Electing the President Chapter 17 Mathematical Modeling
What do these events have in common? 1824 John Quincy Adams defeats Andrew Jackson 1876 Rutherford B. Hayes defeats Samuel Tilden 1888 Benjamin Harrison defeats Grover Cleveland 2000 George Bush defeats Al Gore
Phases of the Election 1. State Primaries seeking nomination how to position the candidate to gather momentum in a set of contests 2. Conventions now mostly for show 3. General Election how to win enough Electoral Votes
Mathematics and Elections Clarify the better and worse campaign strategies in each phase (dominate?) Help understand the effect that different election reforms would have on strategies and outcomes
Two-Candidate Elections (Spatial Models) Modeling the Election: Assume voters respond to positions that candidates take on issues Assume that attitudes of voters can be represented along a left-right continuum from very liberal (on the left) to very conservative (on the right) e.g. government intervention in the economy liberal/left= much intervention conservative/right=little intervention
Symmetric Unimodal Distribution Voter distribution represented by a curve that gives the number of voters who have attitudes at each point along a horizontal axis One peak(mode) = Unimodal Median 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 the right
Two-Candidates Attitudes of the voters are considered a fixed quantity in calculations BUT: The decisions of the voters will depend on the positions that candidates take. Voters will vote for the candidate with attitudes closer to their own
Symmetric Unimodal Distribution Unimodal Median symmetric
Two-Candidates There is an incentive for both candidates to move toward M but not overstep it. Taking a position at M, guarantees A at least 50% of the total votes no matter what B does
Two-Candidates A position is maximini for a candidate if there is no other position that can guarantee a better outcome more votes for that candidate whatever position the other candidate adopts.
Two-Candidates A pair of positions is in equilibrium if, once chosen by both candidates, neither candidate has an incentive to depart from it unilaterally (i.e., by himself)
Two-Candidates Median-voter theorem: In a two-candidate election with an odd number of voters, M is the unique equilibrium position.
Mean Distribution l = 1 n k i= 1 nili k=number of different positions i that voters take on the continuum n i =number of voters at position i l i =location of position i on the continuum n = k i= 1 n 1 + n 2 +... nk = total number of voters
Asymmetric Bimodal Distribution in Which the Median and Mean Are Different The median-voter theorem is applicable whatever the distribution of the attitudes
Median and Mean Bimodal two peaks (and not symmetric) Median half of the voters on each side Mean the average voter position
Mean Distribution The mean l of a voter distribution is: l = 1 n k i= 1 nili
Mean Distribution l = 1 n k i= 1 nili 1 2 3 4 5 6 7 8 9 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 3 2 0 2 3 0 6 2 M=.60 l=.56
Spatial Models: Multicandidate Elections
Spatial Models: Multicandidate Elections
Asymmetric Bimodal Distribution in Which the Median and Mean Are Different The median-voter theorem is applicable whatever the distribution of the electorate s attitudes
Median and Mean Bimodal two peaks (and not symmetric) Median half of the voters on each side Mean the average voter position
Median and Mean Bimodal two peaks Not Symmetric Skewed more voters to the right average position is not best
Mean Distribution l = 1 n k i= 1 nili k=number of different positions i that voters take on the continuum n i =number of voters at position i l i =location of position i on the continuum n = k i= 1 n 1 + n 2 +... nk = total number of voters
Mean Distribution The mean l of a voter distribution is: l = 1 n k i= 1 nili
Mean Distribution l = 1 n k i= 1 nili 1 2 3 4 5 6 7 8 9 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 3 2 0 2 3 0 6 2 M=.60 l=.56
Median and Mean = M=.5 l=.5 Position, i 0 1 2 3 4 5 6 7 8 9 Location (i i ) of position I 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Number of voters (n i ) at position I 2 0 3 4 4 2 0 3 7 1
Spatial Models: Multicandidate Elections
1/3 Separation Obstacle 1/3-separation obstacle. If A and B are distinct positions that are equidistant from the median of a symmetric distribution and separated from each other by no more than 1/3 of the area under the curve C can take no position that will displace both A and B and enable C to win.
2/3 Separation Opportunity 2/3 separation opportunity. If A and B are distinct positions that are equidistant from the median of a symmetric unimodal distribution and separated from each other by at least 2/3 of the area under the curve C can defeat both A and B by taking a position at M.
2/3 Separation Opportunity if there is a wide separation between A and B there may be enough room in the middle for C to win.
Optimal entry Optimal entry of two candidates, anticipating a third entrant- Assume A and B are the first candidates to enter an election and anticipate the later entry of C. Assume that the distribution of voters is uniform. Then the optimal positions of A and B are to enter at ¼ and ¾, whereby ¼ of the voters lie to the left of A and ¼ to the right of B. Then C can do no better than win 25% of the vote.
Winnowing the Field Positioning is important Maximize your vote totals Deter new candidates from entering Beating expectations is the name of the game in the early primaries Result Bandwagon Effect induces voters to vote for the presumed winner, independent of his merit
Poll assumption If necessary, voters adjust their sincere voting strategies to differentiate between the top two candidates as revealed in the poll, voting for the one they prefer
Poll 4 3 2 A B C C C A B A B Poll assumption. If necessary, voters adjust their sincere voting strategies to differentiate between the top two candidates as revealed in the poll, voting for the one they prefer Then A=6, B=3, C=0
Condorcet Winner Unsuccessful Given the poll assumption, a Condorcet winner will always lose if he or she is not one of the top two candidates identified by the poll Note even if C was given serious consideration in this race A would still win with a plurality of the votes
Election Reform: Concern after the 2000 Election has centered on making balloting more accurate and reliable and discrimination among different classes of voters No reforms have tried to address the Plurality flaw of the CWC.
Four Candidates The Poll identifies A, B, and C as the leaders D drops out A=5, B=3, C=4 and A Wins! Is there a Condorcet Winner left? 3 3 4 2 A B C D C C A A B A B B D D D C AvsB = A-9 B-3 AvsC = A-5 C-7 BvsC = B-5 C-7
Condorcet Winner Successful Given the poll assumption, a Condorcet winner will always win if he or she is one of the top two candidates identified by the poll.
Approval Voting Approval Voting voters can vote for as many candidates as they like or find acceptable. The candidate with the most approval votes wins the election. In 2000 polls showed that Gore was the second choice of most Nader voters. Gore would have won Florida and the Election.
Voting for your second choice In a three-candidate election under approval voting, it is never rational for a voter to vote only for a second choice. If a voter finds a second choice acceptable, he should also vote for the first choice. In plurality voting a strategic voter would vote for his second choice if he thought his first couldn t win
Dichotomous preferences Dichotomous preferences if the voter divides the set of candidates into two subsets a preferred subset and a not preferred subset and is indifferent among all candidates in each subset.
Dichotomous Preferences Blue = approve Red = disapprove In pairwise contests? 4: (A B), (C D) 3: (C), (A B D) 2: (B C D), (A) AvsB= A-0 B-2 BvsC B=4 C=3 BvsD B=4 D=0 4 3 2 A C B B A C C B D D D A
Approval Voting A Condorcet winner will always be elected under approval voting if all voters have dichotomous preferences and choose their dominant strategies.