MATH & POLITICSIntroduction to Social Choice

OVERVIEW➤ Introductions

➤ Course coverage and expectations

➤ Introduction to social choice theory

➤ Social choice procedures

➤ Desirable properties of social choice theories

➤ Writing prompt

INTRODUCTIONS

Who Are You and Why Are You Here?(This question is not intended to induce an existential crisis.)

EXPECTATIONS➤ Come to class prepared to

exchange ideas with an open mind.

➤ Come to class ready to collaborate.

➤ Each week, we’ll have a short writing assignment:

➤ A writing prompt and maybe some math

➤ Midterm group project

➤ Final reflection

A FEW MATH PRELIMINARIES➤ Set: collection of mathematical objects

➤ Examples: The set of all even numbers; the set of all quadratic functions; the set of all primes.

➤ Notation:

➤ Function: A rule that assigns to each element of a set, X, a unique element of a set, Y.

➤ Notation: .

➤ X can Y can be the same set.

➤ E.g. We can define as follows:

A = {1,2,3,4,5,6,7,8,9}

f : X → Y

f : {1,2,3,4} → {a, b, c, d}

x 1 2 3 4

f(x) c d b a

LET’S VOTE ON SOMETHING➤ To make things simple, let’s form five coalitions and vote on

something with three alternatives.

➤ Rank your choices.

Coalition 1 2 3 4 5

First choice

Second choice

Third choice

➤ Now, what’s the best way to determine the social choice(s)?

GENERAL FRAMEWORK FOR SOCIAL CHOICE➤ Alternatives: The set A will have elements denoted by a, b, c, d, .

. . etc., which we'll call alternatives - e.g. .

➤ People/Voters: P = .

➤ Individual preference list/ballot: Ordered list of alternatives assigned to a voter.

➤ Profile: A sequence of individual preference lists/ballots.

A = {a, b, c, d, e}

{p1, p2, p3, …}

DEFINITION: Given sets X and Y, a procedure is a function on X whose outputs are members of Y. What

makes a social-choice procedure special is that a typical input is a profile and typical output is a single

alternative (or set of alternatives).

MAY'S THEOREM➤ When there are only two alternatives, there is a seemingly natural social choice

procedure that suggests itself: majority rule.

➤ DEFINITION: Majority Rule is the social choice procedure for two alternatives in which an alternative is a winner if it appears at the top of at least half of the voters' ballots.

➤ THEOREM: If the number of voters is odd and each election produces a unique winner, then majority rule is the only social choice procedure for two alternatives that satisfies the following conditions:

➤ Anonymity: If two voters exchange ballots, the outcome of the election is unaffected.

➤ Neutrality: If every voter reverses his/her vote, the election outcome is reversed as well.

➤ Monotonicity: If a voter were to change his/her vote for the loser to a vote for the winner, the outcome would be unchanged.

WHAT IF THERE ARE MORE THAN TWO ALTERNATIVES➤ We’ll start by looking at six social choice procedures.

1. Condorcet’s Method

2. Plurality Voting

3. Borda Count

4. The Hare System

5. Sequential Pairwise Voting with a Fixed Agenda

6. A Dictatorship

➤ Each of these will fit the definition we provided above.

SOCIAL CHOICE PROCEDURE 1: CONDORCET'S METHOD➤ An alternative x is among the winners if for every other

alternative y, at least half of the voters rank x over y.

➤ Example:

1 2 3 4 5c b b a cb a c b aa c a c b

➤ In one-on-one competitions, b defeats a by a score of 3:2 and also defeats c by a score of 3:2.

1 2 3 4 5c b b a cb a c b aa c a c b

CONDORCET EXAMPLE➤ In your groups, work out the Condorcet winner (if one exists)

for the following voting profile.

1 2 3 4 5 6 7

a a a c c b e

b d d b d c c

c b b d b d d

d e e e a a b

e c c a e e a

➤ Remember, you’re trying to find an alternative (a, b, c, d, or e) that defeats all four other alternatives.

SOCIAL CHOICE PROCEDURE 2: PLURALITY VOTING➤ Quite simply, we declare as the winner(s) the alternative(s)

with the largest number of first-place rankings.

➤ Tie’s are allowed.

1 2 3 4 5 6 7

a a a c c b e

b d d b d c c

c b b d b d d

d e e e a a b

e c c a e e a

SOCIAL CHOICE PROCEDURE 3: BORDA COUNT➤ This is a point system based on rank. We award points to

each of the n alternatives according to each preference list starting with n - 1 points for a first-place vote, and 0 for a last-place vote.

➤ So, a gets a score of 4 + 4 + 4 + 0 + 1 + 1 + 0 = 14

➤ Repeat this for the other four alternatives to determine the social choice.

