PHIL 308S: Voting Theory and Fair Division - Lecture 18epacuit/classes/... · PHIL 308S: Voting...

PHIL 308S: Voting Theory andFair Division

Lecture 18

Eric Pacuit

Department of PhilosophyUniversity of Maryland, College Park

ai.stanford.edu/∼epacuitepacuit@umd.edu

November 13, 2012

PHIL 308S: Voting Theory and Fair Division 1/37

R. Sanver. Approval as an intrinsic part of preference. Handbook of ApprovalVoting, 2010.

Let U(A) be the set of real-valued “utility functions” defined overA, an aggregation function is

F : U(A)n → ℘(A)− {∅}

Assumes the existence of an absolute scale over which the utilitiesof individuals are measured and compared.

At the other extreme, rule out any kind of cardinal information andinterpersonal comparability, partitions U(A)n into cells which areordinally equivalent:

F : W (A)n → ℘(A)− {∅}

where W (A) is the set of weak orderings on A.

F : U(A)n → ℘(A)− {∅}

F : W (A)n → ℘(A)− {∅}

F : U(A)n → ℘(A)− {∅}

F : W (A)n → ℘(A)− {∅}

Approval Voting

F : W (A ∪ {0})n → ℘(A)− {∅}

where 0 separates the “good” and “bad” elements.

“approval is not a strategic action but has an intrinsic meaning: Itrefers to the alternatives which are qualified as good.”

Approval Voting

F : W (A ∪ {0})n → ℘(A)− {∅}

where 0 separates the “good” and “bad” elements.

“approval is not a strategic action but has an intrinsic meaning: Itrefers to the alternatives which are qualified as good.”

Combining Approval and Preference

Under Approval Voting (AV), voters are asked which candidatesthe voter approves

Under preference voting procedures (such as BC), candidates areasked to (linearly) rank the candidates.

The two pieces of information are related, but not derivable fromeach other

What about asking for both pieces of information?

Brams and Sanver . Voting Systems that Combine Approval and Preference.available at the author’s website.

Assumptions

Assume each voter has a (linear) preference over the candidates.

Each voter is asked to rank the candidates from most preferred toleast preferred (ties are not allowed).

Voters are then asked to specify which candidates are acceptable.

Consistency Assumption Given two candidates a and b, if a isapproved and b is disapproved then a is ranked higher than b.

For example, we denote this approval ranking for a set {a, b, c, d}of candidates as follows

a d | c b

Preference Approval Voting (PAV)

1. If no candidate, or exactly one candidate, receives a majorityof approval votes, the PAV winner is the AV winner.

2. If two or more candidates receive a majority of approval votes,then

2.1 If one of these candidates is preferred by a majority to everyother majority approved candidate, then he or she is the PAVwinner.

2.2 If there is not one majority-preferred candidate because of acycle among the majority-approved candidates, then the AVwinner among them is the PAV winner.

PAV vs. Condorcet

Rule 1

I. 1 voter: a b | cII. 1 voter: b | a cIII. 1 voter: c | a b

PAV vs. Condorcet

Rule 1

b is the AV winner.

PAV vs. Condorcet

Rule 1

b is the AV winner.b is also the PAV winner.

PAV vs. Condorcet

Rule 1

b is the AV winner.b is also the PAV winner.a is the Condorcet winner.

PAV vs. Condorcet

Rule 2(a)

I. 1 voter: a b c | dII. 1 voter: b c | a dIII. 1 voter: d | a c b

PAV vs. Condorcet

Rule 2(a)

b is the PAV winner.

PAV vs. Condorcet

Rule 2(a)

b is the PAV winner.a is the Condorcet winner.

PAV vs. Condorcet

Rule 2(b)

PAV vs. Condorcet

Rule 2(b)

a (3 votes), b (3 votes), and c (4 votes) are all majority approved.

PAV vs. Condorcet

Rule 2(b)

a (3 votes), b (3 votes), and c (4 votes) are all majority approved.c is the PAV winner

PAV vs. Condorcet

Rule 2(b)

a (3 votes), b (3 votes), and c (4 votes) are all majority approved.c is the PAV winner.d is the Condorcet winner.

Example

I. 3 voters: a b c | dII. 3 voters: d a c |bIII. 2 voters: b d c | a

Example

c is approved by all 8 voters.

Example

c is approved by all 8 voters.There is a top cycle a > b > d > a which are all preferred bymajorities to c (the AV winner).a is the PAV winner

Example

a is the PAV winner.c is the AV winner.d is the STV winner.

Example

I. 2 voters: a c b | dII. 2 voters: a c d | bIII. 3 voters: b c d | a

Example

I. 2 voters: a c b | dII. 2 voters: a c d | bIII. 3 voters: b c d | a

c is approve by all 7 voters.a is the least approved candidate.a is the PAV winner.BC (a) = 12BC (c) = 14

Fallback Voting (FV)

1. Voters indicate all candidates of whom they approve, whomay range from no candidate to all candidate. Voters rankonly those candidates whom they approve.

2. The highest-ranked candidate of all voters is considered. If amajority agrre on the highest-ranked candidate, this candidateis the FV winner (level 1).

