Chapter 12 - PowerPoint Presentation

Slide1

Chapter 12Electing the President

Paul MooreSlide2

What math can tell us about elections and strategy behind themSpatial Models (Candidate positions on issues)

2 candidates (

Unimodal, Bimodal)2+ candidates (1/3 separation, 2/3 opportunity)Election ReformApproval VotingElectoral CollegeStrategies to maximize:Popular VotesElectoral Votes

Presentation OutlineSlide3

Elections every 4 years35 years oldNative born citizens

US residents for 14 years

No third termRunning for PresidentSlide4

Democratic and Republican PrimariesCandidates campaign for party nomination

Party nominates candidates

National conventionsGeneral Election2-3 serious contendersElectoral CollegeHow to become PresidentSlide5

Campaign strategiesChoosing states to campaign based on electoral college weight

Effects of reform on these strategies

Approval votingPopular voting (without Electoral College)Candidates getting a leg up in the primaries to help them win their party’s nominationSpatial ModelsMath & Presidential ElectionsSlide6

Model Assumptions:Voters respond to positions on issues

Single overriding issue, candidates must chose side

Voter attitudes represented as “left-right continuum” (very liberal to very conservative)Unimodal vs BimodalVoter distribution represented by curve, giving number of voters with attitudes at different points on L-R continuumSpatial Model: 2 CandidatesSlide7

Unimodal Distribution

Unimodal

– one peak, or modePictured as continuous for simplicityMedian, M – of a voter distribution is the point on the horizontal axis where half the voters have attitudes that lie to the left, and half to right

Number of voters

Voter Positions on L-R Continuum

Candidate A

Candidate B

MSlide8

Unimodal

Distribution

Number of voters

Candidate A

Candidate B

M

Voter Positions on L-R Continuum

Number of voters

Voter Positions on L-R Continuum

Attitudes are a fixed quantity, decisions of voters depend on position of candidates

Candidate positions: Candidate A (yellow line), Candidate B (blue line)

Assume voters vote for candidate with attitudes closest to their own (and that all voters vote)

What happens in models above?

M

Candidate A

Candidate BSlide9

“A” attracts all voters to the left of

M

, while “B” attracts all voters to the right of MAny voters on the horizontal distribution between “A” and “B” (when they are not side by side) are split down the middleUnimodal Distribution

Number of voters

Candidate A

Candidate B

MSlide10

Maximin – the position for a candidate at which there is no other position that can

guarantee

a better outcome (more voters), no matter what the other candidate doesAt what position is a candidate in maximin?Is there more than one maximin position?Taking position at M

guarantees a candidate 50% of the votes, no matter what the other candidate does

Is there any other position that can guarantee a candidate more?

No, there is no other position guaranteeing more votes

Unimodal

DistributionSlide11

Further more M is

stable

, meaning that once a candidate chooses this position, the other candidate has no incentive to choose any other position except M. M is a maximin for both candidates, and they are in equilibriumEquilibrium

– when a pair of positions, once chosen by candidates, does not offer any incentive to either candidate to depart from it unilaterally

Is there another equilibrium position(s)?

Unimodal

DistributionSlide12

Equilibrium Positions…unique?2 cases:

Common point – both candidates take the same position

Distinct positions – one taken by eachCase 1: Common PointIf candidates are in at a common point, to the left of M for example, then one candidate can always do better by moving right but staying on the left of M. The same idea can be applied to common points to the right of M. So common position other than M cannot be equilibrium

Case 2: Distinct Positions

If candidates are in two different positions then one candidate may always do better by moving alongside of the other candidate, gathering more voters. So distinct positions cannot be in equilibrium.

Unimodal

DistributionSlide13

…From this we get the Median Voter theorem

Median Voter Theorem

– in 2 candidate elections with an odd number of voters, M is the unique equilibrium positionBimodal distribution?Unimodal DistributionSlide14

Use same logic as unimodal

distribution to examine unique equilibriums

Again at M, a candidate is guaranteed at least 50% of the votes no matter what the other candidate doesIt is a maximin for both candidates, and an equilibriumAny others?