1 2 3 4 5 6 7a a a c c b eb d d b d c cc b b d b d dd e e e a a be c c a e e a

SOCIAL CHOICE PROCEDURE 4: THE HARE SYSTEM➤ Also sometimes called "instant

runoff voting," this procedure arrives at a social choice by deletion of less desirable alternatives. We begin by deleting the alternative that appears on top of the fewest lists. We then repeat this until a single alternative remains.

➤ If an alternative appears at the top of more than half of the ballots, that alternative will be the unique winner.

1 2 3 4 5 6 7a a a c c b eb d d b d c cc b b d b d dd e e e a a be c c a e e a

1 2 3 4 5 6 7a a a c c b eb b b b b c cc e e e a a be c c a e e a

HARE CONT.➤ Repeat the runoff to determine the social choice according to

the Hare System1 2 3 4 5 6 7a a a c c b eb d d b d c cc b b d b d dd e e e a a be c c a e e a

1 2 3 4 5 6 7a a a c c b eb b b b b c cc e e e a a be c c a e e a

1 2 3 4 5 6 7a a a c c c cc c c a a a a

c is the winner

SOCIAL CHOICE PROCEDURE 5: SEQUENTIAL PAIRWISE VOTING WITH A FIXED AGENDA

➤ DEFINITION: An agenda is the order in which a fixed set of things will be discussed or decided upon.

➤ We set an agenda (ordered list of alternatives). Then we pit the first two alternatives in the list against one another in a one-on-one competition. The winner of that competition is pitted against the third choice...so on until we get to the last alternative on the agenda.

1 2 3 4 5 6 7a a a c c b eb d d b d c cc b b d b d dd e e e a a be c c a e e a

Try it here with agenda abcde.➤ a v. b: b wins ➤ b v. c: b wins ➤ b v. d: d wins ➤ d v. e: d wins

SOCIAL CHOICE PROCEDURE 6: A DICTATORSHIP➤ We chose one person to call the dictator. The alternative at

the top of the dictator's ballot is the social choice.

➤ So, for the example we’ve been using, if we take the seventh voter to be our dictator, the social choice is e.

1 2 3 4 5 6 7a a a c c b eb d d b d c cc b b d b d dd e e e a a be c c a e e a

CONCLUSIONS➤ What do we notice?

➤ Condorcet: No winner ➤ Plurality: a ➤ Borda: b ➤ Hare: c ➤ Sequential Pairwise: d ➤ Dictatorship: choosing

voter 7 as our dictator, the winner is e

➤ Each of the six social choice procedures produces a different outcome!

Discuss • Is this surprising? • What’s the best choice? • What are some of the

potentially good and bad features of each voting method?

• What properties might be desirable in a social choice procedure?

FIVE DESIRABLE PROPERTIES OF SOCIAL CHOICE PROCEDURES

Property Definition

Always a Winner (AWA)

Every sequence of individual preference lists produces at least one winner.

Condorcet Winner Condition (CWC)

A social choice procedure is said to satisfy the Condorcet winner criterion (CWC) provided that—if there is a Condorcet winner—then it alone is the social choice.

Pareto Condition If everyone prefers x over y, then y is not a social choice.

MonotonicityA social choice is monotone provided that if is the social choice and a voter changes their preference list by moving up by one spot, then is still the winner.

Independence of Irrelevant Alternatives (IIA)

If the social choice set includes x but not y and one or more voters change their preferences, but no one changes their mind about x and y, then the social choice set should not change to include y.

POSITIVE RESULTS

AAW CWC Pareto Mono IIA

Condorcet Y Y Y Y

Plurality Y Y Y

Borda Y Y Y

Hare Y Y

Seq. Pair Y Y Y

Dictator Y Y Y Y

PROPOSITION: CONDORCET’S METHOD DOES NOT SATISFY THE ALWAYS A WINNER CONDITION

➤ We’ve already seen a counter example, but here’s a simple one known as the Condorcet Voting Paradox:

➤ Alternative a is not a winner because it is defeated by c (2:1)

➤ Alternative b is not a winner because it is defeated by a (2:1)

➤ Alternative c is not a winner because it is defeated by b (2:1)

Voter 1 Voter 2 Voter 3

a b c

b c a

c a b

PROPOSITION: THE PLURALITY PROCEDURE FAILS TO SATISFY THE CONDORCET WINNER CRITERION.

➤ Consider the following sequence of preference groups condensed into blocs.

Voters 1-4 Voters 5-7 Voters 8 & 9

a b c

b c b

c a a

➤ With the plurality procedure, we see that a is the winner.

➤ However, there is a Condorcet winner…

➤ b defeats a by a score of 5:4 and defeats c by a score of 7:2

PROPOSITION: SEQUENTIAL PAIRWISE VOTING WITH A FIXED AGENDA DOES NOT SATISFY THE PARETO CONDITION.

