2Presentation OutlineWhat 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
3Running for President Elections every 4 years 35 years old Native born citizensUS residents for 14 yearsNo third term
4How to become President Democratic and Republican PrimariesCandidates campaign for party nominationParty nominates candidatesNational conventionsGeneral Election2-3 serious contendersElectoral College
5Math & Presidential Elections Campaign strategiesChoosing states to campaign based on electoral college weightEffects of reform on these strategiesApproval votingPopular voting (without Electoral College)Candidates getting a leg up in the primaries to help them win their party’s nominationSpatial Models
6Spatial Model: 2 Candidates Model Assumptions:Voters respond to positions on issuesSingle overriding issue, candidates must chose sideVoter 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 continuum
7Unimodal Distribution Number of votersCandidate AMCandidate BVoter Positions on L-R ContinuumUnimodal – 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
8Unimodal Distribution Number of votersNumber of votersCandidate ACandidate BCandidate ACandidate BMMVoter Positions on L-R ContinuumVoter Positions on L-R ContinuumAttitudes are a fixed quantity, decisions of voters depend on position of candidatesCandidate 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?
9Unimodal Distribution Number of votersCandidate ACandidate BM“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 middle
10Unimodal Distribution 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 doesIs there any other position that can guarantee a candidate more?No, there is no other position guaranteeing more votes
11Unimodal Distribution 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 unilaterallyIs there another equilibrium position(s)?
12Unimodal Distribution Equilibrium Positions…unique?2 cases:Common point – both candidates take the same positionDistinct 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 equilibriumCase 2: Distinct PositionsIf 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.
13Unimodal Distribution …From this we get the Median Voter theoremMedian Voter Theorem – in 2 candidate elections with an odd number of voters, M is the unique equilibrium positionBimodal distribution?
14Bimodal DistributionNumber of votersMUse same logic as unimodal distribution to examine unique equilibriumsAgain 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?
15Bimodal Distribution 2 Cases for possible equilibriums (other than M) Number of votersM2 Cases for possible equilibriums (other than M)Common point – both candidates take the same positionDistinct 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 equilibriumCase 2: Distinct PositionsIf 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.
16Bimodal Distribution 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 attitudesThis 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 attitudes
17Bimodal DistributionHow does M compare with the mean of the distributionMean of Voter Distributionwhere :n = total number of voters = n1+n2+n3+…+nkk = number of different positions i that voters take on continuumni = number of voters at position ili = location of position i on continuumΣ is from i = 1 to kWeighted average – location of each position is weighted by number of voters at that positionExercise 1!
18Bimodal Distribution Exercise 1: Mean need not coincide with median M In 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 distributionEven number of voters?Equilibrium positions?Median: 0.6, Mean: 0.56
19Bimodal Distribution Even # of Voters, Discrete Distribution Discrete 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 = 0.5Median = 0.45 (average of 0.4 and 0.5)Position, i12345678Location (li) of position I0.20.30.40.184.108.40.206Number of voters (ni) at position i
20Bimodal Distribution Mean = 0.5 Median = 0.45 (average of 0.4 and 0.5) Position, i12345678Location (li) of position I0.20.30.40.220.127.116.11Number of voters (ni) at position i
21Bimodal Distribution M Mean = 0.5 Median = 0.45 Both candidates at M, they’re in equilibriumIs 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
22Spatial Models: +2 Candidates Primary elections often have more than 2 candidatesUnder 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 MIs it rational for a 3rd candidate to enter the race? (are there any positions offering the candidate a chance at success?)Number of votersCandidate A(red)Candidate B(blue)M
23Spatial Models: +2 Candidates Candidate A(red)Midway betweenA/B and CNumber of votersCandidate C enters race at position C on graphC’s area of voters is yellowA and B have to split the light blue votersC wins plurality of votesCandidate B(blue)Candidate C(pink)MCA/BUpon entry C gains support of voters to the right, and some to leftBlue votes are split between A and B, and C is left with the majorityC can also enter on the left side of M, still winning by the same logicCan a 4th candidate, D, enter the race and win?