Bimodal Distribution

M

Number of votersSlide15

Bimodal Distribution

M

Number of voters

2 Cases for possible equilibriums (other than

M

)

Common point – both candidates take the same position

Distinct positions – one taken by each

Case 1: Common Point

If candidates are in at a common point, to the left of

M

for example, then one candidate can always do better by moving right but staying on the left of

M

. The same idea can be applied to common points to the right of M. So common position other than

M

cannot be equilibrium

Case 2: Distinct Positions

If candidates are in two different positions then one candidate may always do better by moving alongside of the other candidate, gathering more voters. So distinct positions cannot be in equilibrium.Slide16

Bimodal Distribution

M

Number of voters

So using the same logic, we can see that

M

is the unique equilibrium for bimodal distributions (Median voter theorem)

Extension:

Median Voter Theorem can be applied to any distribution of electorate’s attitudes

This is because the logic for the proof does not rely on any modal characteristics. Only the idea that to the left and right of the median lies an equal distribution of voter attitudesSlide17

How does M compare with the mean of the distribution

Mean of Voter Distribution

where :n = total number of voters = n1+n2+n

3

+…+

n

k

k = number of different positions

i

that voters take on continuum

n

i

= number of voters at position

i

l

i

= location of position

i

on continuum

Σ is from

i

= 1 to k Weighted average – location of each position is weighted by number of voters at that position

Exercise 1!

Bimodal DistributionSlide18

Exercise 1:

Mean need not coincide with median

MIn exercise, distribution is skewed to the leftArea under the curve is less concentrated to the left of M than to the rightFrom Median Voter Theorem: if the distribution is skewed, then it may not be rational for candidates to choose the mean of the distribution

Even number of voters?

Equilibrium positions?

Bimodal Distribution

Median: 0.6, Mean: 0.56Slide19

Even # of Voters, Discrete DistributionDiscrete Distribution of voters -

where voters are located at only certain positions along the left-right continuum (like in the exercise)

Consider example:n = 26k = 8 different positions over interval [0, 1]

Mean = 0.5

Median = 0.45 (average of 0.4 and 0.5)

Bimodal Distribution

Position,

i

1

2

3

4

5

6

7

8

Location (

l

i

) of position

I

0

0.2

0.3

0.4

0.5

0.7

0.8

0.9

Number of voters (

n

i

) at position

i

2

3

4

4

2

3

7

1Slide20

Mean = 0.5

Median = 0.45 (average of 0.4 and 0.5)

Bimodal Distribution

Position,

i

1

2

3

4

5

6

7

8

Location (

l

i

) of position

I

0

0.2

0.3

0.4

0.5

0.7

0.8

0.9

Number of voters (

n

i

) at position

i

2

3

4

4

2

3

7

1Slide21

Both candidates at M

, they’re in equilibrium

Is this equilibrium position still unique?Any pair of positions between 0.4 and 0.5 is in equilibriumFollowing that, distinct positions 0.4 and 0.5 are also in equilibriumIn generalWith even number of voters and 2 middle voters have different positions then the candidates can choose those 2 positions, or any in between, and be in equilibrium

Bimodal Distribution

Mean = 0.5

Median = 0.45

MSlide22

Primary elections often have more than 2 candidates

Under what conditions is a multicandidate race “attractive”?

Using a similar model, will examine the different positions of an entering 3rd candidateConsider the unimodal 2 candidate race with both at M

Spatial Models: +2 Candidates

Number of voters

Candidate A

(red)

Candidate B

(blue)

M

Is it rational for a 3

rd

candidate to enter the race? (are there any positions offering the candidate a chance at success?)Slide23

Candidate C enters race at position C on graph

C’s area of voters is yellow

A and B have to split the light blue votersC wins plurality of votesSpatial Models: +2 CandidatesNumber of voters

Candidate A

(red)

Candidate B

(blue)

Candidate C

(pink)

M

A/B

C

Upon entry C gains support of voters to the right, and some to left

Blue votes are split between A and B, and C is left with the majority

C can

also enter on the left side of

M

, still winning by the same logic

Can a 4

th

candidate, D, enter the race and win?

