Rock the Vote or Vote The Rock

Rock the Vote or Vote The Rock Tom Edgar Department of Mathematics University of Notre Dame Notre Dame, Indiana October 27, 2008 Graduate Student Seminar

Introduction Basic Counting Extended Counting Introduction November 4, 2008: The Election of a Lifetime Senator John McCain Senator Barack Obama Future

Introduction Basic Counting Extended Counting Introduction November 4, 2008: The Election of a Lifetime Senator John McCain Senator Barack Obama But no one saw the real threat coming: Future

Introduction Basic Counting Extended Counting Introduction November 4, 2008: The Election of a Lifetime Senator John McCain Senator Barack Obama But no one saw the real threat coming: Dwayne Johnson (A.K.A. The Rock) Future

Introduction It s not like it is unheard of:

Schwarzenegger Introduction It s not like it is unheard of:

Schwarzenegger Introduction It s not like it is unheard of: Governor of California

Introduction It s not like it is unheard of: Schwarzenegger Ventura Governor of California

Introduction It s not like it is unheard of: Schwarzenegger Ventura Governor of California Governor of Minnesota

Introduction It s not like it is unheard of: Schwarzenegger Ventura Reagan Governor of California Governor of Minnesota Kevin Bacon Fact: Both starred in the movie The Running Man

Introduction It s not like it is unheard of: Schwarzenegger Ventura Reagan Governor of California Governor of Minnesota President Kevin Bacon Fact: Both starred in the movie The Running Man

Outline Four Basic Methods of Counting Votes: What? There are Four? An Infinite Class of Voting Methods: How Bad Can it Be? What Should We Do About This?

Non-Outline 1. Strategic Voting 2. Serious Impossibility 3. Politics 4. Electoral College

Non-Outline 1. Strategic Voting 2. Serious Impossibility 3. Politics 4. Electoral College 2 3 in both houses or Constitutional Convention Ratification takes 75% of states At least 15 states would not vote for it

Impossibility Kenneth Arrow : 1948 Isolated proper fairness criteria. With more than 3 candidates, no election process will satisfy all the fairness criteria. Except for a Dictatorship.

What is an Election An Election will consist of the following data: 1. A finite set of candidates 2. A finite set of voters 3. Each voter provides a totally-ordered ranking of candidates 3.1 Preference Schedule: List of each voter s ranking (Most of the time we will consider equal voters) 4. An Election Procedure: A method to aggregate the preference schedule into a single transitive ranking (ties allowed)

What is an Election Example Candidates Possible Voter Rankings: {McCain, Obama, The Rock} p 1 p 2 p 3 p 4 p 5 p 6 M M R R O O O R M O R M R O O M M R Preference Schedule for 21 voters: (1, 2, 3, 4, 5, 6) Election Procedure: Tom s Ranking (denoted by *) wins. This is a dictatorship: The Rock Wins!

Four Basic Election Procedures 1. Plurality Current Process in the U.S. for choosing state electors 2. Pairwise Comparisons Think round robin tournament 3. Borda Count Think G.P.A. s in school. 4. Instant Runoff (Plurality with Elimination) A modification on Plurality by successive eliminations

Basic Methods Example (Plurality) Candidates {McCain, Obama, The Rock} Possible Voter Rankings: p 1 p 2 p 3 p 4 p 5 p 6 M M R R O O O R M O R M R O O M M R Preference Schedule for 100 voters: (33, 0, 20, 14, 0, 33) Election Procedure: Count First Place Votes Only

Basic Methods Example (Pairwise Comparisons) Candidates {McCain, Obama, The Rock} Possible Voter Rankings: p 1 p 2 p 3 p 4 p 5 p 6 M M R R O O O R M O R M R O O M M R Preference Schedule for 30 voters: (10, 0, 10, 0, 10, 0) Election Procedure: Run all head-to-head matchups (3 matches) Outcome: M vs. O (20-10) M vs. R (10-20) O vs. R (20-10) So each candidate gets 1 point for his 1 win. We have a tie! Computationally Hard

