CSC304 Lectures 17-18 Voting 3: Axiomatic, Statistical, and - - PowerPoint PPT Presentation

CSC304 Lectures 17-18

Voting 3: Axiomatic, Statistical, and Utilitarian Approaches to Voting

CSC304 - Nisarg Shah 1

Recap

CSC304 - Nisarg Shah 2

• We introduced a plethora of voting rules

➢ Plurality ➢ Borda ➢ Veto ➢ 𝑙-Approval ➢ STV ➢ Plurality with

runoff

➢ Kemeny ➢ Copeland ➢ Maximin

• Which is the right way to aggregate preferences?

➢ GS Theorem: There is no good strategyproof voting rule. ➢ For now, let us forget about incentives. Let us focus on

how to aggregate given truthful votes.

Recap

CSC304 - Nisarg Shah 3

• Set of voters 𝑂 = {1, … , 𝑜}
• Set of alternatives 𝐵, 𝐵 = 𝑛
• Voter 𝑗 has a preference

ranking ≻𝑗 over the alternatives

1 2 3 a c b b a a c b c

• Preference profile ≻ = collection of all voter rankings
• Voting rule (social choice function) 𝑔

➢ Takes as input a preference profile ≻ ➢ Returns an alternative 𝑏 ∈ 𝐵

Axiomatic Approach

CSC304 - Nisarg Shah 4

• Goal: Define a set of reasonable desiderata, and

find voting rules satisfying them

➢ Ultimate hope: a unique voting rule satisfies the axioms

we are interested in!

• Sadly, it’s often the opposite case.

➢ Many combinations of reasonable axioms cannot be

satisfied by any voting rule.

➢ GS theorem: nondictatorship + ontoness +

strategyproofness = ∅

➢ Arrow’s theorem: we’ll see ➢ …

Axiomatic Approach

CSC304 - Nisarg Shah 5

• Unanimity: If all voters have the same top choice,

that alternative is the winner.

𝑢𝑝𝑞 ≻𝑗 = 𝑏 ∀𝑗 ∈ 𝑂 ⇒ 𝑔 ≻ = 𝑏

➢ I used 𝑢𝑝𝑞 ≻𝑗 = 𝑏 to denote 𝑏 ≻𝑗 𝑐 ∀𝑐 ≠ 𝑏

• Pareto optimality: If all voters prefer 𝑏 to 𝑐, then 𝑐 is

not the winner. 𝑏 ≻𝑗 𝑐 ∀𝑗 ∈ 𝑂 ⇒ 𝑔 ≻ ≠ 𝑐

• Q: What is the relation between these axioms?

➢ Pareto optimality ⇒ Unanimity

Axiomatic Approach

CSC304 - Nisarg Shah 6

• Anonymity: Permuting votes does not change the

winner (i.e., voter identities don’t matter).

➢ E.g., these two profiles must have the same winner:

{voter 1: 𝑏 ≻ 𝑐 ≻ 𝑑, voter 2: 𝑐 ≻ 𝑑 ≻ 𝑏} {voter 1: 𝑐 ≻ 𝑑 ≻ 𝑏, voter 2: 𝑏 ≻ 𝑐 ≻ 𝑑}

• Neutrality: Permuting the alternative names

permutes the winner accordingly.

➢ E.g., say 𝑏 wins on {voter 1: 𝑏 ≻ 𝑐 ≻ 𝑑, voter 2: 𝑐 ≻ 𝑑 ≻ 𝑏} ➢ We permute all names: 𝑏 → 𝑐, 𝑐 → 𝑑, and 𝑑 → 𝑏 ➢ New profile: {voter 1: 𝑐 ≻ 𝑑 ≻ 𝑏, voter 2: 𝑑 ≻ 𝑏 ≻ 𝑐} ➢ Then, the new winner must be 𝑐.

Axiomatic Approach

CSC304 - Nisarg Shah 7

• Neutrality is tricky

➢ As we defined it, it is inconsistent with anonymity!