Fallback Voting (FV)

1. If there is no level 1 winner, the next-highest ranked candidateof all voters in considered. If a majority of voters agree on onecandidate as either their highest or their next-highest rankedcandidate, this candidate is the FV winner (level 2). If morethan one receive majority approval, then the candidate withthe largest majority is the FV winner.

2. If no level 2 winner, the voters descend – one level at a time— to lower ranks of approved candidates stopping when oneor more candidates receives majority approval. If more thanone receives majority approval then the candidate with thelargest majority is the FV winner. If the descent reaches thebottom and no candidate has won, then the candidate withthe most approval is the FV winner.

Example

I. 4 voters: a b c | dII. 3 voters: b c | a dIII. 2 voters: d a c | b

Example

I. 4 voters: a b c | dII. 3 voters: b c | a dIII. 2 voters: d a c | b

b is the FV winner.c is the AV winner.a is the PAV winner.

PAV and FV

I Neither PAV nor FV may elect the Condorcet winner.

I Both PAV and FV are monotonic (approval-monotonic andrank-monotonic)

I Truth-telling strategies of voters under PAV and FV may notbe in equilibrium

qi (x ;R) = G iff xPi0

nG (x ;R) = |{i ∈ N | qi (x ;R) = G}|

γ(R) = {x ∈ A | nG (x ;R) ≥ n/2}

f : W (A ∪ {0)n → ℘(A)− ∅ satisfies majoritarian approval iff wehave f (R) ⊆ γ(R) for every R ∈W (A ∪ {0)n where γ(R) 6= ∅

nG (x ;R) = |{i ∈ N | qi (x ;R) = G}|

γ(R) = {x ∈ A | nG (x ;R) ≥ n/2}

nG (x ;R) = |{i ∈ N | qi (x ;R) = G}|

γ(R) = {x ∈ A | nG (x ;R) ≥ n/2}

nG (x ;R) = |{i ∈ N | qi (x ;R) = G}|

γ(R) = {x ∈ A | nG (x ;R) ≥ n/2}

f : W (A ∪ {0)n → ℘(A)− ∅ satisfies approval independence iffwe have f (R) = f (R ′) for every R,R ′ ∈W (A ∪ {0)n where xRiyiff xR ′i y for all x , y ∈ A and i ∈ N.

Fact. Majoritarian approval and approval independence arelogically incompatible.

f : W (A ∪ {0)n → ℘(A)− ∅ satisfies approval independence iffwe have f (R) = f (R ′) for every R,R ′ ∈W (A ∪ {0)n where xRiyiff xR ′i y for all x , y ∈ A and i ∈ N.

Fact. Majoritarian approval and approval independence arelogically incompatible.

Two Questions

1. How to refine the set of socially good alternatives, when thisset contains more than one alternative?

2. Which alternative to choose when none of them is sociallygood?

Majoritarian Compromise

1. The highest-ranked candidate of all voters is considered. If amajority of voters agree on one highest-ranked candidate, thiscandidate is the MC winner.

2. If there is no level 1 winner, the next-highest ranked candidateof all voters is considered. If a majority of voters agree on onecandidate as either their highest or their next-highest rankedcandidate, this candidate is the MC winner. If more than onecandidate receives a majority support, then the candidate withhighest support is the MC winner. The procedure stops.

3. If there is no level 2 winner, the voters descend one level at atime to lower and lower ranks, stopping when, for the firsttime, one or more candidates receive a majority support. Ifmore than one candidate receives a majority support, then thecandidate with the highest majority support is the MC winner.

Majoritarian Approval Compromise

3 voters a | b c d2 voters b a c | d2 voters c | a b d2 voters d b c | a

a is the MAC winner.

Majoritarian Approval Compromise

3 voters a | b c d2 voters b a c | d2 voters c | a b d2 voters d b c | a

a is the MAC winner.

Approval Voting with Runoff

Given R ∈W (A ∪ {0})n, let ρ(R) = {x , y} be the pair ofalternatives which receive the highest approval. AVR picks thepairwise majority winner among the runoff winners.

R ′ is a lifting of x with respect to R provided for all i ∈ N,xRiy implies xR ′i y for all y ∈ AxPiy implies xP ′i y for all y ∈ AxPi0 implies xP ′i0and yPiz iff yP ′i z for all y , z ∈ A− {x} ∪ {0}.

f is monotonic iff x ∈ f (R) implies x ∈ f (R ′) for all R ′ that areliftings of x with respect to R.

Theorem. Approval Voting, Majoritarian Approval Compromise,Preference- Approval Voting and Approval Voting with a runoff areall monotonic.

B = A ∪ {x∗}.

We say that R ∈W (A ∪ {0})n and R ′ ∈W (B ∪ {0})n agreeprovided for all x , y ∈ A, xRiy iff xR ′i y and xRi0 iff xR ′i0.

We call x∗ a spoiler if its presence can change the social choicewithout x∗ being chosen.

x∗ 6∈ f (R ′) 6= f (R) at some R ∈W (A ∪ {0}) that agrees withR ′ ∈W (B ∪ {0})

f is independent if it does not admit a spoiler.

Theorem. Majoritarian Approval Compromise,Preference-Approval Voting and Approval Voting with a runoff failindependence.

B = A ∪ {x∗}.

PHIL 308S: Voting Theory and Fair Division - Lecture 18epacuit/classes/... · PHIL 308S: Voting...

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