Basic Methods Example (Borda Count) Candidates {McCain, Obama, The Rock} Possible Voter Rankings: p 1 p 2 p 3 p 4 p 5 p 6 M M R R O O O R M O R M R O O M M R Preference Schedule for 101 voters: (0, 50, 0, 1, 50, 0) Election Procedure: 2 (# first place votes) + 1 (# second place votes)

Basic Methods Example (Instant Runoff) Candidates {McCain, Obama, The Rock} Possible Voter Rankings: p 1 p 2 p 3 p 4 p 5 p 6 M M R R O O O R M O R M R O O M M R Preference Schedule for 100 voters: (0, 32, 33, 0, 20, 15) Election Procedure: Count first place votes only, drop candidate with fewest, reassign dropped votes, and repeat.

Instant Runoff is Bad Possible Voter Rankings: p 1 p 2 p 3 p 4 p 5 p 6 M M R R O O O R M O R M R O O M M R Initial preference schedule: (4, 7, 0, 8, 0, 10) Winner : Obama

Instant Runoff is Bad Possible Voter Rankings: p 1 p 2 p 3 p 4 p 5 p 6 M M R R O O O R M O R M R O O M M R Initial preference schedule: (4, 7, 0, 8, 0, 10) Winner : Obama Poll Projects Obama

Instant Runoff is Bad Possible Voter Rankings: p 1 p 2 p 3 p 4 p 5 p 6 M M R R O O O R M O R M R O O M M R Initial preference schedule: (4, 7, 0, 8, 0, 10) Winner : Obama Poll Projects Obama New Preference Schedule: (0, 7, 0, 8, 0, 14) (only favors Obama)

Major Problem United States Election: John McCain, Barack Obama, The Rock, and Don Brower. Of the 24 ranked ballot types, only 5 appear as below (numbers in millions): Ballot Type Number MORD 14 RODM 10 DROM 8 ODRM 4 RDOM 1 1. Plurality: McCain 2. Pairwise: The Rock 3. Borda: Obama

1700 s: Characters Marquis de Condorcét Jean-Charles de Borda

1700 s: Characters Marquis de Condorcét Jean-Charles de Borda 1900 s-2000 s: Steven Brams Donald Saari

The Old (New) Fight 1700 s Condorcét Borda Pairwise Comparisons Borda Count 2000 s Brams Saari Approval Voting Borda Count Wait: What is Approval Voting?

The Old (New) Fight 1700 s Condorcét Borda Pairwise Comparisons Borda Count 2000 s Brams Saari Approval Voting Borda Count Wait: What is Approval Voting? 1. Voters give each candidate approval or non-approval. 2. Count total number of approvals. 3. Technicality: Not actually an election by our definition.

Mathematics Time 1. Fix an election with n candidates. 2. A preference schedule, p, is a list of n! numbers, i.e. p R n!. 3. Election procedure is then a map ϕ : R n! R n. 4. Saari: Use voting vector to generalize Borda Count. 5. Voting vector gives a linear map. 6. Use convexity to understand possible elections.

Positional Voting Consider an election with N candidates. Definition A voting vector is a vector w = (w 1,..., w N ) R N such that the following hold w 1 w 2 w N 1 w N w 1 > w N = 0 Definition Suppose we are given w a voting vector. Suppose a candidate receives q i i th place votes. The positional voting procedure for w is given by giving each candidate N w i q i i=1 points. The candidate with the most points wins.