• Imagine {voter 1: 𝑏 ≻ 𝑐, voter 2: 𝑐 ≻ 𝑏}
• Without loss of generality, say 𝑏 wins
• Imagine a different profile: {voter 1: 𝑐 ≻ 𝑏, voter 2: 𝑏 ≻ 𝑐}
• Neutrality: We just exchanged 𝑏 ↔ 𝑐, so winner is 𝑐.
• Anonymity: We just exchanged the votes, so winner stays 𝑏.

➢ Typically, we only require neutrality for…

• Randomized rules: E.g., a rule could satisfy both by choosing 𝑏 and

𝑐 as the winner with probability ½ each, on both profiles

• Deterministic rules allowed to return ties: E.g., a rule could return

{𝑏, 𝑐} as tied winners on both profiles.

Axiomatic Approach

CSC304 - Nisarg Shah 8

• Majority consistency: If a majority of voters have the same

top choice, that alternative wins. 𝑗: 𝑢𝑝𝑞 ≻𝑗 = 𝑏 > 𝑜 2 ⇒ 𝑔 ≻ = 𝑏

➢ Satisfied by plurality, but not by Borda count

• Condorcet consistency: If 𝑏 defeats every other alternative

in a pairwise election, 𝑏 wins. 𝑗: 𝑏 ≻𝑗 𝑐 > 𝑜 2 , ∀𝑐 ≠ 𝑏 ⇒ 𝑔 ≻ = 𝑏

➢ Condorcet consistency ⇒ Majority consistency ➢ Violated by both plurality and Borda count

Axiomatic Approach

CSC304 - Nisarg Shah 9

• Is even the weaker axiom majority consistency a

reasonable one to expect?

1 2 3 4 5 a a a b b b b b a a

Axiomatic Approach

CSC304 - Nisarg Shah 10

• Consistency: If 𝑏 is the winner on two profiles, it

must be the winner on their union.

𝑔 ≻1 = 𝑏 ∧ 𝑔 ≻2 = 𝑏 ⇒ 𝑔 ≻1+≻2 = 𝑏

➢ Example: ≻1= 𝑏 ≻ 𝑐 ≻ 𝑑 , ≻2= 𝑏 ≻ 𝑑 ≻ 𝑐, 𝑐 ≻ 𝑑 ≻ 𝑏 ➢ Then, ≻1+≻2= 𝑏 ≻ 𝑐 ≻ 𝑑, 𝑏 ≻ 𝑑 ≻ 𝑐, 𝑐 ≻ 𝑑 ≻ 𝑏

• Is this reasonable?

➢ Young [1975] showed that subject to mild requirements, a voting rule

is consistent if and only if it is a positional scoring rule!

➢ Thus, plurality with runoff, STV, Kemeny, Copeland, Maximin, etc are

not consistent.

Axiomatic Approach

CSC304 - Nisarg Shah 11

• Weak monotonicity: If 𝑏 is the winner, and 𝑏 is

“pushed up” in some votes, 𝑏 remains the winner.

➢ 𝑔 ≻ = 𝑏 ⇒ 𝑔 ≻′ = 𝑏 if

• 1. 𝑐 ≻𝑗 𝑑 ⇔ 𝑐 ≻𝑗

′ 𝑑, ∀𝑗 ∈ 𝑂, 𝑐, 𝑑 ∈ 𝐵\{𝑏}

“Order among other alternatives preserved in all votes”

• 2. 𝑏 ≻𝑗 𝑐 ⇒ 𝑏 ≻𝑗

′ 𝑐, ∀𝑗 ∈ 𝑂, 𝑐 ∈ 𝐵\{𝑏}

(𝑏 only improves) “In every vote, 𝑏 still defeats all the alternatives it defeated”

• Contrast: strong monotonicity requires 𝑔 ≻′ = 𝑏

even if ≻′ only satisfies the 2nd condition

➢ It is thus too strong. Equivalent to strategyproofness! ➢ Only satisfied by dictatorial/non-onto rules [GS theorem]

Axiomatic Approach

CSC304 - Nisarg Shah 12

• Weak monotonicity: If 𝑏 is the winner, and 𝑏 is

“pushed up” in some votes, 𝑏 remains the winner.

