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

CSC304 Lecture 17

Voting 4: Impartial selection

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Recap

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• 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

Impartial Selection

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• “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 = {𝑤1, … , 𝑤𝑜} are the 𝑜 people ➢ Edge 𝑓 = 𝑤𝑗, 𝑤𝑘 ∈ 𝐹: 𝑤𝑗 supports/approves of 𝑤𝑘

• We do not allow or ignore self-edges (𝑤𝑗, 𝑤𝑗)

➢ Output: a subset 𝑊′ ⊆ 𝑊 with 𝑊′ = 𝑙 ➢ 𝑙 ∈ {1, … , 𝑜 − 1} is given

Impartial Selection

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• Impartiality: A 𝑙-selection rule 𝑔 is impartial if 𝑤𝑗 ∈

𝑔(𝐻) does not depend on the outgoing edges of 𝑤𝑗

➢ 𝑤𝑗 cannot manipulate his outgoing edges to get selected ➢ Q: But the definition says 𝑤𝑗 can neither go from 𝑤𝑗 ∉ 𝑔(𝐻)

to 𝑤𝑗 ∈ 𝑔(𝐻), nor from 𝑤𝑗 ∈ 𝑔(𝐻) to 𝑤𝑗 ∉ 𝑔(𝐻). Why?

• Societal goal: maximize the sum of in-degrees of

selected agents σ𝑤∈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

Optimal ≠ Impartial

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• An optimal 1-selecton rule must select 𝑤1 or 𝑤2
• The other node can remove his edge to the winner,

and make sure the optimal rule selects him instead

• This violates impartiality

𝑤1 𝑤2 𝑤3 𝑤𝑜

…

Goal: Approximately Optimal

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• 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 𝑤1 . If 𝑝𝑣𝑢 𝑤1 = ∅, select 𝑤2.”

➢ A. Impartial + constant approximation ➢ B. Impartial + bad approximation ➢ C. Not impartial + constant approximation ➢ D. Not impartial + bad approximation

No Finite Approximation 

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• 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.

• What if 𝐻 has two nodes 𝑤1 and 𝑤2 that point to each other, and

there are no other edges?

• For finite approximation, the rule must choose either 𝑤1 or 𝑤2
• Say it chooses 𝑤1. If 𝑤2 now removes his edge to 𝑤1, the rule must

choose 𝑤2 for any finite approximation.

• Same argument as before. But applies to any “finite approximation

rule”, and not just the optimal rule.

No Finite Approximation 

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• 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.

• 𝑙 = 𝑜 − 1: given a graph, “eliminate” a node.

➢ Suppose for contradiction that there is such a rule 𝑔. ➢ W.l.o.g., say 𝑤𝑜 is eliminated in the empty graph. ➢ Consider a family of graphs in which a subset of

{𝑤1, … , 𝑤𝑜−1} have edges to 𝑤𝑜.

No Finite Approximation 

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• Proof (𝑙 = 𝑜 − 1 continued):

➢ Consider star graphs in which a non-empty

subset of {𝑤1, … , 𝑤𝑜−1} have edge to 𝑤𝑜, and there are no other edges

• Represented by bit strings 0,1 𝑜−1\{0}

➢ 𝑤𝑜 cannot be eliminated in any star graph

• Otherwise we have infinite approximation

➢ 𝑔 maps 0,1 𝑜−1\{0} to {1, … , 𝑜 − 1}

• “Who will be eliminated?”

➢ Impartiality: 𝑔 Ԧ

𝑦 = 𝑗 ⇔ 𝑔 Ԧ 𝑦 + Ԧ 𝑓𝑗 = 𝑗

• Ԧ

𝑓𝑗 has 1 at 𝑗𝑢ℎ coordinate, 0 elsewhere

• In words, 𝑗 cannot prevent elimination by adding
• r removing his edge to 𝑤𝑜

𝑤1 𝑤2 𝑤3 𝑤𝑜 𝑤𝑜−1 𝑤4 𝑤1 𝑤2 𝑤3 𝑤𝑜 𝑤𝑜−1 𝑤4

No Finite Approximation 

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• Proof (𝑙 = 𝑜 − 1 continued):

➢ 𝑔: 0,1 𝑜−1\{0} → {1, … , 𝑜 − 1} ➢ 𝑔 Ԧ

𝑦 = 𝑗 ⇔ 𝑔 Ԧ 𝑦 + Ԧ 𝑓𝑗 = 𝑗

• Ԧ

𝑓𝑗 has 1 only in 𝑗𝑢ℎ coordinate

➢ Pairing implies…

• The number of strings on which 𝑔 outputs 𝑗 is

even, for every 𝑗.

• Thus, total number of strings in the domain

must be even too.