➤ Consider the four alternatives a, b, c, d with fixed agenda abcd. What happens if this is our preference list?

Voter 1 Voter 2 Voter 3

a c b

b a d

d b c

c d a

➤ Every voter prefers b to d, but, with this fixed agenda, a defeats b, a loses to c, and then d defeats c.

PROPOSITION: THE PLURALITY PROCEDURE DOES NOT SATISFY INDEPENDENCE OF IRRELEVANT ALTERNATIVES.

➤ Consider the following sequence of preference lists:

Voter 1 Voter 2 Voter 3 Voter 4a a b cb b c bc c a a

➤ What if Voter 4 changes their mind regarding alternatives c and b?

Voter 1 Voter 2 Voter 3 Voter 4a a b bb b c cc c a a

➤ Now a is tied with b, even though Voter 4 did not change their mind regarding alternatives a and b - I.e. they still prefer b to a.

IMPOSSIBILITY➤ We can show that none of the five social choice procedures

we’ve discussed satisfies all five of the criteria we suggested.

➤ Can we modify any of the aforementioned procedures to satisfy the conditions they don’t meet?

➤ Or is it actually impossible to come up with such a procedure?

➤ THEOREM: There is no social choice procedure for three or more alternatives that satisfies the always-a-winner criterion, independence of irrelevant alternatives, and the Condorcet winner criterion.

PROOF

➤ Assume there is a social choice procedure that satisfies both independence of irrelevant alternatives and the Condorcet winner criterion. We’ll consider the Condorcet voting paradox profile:

Voter 1 Voter 2 Voter 3

a c b

b a c

c b a

➤ We want to show that a procedure satisfying CWC and IIR will fail to satisfy AAW. We’ll do so with three claims.

CLAIM 1: ALTERNATIVE a IS A NON WINNER➤ Because we’re assuming

that IIR is satisfied, if a voter changed their mind about b and c, but not about a and b or a and c, then a should still be a non winner.

➤ Assume that Voter 3 originally had c over b:

Voter 1 Voter 2 Voter 3a c cb a bc b a

➤ In this case, c is a Condorcet winner, making a a non winner.

➤ When Voter 3 changes their mind regarding b and c, a must still be a non winner (b/c IIR).

Voter 1 Voter 2 Voter 3a c bb a cc b a

CLAIM 2: ALTERNATIVE b IS A NON WINNER➤ Can you repeat the logic of Claim 1 to show that b must be a

non winner if both CWC and IIR hold?

Voter 1 Voter 2 Voter 3a a bb c cc b a

➤ Suppose we started with this profile:

➤ So, a is a Condorcet winner and b is therefore a non winner

Voter 1 Voter 2 Voter 3a c bb a cc b a

➤ Suppose Voter 2 changes their mind about a and c:

➤ Then b must still be a non winner.

CLAIM 3: ALTERNATIVE c IS A NON WINNER➤ Suppose we started with

this profile:

➤ So, b is a Condorcet winner and c is therefore a non winner

Voter 1 Voter 2 Voter 3a c bb a cc b a

➤ Suppose Voter 1 changes their mind about a and b:

➤ Then c must still be a non winner.

Voter 1 Voter 2 Voter 3b c ba a cc b a

Having shown that none of the alternatives is a winner we can conclude that any social choice procedure satisfying

CWC and IIA fails to satisfy AAW.

APPROVAL VOTING➤ Recall that a social choice procedure is a function from a set

of individual preference lists (ballots) to a set of alternatives.

➤ That’s not the only way to conduct a vote.

➤ One popular alternative is called approval voting.

➤ Given a set of alternatives, each voter votes for (i.e. approves of) as many alternatives as he/she chooses. The alternatives are not ranked - so it's an unordered ballot.

➤ This method is used by many professional societies and is how the Secretary-General of the UN is elected.

EXAMPLE

Voter 1 Voter 2 Voter 3 Voter 4 Voter 5

a a a b c

c b c

c➤ All we need to do is tally the votes:

➤ a: 3, b: 2, c: 4

➤ So, c is the social choice.

DISCUSSION➤ What are some arguments for approval voting?

➤ Simplicity

➤ Ability to favor two or more alternatives equally

➤ May reduce negative campaigning

➤ Lessens the effect of spoiler candidates

➤ More flexibility than plurality

DISCUSSION➤ We looked at six social choice procedures, five properties that

those procedures can have, and an alternative voting method.

➤ What are your biggest take-aways?

➤ Can you envision any other social choice procedures?

➤ Do you think any of these would be good to use for, say, our presidential elections (assuming the same general framework of the electoral college, which we’ll discuss later in the term)?