➢ 𝑔 ≻ = 𝑏 ⇒ 𝑔 ≻′ = 𝑏, where

• 𝑐 ≻𝑗 𝑑 ⇔ 𝑐 ≻𝑗

′ 𝑑, ∀𝑗 ∈ 𝑂, 𝑐, 𝑑 ∈ 𝐵\{𝑏} (Order of others preserved)

• 𝑏 ≻𝑗 𝑐 ⇒ 𝑏 ≻𝑗

′ 𝑐, ∀𝑗 ∈ 𝑂, 𝑐 ∈ 𝐵\{𝑏}

(𝑏 only improves)

• Weak monotonicity is satisfied by most voting rules

➢ Only exceptions (among rules we saw):

STV and plurality with runoff

➢ But this helps STV be hard to manipulate

• [Conitzer & Sandholm 2006]: “Every weakly monotonic voting rule is

easy to manipulate on average.”

Axiomatic Approach

CSC304 - Nisarg Shah 13

• STV violates weak monotonicity

7 voters 5 voters 2 voters 6 voters a b b c b c c a c a a b

• First 𝑑, then 𝑐 eliminated
• Winner: 𝑏

7 voters 5 voters 2 voters 6 voters a b a c b c b a c a c b

• First 𝑐, then 𝑏 eliminated
• Winner: 𝑑

Axiomatic Approach

CSC304 - Nisarg Shah 14

• For social welfare functions that output a ranking:
• Independence of Irrelevant Alternatives (IIA):

➢ If the preferences of all voters between 𝑏 and 𝑐 are

unchanged, then the social preference between 𝑏 and 𝑐 should not change.

• Arrow’s Impossibility Theorem

➢ No voting rule satisfies IIA, Pareto optimality, and

nondictatorship.

➢ Proof omitted. ➢ Foundations of the axiomatic approach to voting

NOT IN SYLLABUS

Statistical Approach

CSC304 - Nisarg Shah 15

• Assume that there is a “true” ranking of

alternatives

➢ Unknown to us apriori

• Votes {≻𝑗} are generated i.i.d. from a distribution

parametrized by a ranking 𝜏∗

➢ Pr[≻ |𝜏∗] denotes the probability of drawing a vote ≻

given that the ground truth is 𝜏∗

• Maximum likelihood estimate (MLE):

➢ Given ≻, return argmax𝜏 Pr ≻ 𝜏 = ς𝑗=1

𝑜

Pr ≻𝑗 𝜏

NOT IN SYLLABUS

Statistical Approach

CSC304 - Nisarg Shah 16

• Example: Mallows’ model

➢ Recall Kendall-tau distance 𝑒 between two rankings:

#pairs of alternatives on which they disagree

➢ Malllows’ model: Pr ≻ 𝜏∗ ∝ 𝜒𝑒 ≻,𝜏∗ , where

• 𝜒 ∈ (0,1] is the “noise parameter”
• 𝜒 → 0 : Pr 𝜏∗ 𝜏∗ → 1
• 𝜒 = 1 : uniform distribution
• Normalization constant 𝑎𝜒 = σ≻ 𝜒𝑒 ≻,𝜏∗ does not depend on 𝜏∗

➢ The greater the distance from the ground truth, the

smaller the probability

NOT IN SYLLABUS

Statistical Approach

CSC304 - Nisarg Shah 17

• Example: Mallows’ model

➢ What is the MLE ranking for Mallows’ model?

max

𝜏

ෑ

𝑗=1 𝑜

Pr ≻𝑗 𝜏∗ = max

𝜏

ෑ

𝑗=1 𝑜 𝜒𝑒 ≻𝑗,𝜏∗

𝑎𝜒 = max

𝜏

𝜒σ𝑗=1

𝑜

𝑒 ≻𝑗,𝜏∗

𝑎𝜒

➢ The MLE ranking 𝜏∗ minimizes σ𝑗=1

𝑜

𝑒(≻𝑗, 𝜏∗)

➢ This is precisely the Kemeny ranking!