24Spatial Models: +2 Candidates Median M no longer appealing to candidatesVulnerableHowever, a 3rd candidate C will not necessarily win against both A and B1/3-Separation ObstacleIf 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 winABM
25Spatial Models: +2 Candidates BM2/3 Separation OpportunityIf 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 M
26Spatial Models: +2 Candidates Not exactly to scale1/6 eachMAB1/62/31/6
28Election Reform Abolition of the Electoral College More accurate and reliable ballotsEliminating election irregularitiesMost reforms ignore problem with multicandidate electionsCandidate who wins is not always a Condorcet winnerApproval Voting
29Election Reform Approval Voting 2000 Election Voters can vote for as many candidates as they like or find acceptable. Candidate with the most approval votes wins.2000 ElectionCame 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 Gore
30Electoral CollegeOriginal 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 worksEach 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
31Electoral College Advantages in small states? Technically California voters are about three times more powerful as individuals than those in the smallest statesThough 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 electionLook at it as a game between 2 major party candidatesDevelop 2 modelsCandidates seek to max their expected popular voteCandidates seek to max their expected electoral vote
32Maximizing Popular Vote 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, respectivelyThe probability of a voter voting Republican is 1 – piExpected Popular Vote (EPV)Is toss up states, the EPV of the Democratic candidate (EPVD) is the number of voters, ni, in toss up state i, multiplied by the probability, pi, that a voter in this toss up state votes Democrat, summed up across all toss up states isBasically a weighted average, weighted byprobabilities
33Electoral CollegeCandidates allocate resources across toss up states and attempt to do so in an optimal fashionDemocratic candidate seeks strategy di 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 EPVD, given Republican candidate also chooses maximizing strategy is:Summed up across toss up statesCandidate allocates resources in proportion to the size of each state (ni/N)Exercise 3
34Maximizing Popular VoteCollege The optimal spending strategy for each state (from di* 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%pi = 50% for all i toss up statesSo at optimal strategy, the EPV is the same for both candidates (D = R)EPVD or R = 2[14/(14+14)] + 3[21/(21+21)] + 4[28/( )]Strategy sound familiar?Candidates are at equilibriumNow Exercise…what happens when departing from equilibrium?
35Maximizing Popular Vote Exercise 3Calculating p for the Republican candidate in 3 statesp1 = 14/14p2 = 21/(21+27)p3 = 28/(28+36)EPV REPV R= 2[14/14] + 3[21/(21+27)] + 4[28/( )] = 5.06 votesor 56% of the 9 votes in those 3 statesCan the Republican candidate do even better?Change his spending to ($2M : $26M : $35M) to achieve anEPVR = 5.44 votes*Departure from popular-vote maximizing strategy lowers candidates expected popular vote*Way to Go!!
36Maxing Electoral Votes Assume now goal is to max electoral votesCandidate may think of throwing all resources into 11 largest states11 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 vi = number of electoral votes of toss-up statePi = probability that the Democrat wins more than 50% of popular votes in state i
37Maxing Electoral Votes Expected Electoral Votes (EEV)Calculating PiMust determine all probabilities that majority of voters in i will vote Democratic3 states: A, B, C with 2, 3, 4 electoral votesAgain, assume number of pop votes = number of electoral votesState A: both votersPA = (pA)(pA)) = (pA)2State B: 2 of 3 (3 ways) or allPB = 3[(pB)2(1 – pB)] + (pB )3State C: 3 of 4 (4 ways) or all 4PC = 4[(pC)3(1 – pC)] + (pC)4
38Maxing Electoral Votes Strategies to maximize ( 3/2’s Rule )Candidates should allocate resources in proportion to number of electoral votes of each state (vi ) multiplied by the square root of its size (ni).Can also be used to approximate maximizing strategiesNumber of electoral voters is roughly proportional to number of voters in each stateSo, 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 EEV
39Maxing Electoral Votes Applying 3/2’s Rule3 States: A, B, C with 9, 16, 25 electoral voters respectivelyCandidates want to know how much to use in each stateUse 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)d1* = [ 93/2 / S ]*D= 93/2 = 9 √(9) = 9(3) = 27d2* = 163/2 = 16 √(16) = 16(4) = 64d3* = 253/2 = 25√(25) = 25(5) = 125Optimal allocation of resources:(27 : 64 : 125)
40Conclusions & Discussion Mathematics is certainly used, though not obviously, in strategic aspects of campaigning and voting in presidential electionsIs there a better way to elect president?Many believe in approval votingBelieve it would better enable voters toexpress their preferencesWhat do you think?Electoral College creates a large statebiasHW: Chapter 12(45, 51)(7th Ed)