Midway between

A/B and CSlide24

Median

M

no longer appealing to candidatesVulnerableHowever, a 3rd candidate C will not necessarily win against both A and B

1/3-Separation Obstacle

If A and B are distinct positions equidistant from

M

of symmetric distribution, and separated from each other by at most 1/3 of total area, then C can take no position that will displace A and B and enable C to win

Spatial Models: +2 Candidates

M

A

BSlide25

2/3 Separation Opportunity

If A and B are distinct positions equidistant from

M on a symmetric distribution and separated by at least 2/3 of the area, then C can defeat both candidates by taking position at MSpatial Models: +2 Candidates

M

A

BSlide26

Spatial Models: +2 Candidates

A

B

M

2/3

1/6

1/6

1/6 each

Not exactly to scaleSlide27

Exercise 2

Spatial Models: +2 Candidates

M

A

B

C

M

A

B

CSlide28

Abolition of the Electoral CollegeMore accurate and reliable ballotsEliminating election irregularities

Most reforms ignore problem with multicandidate elections

Candidate who wins is not always a Condorcet winnerApproval VotingElection ReformSlide29

Approval VotingVoters can vote for as many candidates as they like or find acceptable. Candidate with the most approval votes wins.

2000 Election

Came down to the “toss-up” state of Florida, where Bush won the electoral votes by beating Gore by a little over 300 popular votesAccording to polls, Gore was the second choice of most Nader votersIn an approval voting system Gore would have almost certainly won the election since Nader supporters could have also given a vote of approval to Al GoreElection ReformSlide30

Original purpose was to place the selection of a president in the hands of a body that, while its members would be chosen by the people, would be sufficiently removed from them so that it could make more deliberative choices.

How it works

Each state gets 2 electoral votes (for the 2 senators)Also receives 1 additional electoral vote for each of its representatives in the House of Representatives (number of members for each state based on population)Ranges from 1 in the smallest states to 53 in CaliforniaAltogether there are 538 electoral votes, so candidate needs 270 to winIn 2000, Bush received 271

Electoral CollegeSlide31

Advantages in small states?Technically California voters are about three times more powerful as individuals than those in the smallest states

Though in smallest states with 1 representative and 2 senators (3 electoral votes), the population receives a 200% (2/1) boost from having 2 senatorial electoral votes automatically.

California receives less than 4% (2/53) boost from senatorial electoral votesNow that we know how it works, let’s examine it’s role in the 2000 electionLook at it as a game between 2 major party candidatesDevelop 2 modelsCandidates seek to max their

expected popular vote

Candidates seek to max their

expected electoral vote

Electoral CollegeSlide32

Both models use the assumption that the probability of a voter in a “toss up state” i

votes for the Democratic candidate is:where di and ri represent the proportion of campaign resources spent in state i by the Democratic and Republican candidates, respectively

The probability of a voter voting Republican is 1 – p

i

Expected Popular Vote (EPV)

Is toss up states, the EPV of the Democratic candidate (EPV

D

) is the number of voters,

n

i

, in toss up state

i

, multiplied by the probability, p

i

, that a voter in this toss up state votes Democrat, summed up across all toss up states is

Maximizing Popular Vote

Basically a weighted average, weighted by

probabilitiesSlide33

Candidates allocate resources across toss up states and attempt to do so in an optimal fashionDemocratic candidate seeks strategy

d

i to maximize EPVDMuch like profit maximization among feasibility regions in Chapter 4Here some of the constraints are amount of campaign resources, timeProportional RuleStrategy of Democratic candidate to maximize EPV

D

, given Republican candidate also chooses maximizing strategy is:

Summed up across toss up states

Candidate allocates resources in proportion to the size of each state (

n

i

/N)

Exercise 3

Electoral CollegeSlide34

The optimal spending strategy for each state (from d

i

* and ri*) is ($14M : $21M : $28M) on states 2, 3, and 4 respectivelyMeaning the probability that either candidate will win any state i

is 50%

p

i

= 50% for all

i

toss up states

So at optimal strategy, the EPV is the same for both candidates (D = R)

EPV

D

or

R

= 2[14/(14+14)] + 3[21/(21+21)] + 4[28/(28 + 28)]

Strategy sound familiar?