• Statistical approach yields a unique rule, but is

specific to the assumed distribution of votes

NOT IN SYLLABUS

Utilitarian Approach

CSC304 - Nisarg Shah 18

• Each voter 𝑗 still submits a ranking ≻𝑗

➢ But the voter has “implicit” numerical utilities {

} 𝑤𝑗 𝑏 ≥ Σ𝑏 𝑤𝑗 𝑏 = 1 𝑏 ≻𝑗 𝑐 ⇒ 𝑤𝑗 𝑏 ≥ 𝑤𝑗 𝑐

• Goal:

➢ Select 𝑏∗ with the maximum social welfare σ𝑗 𝑤𝑗 𝑏∗

• Cannot always find this given only rankings from voters

➢ Refined goal: Select 𝑏∗ that gives the best worst-case

approximation of welfare

Distortion

CSC304 - Nisarg Shah 19

• The distortion of a voting rule 𝑔 is its

approximation ratio of social welfare, on the worst preference profile.

𝑒𝑗𝑡𝑢 𝑔 = sup

𝑤𝑏𝑚𝑗𝑒 {𝑤𝑗}

max

𝑐

σ𝑗 𝑤𝑗 𝑐 σ𝑗 𝑤𝑗 𝑔(≻)

➢ where each 𝑤𝑗 is valid if Σ𝑏 𝑤𝑗 𝑏 = 1 ➢ ≻ = ≻1, … , ≻𝑜 where ≻𝑗 represents the ranking of

alternatives according to 𝑤𝑗

Example

CSC304 - Nisarg Shah 20

• Suppose there are 2 voters and 3 alternatives
• Suppose our 𝑔 returns 𝑑 on this profile

1 2 a c b a c b

Rankings

1 2 a : 1.0 c : 0.5 b : 0.0 a : 0.5 c : 0.0 b : 0.0

Utilities

1 2 a : 0.4 c : 0.7 b : 0.3 a : 0.2 c : 0.3 b : 0.1

Utilities

Social welfare 𝑏 = 1.5 (optimal) 𝑑 = 0.5 𝑒𝑗𝑡𝑢(𝑔) ≥ 3

…

Social welfare 𝑑 = 1.0 (optimal) 𝑒𝑗𝑡𝑢(𝑔) ≥ 1 𝑒𝑗𝑡𝑢(𝑔) is the largest such number you can find by constructing consistent utility profiles

Optimal Deterministic Rules

CSC304 - Nisarg Shah 21

• Theorem [Caragiannis et al. ‘17]:

Plurality achieves 𝑃 𝑛2 distortion.

• Proof:

➢ The winner is the top choice of at least 𝑜/𝑛 voters. ➢ Each voter must have utility at least 1/𝑛 for her top

• choice. (WHY?)

➢ Plurality achieves social welfare at least

𝑜 𝑛 ⋅ 1 𝑛 = 𝑜 𝑛2

➢ No alternative can achieve social welfare more than 𝑜

(WHY?)

➢ QED!

Optimal Deterministic Rules

CSC304 - Nisarg Shah 22

• Theorem [Caragiannis et al. ‘17]:

Every deterministic voting rule has Ω 𝑛2 distortion.

• Proof:

➢ 𝑜 voters divided into 𝑛 − 1 blocks of equal size ➢ Preference profile:

• voters in block 𝑗 put 𝑏𝑗 first, 𝑏𝑛 next, and the rest arbitrarily

➢ If output = 𝑏𝑛 ⇒ ∞ distortion (WHY?) ➢ If output ∈ {𝑏1, … , 𝑏𝑛−1} ⇒ Ω 𝑛2 distortion

• Derivation on the board!

