to approximate 𝐽 = # edges incoming to nodes in 𝑃𝑄𝑈 . o Let 𝑃𝑄𝑈 1 = 𝑃𝑄𝑈 ∩ 𝑊 1 , and 𝑃𝑄𝑈 2 = 𝑃𝑄𝑈 ∩ 𝑊 2 . o Let 𝐽 1 = # edges incoming to 𝑃𝑄𝑈 1 from 𝑊 2 . o Let 𝐽 2 = # edges incoming to 𝑃𝑄𝑈 2 from 𝑊 1 . ➢ Note that 𝐹 𝐽 1 + 𝐽 2 = 𝐽/2 . (WHY?) ➢ With probability ½ , mechanism picks 𝑙 nodes from 𝑊 1 that have most incoming edges from 𝑊 2 (thus at least 𝐽 1 incoming edges). o Because they’re at least as good as 𝑃𝑄𝑈 1 . ➢ With probability ½ , mechanism picks 𝑙 nodes from 𝑊 2 that have most incoming edges from 𝑊 1 (thus at least 𝐽 2 incoming edges). ➢ The expected total incoming edges is at least 1 1 1 2 ⋅ 𝐽 1 2 = 𝐽 o 𝐹 2 ⋅ 𝐽 1 + 2 ⋅ 𝐽 2 = 2 ⋅ 𝐹 𝐽 1 + 𝐽 2 = 4 CSC304 - Nisarg Shah 13

Random Partition • Improvement ➢ The approximation ratio improves if we pick 𝑙/2 nodes from each part instead of picking all 𝑙 from one part. ➢ More generally, we can divide into ℓ parts, and pick 𝑙/ℓ nodes from each part based on incoming edges from all other parts. • Theorem [Alon et al. 2011]: ➢ ℓ = 2 gives a 4 -approximation. 1 ➢ For 𝑙 ≥ 2 , ℓ~𝑙 1/3 gives 1 + 𝑃 𝑙 1/3 approximation. CSC304 - Nisarg Shah 14

Better Approximations • [Alon et al. 2011] conjectured that for randomized impartial 1 - selection… ➢ (For which their mechanism is a 4 -approximation) ➢ It should be possible to achieve a 2 -approximation. ➢ Recently proved by [Fischer & Klimm, 2014] ➢ Permutation mechanism: o Select a random permutation (𝜌 1 , 𝜌 2 , … , 𝜌 𝑜 ) of the vertices. o Start by selecting 𝑧 = 𝜌 1 as the “current answer”. o At any iteration 𝑢 , let 𝑧 ∈ {𝜌 1 , … , 𝜌 𝑢 } be the current answer. o From {𝜌 1 , … , 𝜌 𝑢 }\{𝑧} , if there are more edges to 𝜌 𝑢+1 than to 𝑧 , change the current answer to 𝑧 = 𝜌 𝑢+1 . CSC304 - Nisarg Shah 16

Better Approximations • 2-approximation is tight. ➢ In an 𝑜 -node graph, fix 𝑣 and 𝑤 , and suppose no other nodes have any incoming/outgoing edges. ➢ Three cases: only 𝑣 → 𝑤 edge, only 𝑤 → 𝑣 , or both. o The best impartial mechanism selects 𝑣 and 𝑤 with probability ½ in every case, and achieves 2 -approximation. • But this is because 𝑜 − 2 nodes are not voting! ➢ What if every node must have an outgoing edge? ➢ [Fischer & Klimm]: 12 7 = 1.714 approximation. o Permutation mechanism gives Τ o No mechanism gives better than 2/3 approximation. 12 7 . o Open question to achieve better than Τ CSC304 - Nisarg Shah 17

The rest of this lecture is not part of the syllabus. CSC304 - Nisarg Shah 18