• But total number of strings is 2𝑜−1 − 1 (odd)

➢ So impartiality must be violated for some

pair of Ԧ 𝑦 and Ԧ 𝑦 + Ԧ 𝑓𝑗

𝑤1 𝑤2 𝑤3 𝑤𝑜 𝑤𝑜−1 𝑤4 𝑤1 𝑤2 𝑤3 𝑤𝑜 𝑤𝑜−1 𝑤4

Back to Impartial Selection

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• 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

• Sadly, this still has arbitrarily bad approximation.
• Imagine having 𝑙 special nodes with indegree 𝑜 − 1, and all other

nodes having indegree 0.

• Mechanism achieves

Τ 𝑙 𝑜 ∗ 𝑃𝑄𝑈 ⇒ approximation = 𝑜/𝑙

• Good when 𝑙 is comparable to 𝑜, but bad when 𝑙 is small.

Random Partition

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• 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

Random Partition

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• Analysis:

➢ We want to approximate 𝐽 = # edges incoming to nodes in 𝑃𝑄𝑈.

• Let 𝑃𝑄𝑈

1 = 𝑃𝑄𝑈 ∩ 𝑊 1, and 𝑃𝑄𝑈2 = 𝑃𝑄𝑈 ∩ 𝑊 2.

• Let 𝐽1 = # edges incoming to 𝑃𝑄𝑈

1 from 𝑊 2.

• Let 𝐽2 = # edges incoming to 𝑃𝑄𝑈2 from 𝑊

1. ➢ Note that 𝐹 𝐽1 + 𝐽2 = 𝐽/2. (WHY?) ➢ With probability ½, mechanism picks 𝑙 nodes from 𝑊

1 that have

most incoming edges from 𝑊

2 (thus at least 𝐽1 incoming edges).

• 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 𝐽2 incoming edges).

➢ The expected total incoming edges is at least

• 𝐹

1 2 ⋅ 𝐽1 + 1 2 ⋅ 𝐽2 = 1 2 ⋅ 𝐹 𝐽1 + 𝐽2 = 1 2 ⋅ 𝐽 2 = 𝐽 4

Random Partition

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• 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

• ther parts.
• Theorem [Alon et al. 2011]:

➢ ℓ = 2 gives a 4-approximation. ➢ For 𝑙 ≥ 2, ℓ~𝑙1/3 gives 1 + 𝑃

1 𝑙1/3 approximation.

Better Approximations

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• [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:

• Select a random permutation (𝜌1, 𝜌2, … , 𝜌𝑜) of the vertices.
• Start by selecting 𝑧 = 𝜌1 as the “current answer”.
• At any iteration 𝑢, let 𝑧 ∈ {𝜌1, … , 𝜌𝑢} be the current answer.
• From {𝜌1, … , 𝜌𝑢}\{𝑧}, if there are more edges to 𝜌𝑢+1 than to 𝑧,

change the current answer to 𝑧 = 𝜌𝑢+1.

Better Approximations

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• 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.

• 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]:

• Permutation mechanism gives Τ

12 7 = 1.714 approximation.

• No mechanism gives better than 2/3 approximation.
• Open question to achieve better than Τ

12 7.

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The rest of this lecture is not part of the syllabus.

PageRank

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• An extension of the impartial selection problem

➢ Instead of selecting 𝑙 nodes, we want to rank all nodes

• The PageRank Problem: Given a directed graph,

rank all nodes by their “importance”.

➢ Think of the web graph, where nodes are webpages, and

a directed (𝑣, 𝑤) edge means 𝑣 has a link to 𝑤.

• Questions:

➢ What properties do we want from such a rule? ➢ What rule satisfies these properties?

PageRank

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• Here is the PageRank Algorithm:

➢ Start from any node in the graph. ➢ At each iteration, choose an outgoing edge of the current

node, uniformly at random among all its outgoing edges.

➢ Move to the neighbor node on that edge. ➢ In the limit of 𝑈 → ∞ iterations, measure the fraction of

time the “random walk” visits each node.

➢ Rank the nodes by these “stationary probabilities”.

• Google uses (a version of) this algorithm

➢ It’s seems a reasonable algorithm. ➢ What nice axioms might it satisfy?

Axioms

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• Axiom 1 (Isomorphism)

➢ Permuting node names permutes the final

ranking.

• Axiom 2 (Vote by Committee)

➢ Voting through intermediate fake nodes

cannot change the ranking.

• Axiom 3 (Self Edge)

➢ 𝑤 adding a self edge cannot change the

• rdering of the other nodes.
• Axiom 4 (Collapsing)

➢ Merging identically voting nodes cannot change the

• rdering of the other nodes.
• Axiom 5 (Proxy)

➢ If a set of nodes with equal score vote for 𝑤 through a

proxy, it should not be different than voting directly.

𝑏 𝑏

PageRank

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• Theorem [Altman and Tennenholtz, 2005]:

The PageRank algorithm satisfies these five axioms, and is the unique algorithm to satisfy all five axioms.

• That is, any algorithm that satisfies all five axioms

must output the ranking returned by PageRank on every single graph.