𝑏1 ≻ 𝑏𝑛 ≻ ⋯ 𝑏2 ≻ 𝑏𝑛 ≻ ⋯ 𝑏3 ≻ 𝑏𝑛 ≻ ⋯ ⋮ 𝑏𝑛−1 ≻ 𝑏𝑛 ≻ ⋯ 𝑜/(𝑛 − 1) times

Optimal Randomized Rules

CSC304 - Nisarg Shah 23

• Theorem [Boutilier et al. ‘15]:

There is a randomized rule with O

𝑛 ⋅ log 𝑛 distortion.

• Proof:

➢ Given profile ≻, define the harmonic score sc(𝑏, ≻):

• Each voter gives Τ

1 𝑙 points to her 𝑙𝑢ℎ most preferred alternative

• sc(𝑏, ≻) = sum of points received by 𝑏 from all voters

➢ Want to compare sc 𝑏, ≻ to social welfare sw 𝑏, Ԧ

𝑤

• sw 𝑏, Ԧ

𝑤 ≤ sc(𝑏, ≻) (WHY?)

• σ𝑏 𝑡𝑑(𝑏, ≻) = 𝑜 ⋅ σ𝑙=1

𝑛

Τ 1 𝑙 ≤ 𝑜 ⋅ (ln 𝑛 + 1)

Optimal Voting Rules

CSC2556 - Nisarg Shah 24

• Proof (continued):

➢ Golden voting rule:

• Rule 1: Choose every 𝑏 w.p. proportional to sc(𝑏, ≻)
• Rule 2: Choose every 𝑏 w.p. Τ

1 𝑛 (uniformly at random)

• Execute rule 1 and rule 2 with probability ½ each

➢ Distortion ≤ 2 𝑛 ⋅ (ln 𝑛 + 1) (proof on the board!) ➢ Trick: Take optimal alternative 𝑏∗ ∈ argmax𝑏∈𝐵 sw 𝑏, Ԧ

𝑤

• If sc 𝑏∗, ≻ ≥ 𝑜 (ln 𝑛 + 1)/𝑛 :
• Rule 1 picks 𝑏∗ with enough probability
• Otherwise, we know sw 𝑏∗, Ԧ

𝑤 ≤ sc 𝑏∗, ≻ ≤ 𝑜 (ln 𝑛 + 1)/𝑛 :

• Rule 2 generates enough social welfare (𝑜/𝑛).

Optimal Randomized Rules

CSC304 - Nisarg Shah 25

• Theorem [Boutilier et al. ‘15]:

No randomized rule has distortion better than 𝑛/3.

• Proof:

➢ Pick

𝑛 special alternatives: 𝑏1, … , 𝑏 𝑛

➢ 𝑜 voters divided into 𝑛 equal-size blocks ➢ Preference profile:

• For 𝑗 ∈ 1, … ,

𝑛 , voters in block 𝑗 put 𝑏𝑗 first, and others arbitrarily

➢ Pigeonhole principle:

• ∃𝑏𝑗 ∈ 𝑏1, … , 𝑏 𝑛 that the voting rule picks with probability at most

1/ 𝑛

• Construct worst-case valuation to make 𝑏𝑗 look as good as possible in

hindsight to derive 𝑛/3 distortion bound (proof on the board!)

Utilitarian Approach

CSC304 - Nisarg Shah 26

• Pros: Uses minimal assumptions and yields a

uniquely optimal voting rule

• Cons: The optimal rule is difficult to compute and

unintuitive to humans

• This approach is currently deployed on

RoboVote.org

➢ It has been extended to select a set of alternatives, select

a ranking, select public projects subject to a budget constraint, etc.

CSC304 - Nisarg Shah 27