Candidates are at equilibriumNow Exercise…

what happens when departing from equilibrium?

Maximizing Popular

VoteCollegeSlide35

Exercise 3Calculating

p

for the Republican candidate in 3 statesp1 = 14/14p2 = 21/(21+27)p3

= 28/(28+36)

EPV

R

EPV

R

= 2[14/14] + 3[21/(21+27)] + 4[28/(28 + 36)] = 5.06 votes

or 56% of the 9 votes in those 3 states

Can the Republican candidate do even better?

Change his spending to ($2M : $26M : $35M) to achieve an

EPV

R

= 5.44 votes

*Departure from popular-vote maximizing strategy lowers candidates expected popular vote*

Maximizing Popular Vote

Way to Go!!Slide36

Assume now goal is to max electoral votes

Candidate may think of throwing all resources into 11 largest states

11 largest states have majority of electoral voters (271)However, opponent may simply spend enough in 1 big state to defeat and use rest to spend small amounts in other 39, winning themExpected Electoral Votes (EEV)

Where v

i

= number of electoral votes of toss-up state

P

i

= probability that the Democrat wins

more than

50% of popular votes in state

i

Maxing Electoral VotesSlide37

Expected Electoral Votes (EEV)

Calculating P

i Must determine all probabilities that majority of voters in i

will vote Democratic

3 states

: A, B, C with 2, 3, 4 electoral votes

Again, assume number of pop votes = number of electoral votes

State A: both voters

P

A

= (

p

A

)(

p

A

)) = (

p

A

)

2

State B: 2 of 3 (3 ways) or all

P

B

= 3[(

p

B

)

2

(1 –

p

B

)] + (

p

B

)

3

State C: 3 of 4 (4 ways) or all 4

P

C

= 4[(

p

C

)

3

(1 –

p

C

)] +

(

p

C

)

4

Maxing Electoral VotesSlide38

Strategies to maximize ( 3/2’s Rule )

Candidates should allocate resources in proportion to number of electoral votes of each state (v

i ) multiplied by the square root of its size (ni).

Can also be used to approximate maximizing strategies

Number of electoral voters is roughly proportional to number of voters in each state

Maxing Electoral Votes

So, if the candidates allocate same amount to each toss up, 3/2’s rule says they should spend approximately in proportion to 3/2’s power of the # of electoral votes in order to maximize EEVSlide39

Applying 3/2’s Rule3 States: A, B, C with 9, 16, 25 electoral voters respectively

Candidates want to know how much to use in each state

Use approximationIf all states are toss ups, then 3/2’s rule says candidates should allocate resources accordingly, spending, in total, the approximate value of S (S = D)d

1

*

=

[ 9

3/2

/

S

]*

D

=

9

3/2

= 9 √(9) = 9(3) = 27

d

2

*

=

163/2 = 16 √(16) = 16(4) = 64

d3* = 25

3/2

= 25√(25) = 25(5) = 125

Optimal allocation of resources:

(27 : 64 : 125)

Maxing Electoral VotesSlide40

Mathematics is certainly used, though not obviously, in strategic aspects of campaigning and voting in presidential elections

Is there a better way to elect president?

Many believe in approval votingBelieve it would better enable voters to express their preferencesWhat do you think?Electoral College creates a large state bias

Conclusions & Discussion

HW: Chapter 12

(45, 51)

(7

th

Ed)