Positional Voting Suppose we have n candidates. Example (Borda Count) General: w BC = (n 1, n 2,..., 1, 0) n = 3: w BC = (2, 1, 0) Example (Plurality) w P = (1, 0, 0,..., 0, 0) Example (Anti-plurality) w AP = (1, 1,..., 1, 0) Example (4 Candidates) w = (10, 9, 1, 0)

Positional Voting Suppose we have a voting vector w. Question: Where do we get a map ϕ : R N! R N? Answer: Create the appropriate N N! matrix with columns given by permuting w. Example w BC = (2, 1, 0) then we get the matrix: 2 2 1 0 0 1 1 0 0 1 2 2 0 1 2 2 1 0 Note: we have to fix some order of the permutations when discussing preference schedules.

Linear Algebra Could do the Trick Let s only consider 3 candidate elections. Let w be a voting vector. We get a 3 6 matrix ϕ : R 6 R 3. Normalizing, we can restrict range to 2-simplex. M R O

Linear Algebra Could do the Trick Let s only consider 3 candidate elections. Let w be a voting vector. We get a 3 6 matrix ϕ : R 6 R 3. Normalizing, we can restrict range to 2-simplex. M R O Which region the point lands on decides the final society ranking.

Why Not Use The Geometry Method: 1. For given profile p, write p i in corresponding region. 2. Given voting vector w, normalize so that w = (1, s, 0). 3. Can easily fill in information into triangle as follows: Election: w = (1, s, 0) p = (2, 3, 4, 5, 6, 7) (4+5)+(3+6)s R 15 4 5 12 3 6 2 7 M (3+2)+(4+7)s 9 18 12 15 O (6+7)+(2+5)s

The Picture Says it All Important Points: 9+9s R 15 4 5 12 12 15 3 6 2 7 M 9 18 5+11s 1. Easily see Pairwise Comparisons 2. Plurality (s = 0) is easily read off 3. Anti-plurality (s = 1) is easily read off O 13+7s

The Picture Says it All 9+9s R 15 4 5 12 12 15 3 6 2 7 M 9 18 5+11s Proposition For any voting vector w we have w = t w P + (1 t)w AP O 13+7s

Procedure Line Remark All positional voting outcomes for three candidate election lie on the line between the plurality point and the anti-plurality point. M R O M R O Plurality and Anti-plurality Hulls Procedure Line Theorem For N 3 candidates {c 1,..., c N }, there exist preference schedules so that c j wins when the voters for for j candidates (j = 1,..., N 1) and c N wins with the Borda count. Point: Single data set can lead to each candidate winning

What about Approval Voting? Approval Voting has even stranger consequences: Example Possible Voter Rankings: p 1 p 2 p 3 p 4 p 5 p 6 M M R R O O O R M O R M R O O M M R p = (1, 2, 3, 4, 5, 6) For each candidate, there are two extreme cases: Candidate receives minimal possible approvals (i.e. O gets 11) Candidate receives all possible approvals (i.e. O gets 16) This leads to eight extreme points of approval voting based on a preference schedule.

Approval Hull An approval election result lies inside the convex hull of the extreme points. We call this the approval hull. Example Suppose p = (13, 11, 0, 9, 8, 11). 19 s 9 C 17 0 9 20 32 11 8 32 13 11 A 24 28 B 11 s 24 22 s 19

The Debate Continues. So What Now? 1. Saari has many results showing that Borda count is the best. 1.1 Base point of procedure line. 1.2 Cancels out ties. 1.3 Requires ranking: almost impossible for large elections. 2. Brams believes approval hull doesn t happen practically. 2.1 Approval voting is computationally easy. 2.2 Current voting machines can handle it. 3. Arrow s Theorem 3.1 Strong fairness assumptions make fair counting impossible 3.2 Which assumptions should we weaken? 4. Range Voting? 4.1 Imdb or Amazon. 4.2 Hotornot.com. 4.3 Bayesian regret.

Thanks! For Further Self-Guided Learning: http://www.nd.edu/ tedgar/ret2008/voting.html For Reading Poundstone, William, Gaming the Vote. Hill and Wang, 2008. Saari, Donald, Chaotic Elections! A Mathematician Looks at Voting. The American Mathematical Society, 2001