Equilateral Triangle & Ranking Regions

1. More Voting Theory: Possible Election Outcomes in Positional and Approval Voting & Classifying ElectionManipulations

2. Equilateral Triangle & Ranking Regions • Refer to the handout for the different ranking regions in an equilateral triangle for a three-candidate election – this was discussed on Wednesday. • A voter profile is a vector-type quantity p = (p1, p2, p3, p4, p5, p6) where each pi represents the number of voters with the i-th type ranking. • For example, p2 is the number of voters with the Type 2 ranking of “A > C > B.”

3. Weightings of Positional Elections • Remember the Borda Count from Wednesday’s presentation. This is a type of election procedure that falls under the general category of positional elections. • Positional elections assign points to each candidate based on their rankings. For example, we can assign 3 points to the top-ranked candidate, 2 points to the second-ranked, and 1 point to the last-ranked. • We can begin to normalize these positional weightings by always assigning 0 points to the third-ranked candidate. In the example above, this would be the same as using the weighting (2, 1, 0). Note that this does not change the final outcome.

4. To fully normalize these positional rankings and weightings, we can make another vector-type quantity ws = (1, s, 0) where s corresponds to the positional method used. In the (2, 1, 0) procedure, we would use s = 1/2; In the (6, 5, 0) procedure, we would use s = 5/6. • Note that the plurality tally is just a special case of this ws weighting with s = 0. • Another type of tally is known as anti-plurality, which uses s = 1. This is the same thing as “voting against” a candidate.

5. Calculating Outcomes Using ws • We can very quickly and easily calculate election outcomes for different procedures using the value of s in the ws weighting vector. • A ws weighting only takes into account the profiles where a particular candidate is first or second ranked. • So, to calculate a particular candidate’s tally, use the general formula a + bs Where a is the number of voters who rank a candidate first, and b is the number of voters who rank that candidate second.

6. Classifying All Outcomes in Positional Three Candidate Elections • A normalized tallyqs=(q1, q2, q3) can also be calculated for any positional election procedure with three candidates so that q1+q2+q3 = 1. Just divide each candidate’s tally by the total number of voters. • This qs is a point in the plane x+y+z =1 bounded by the positive axes of R3, but we can also think of this as a point in the equilateral triangle representation discussed on Wednesday. We simply suppress the axes and “flatten” the plane to get the equilateral triangle.

7. It is possible to classify all possible positional election outcomes for a particular profile p by constructing what is known as the procedure line. • After a bit of algebra, it can be shown that a tally of votes qs for a particular positional procedure s always obeys the following equation: qs = (1-2t)q0 + 2tq1 where q0 is the plurality tally and q1 is the antiplurality tallyfor the profilep, and t = s/(1+s). This equation is the procedure line of a particular profile p.

8. Geometric Intuition of Procedure Line • The procedure line can be thought of as the line in the equilateral triangle representation of voter preferences for a particular profile p that stretches from q0 to q1. • Any ranking regions that this line crosses are possible election outcomes! • Thus, for varying procedures, we can get very different election outcomes. For instance, it is entirely possible that in some voter profiles, the commonly-used Borda Count of (2, 1, 0) will lead to a different winner than simple plurality vote.

9. Example: 1992 Election • An economic researcher (Tabarrok) wrote a paper pertaining to the possible outcomes in the 1992 Clinton-Bush-Perot election. • Using polling data, he calculated the profile p = (.2085, .2210, .135, .0615, .1635, .2105) p1 is C > B > P, p2 is C > P > B, p3 is P > C > B, p4 is P > B > C, p5 is B > P > C, p6 is B > C > P. • He then constructed the procedure line for the 1992 election using the points q0 and q1 and found …

10. Procedure Line for the 1992 Election The procedure line is completely within the “Clinton > Bush > Perot” region!

11. Interesting Paradoxes • Recall procedure line: qs = (1-2t)q0 + 2tq1 • Suppose q0 = q1, then plurality and anti-plurality tallies are the same. In this case, the procedure line is just a point, meaning all procedures give the same result. • Suppose the procedure line passes through the intersection of the three perpendicular bisectors in the equilateral triangle. Then some procedure allows for an exact tie of A ≈ B ≈ C, and the only other two possible outcomes have opposite election rankings.

12. Geometric Classification of Approval Voting Outcomes • Recall that in Approval Voting, you can cast a vote for each candidate of whom you “approve.” Note that casting an approval for each candidate would be like not voting. • Using similar geometric ideas to the construction of the procedure line, we can classify all possible outcomes for approval voting. • In a three-candidate election, we first look at the highest possible tally for each candidate (s = 1, antiplurality tally) and the lowest possible tally for each candidate (s = 0, plurality tally).

13. This makes sense because nobody would reasonably choose to vote for all three candidates in an AV scenario with three candidates. • Now, look at the “extreme points” based on combinations of these plurality and antiplurality tallies. There will be up to 8 possible choices for these extreme points. • Normalizing these extreme points, we can again plot them in the equilateral triangle. • The set of all possible AV outcomes lies within the convex hull defined by these points. • This gives us a much wider range of possible outcomes than with positional procedures!

14. Back to the1992 Election • “For any profile, any undesired or troubling election outcome that results from the use of some positional method must be an admissible AV outcome. Moreover, AV allows many other election outcomes that never can occur with a positional method” (Saari, Chaotic Elections 56-57). • The same researcher (Tabarrok) who calculated the procedure line for the 1992 election also figured out all possible AV outcomes. • AV outcomes can vary widely, but just how widely?

15. Possible AV Outcomes in 1992 Yes, both Bush and Perot could have been elected using AV! While highly unlikely, it is still theoretically possible.

16. Using Multivariable Calculus to Classify Manipulation As this political cartoon alludes to, close elections can be altered by a small number of voters who choose to vote strategically. Ideas from multivariable calculus can help us to classify the effects of this type of strategic behavior.

17. Altering Voter Profiles • Suppose a certain three-candidate election has a given profile p, but some of the voters choose to alter their preferences for strategic means. • For example, if a voter chooses to change from a Type II voter to a Type IV voter, the change on the profile can be given by the vector quantity v = (0, -1, 0, +1, 0, 0) • The resulting altered profile will then be p + v.

18. Effect on Election Outcomes • Define f (p) to be election procedure which determines the winner or election ranking. This is a function mapping profiles to candidates/rankings. • The change that a particular voter (or voters) can effect on a election by switching types can be given by the expression f (p+v) – f (p) • Suppose f (p+v) elects Gore, and f (p) elects Bush. But what the heck does {Gore} – {Bush} mean?

19. Making Sense of {Gore} – {Bush} • Notice that this equation resembles the definition of the gradient and directional derivative in that f (p+v) – f (p) ≈ gradf (p) • v • If the gradient gradf existed, it would define a vector orthogonal to a level set of f (e.g. f = {Bush} or f = {Gore} ) in the direction of greatest change. • The direction of greatest change is given by a normal vector N between the boundary of two f level sets (e.g. those profiles which elect Bush and those profiles which elect Gore).

20. Determining the Effect of v • Note that the boundary between two level sets of f (e.g. {Gore} and {Bush}) is essentially where the two candidates are tied. • Suppose N points into the set that elects Gore. The effect of v can be loosely classified as follows: N • v > 0 if the change helps Gore N • v < 0 if the change helps Bush N • v = 0 if the change is neutral.

21. A Particularly Manipulative Scenario • Suppose a selection committee of 30 members is split evenly (10 each) among the following profiles for a new candidate: A > B > C > D > E > F > G > H B > C > D > E > F > G > H > A C > D > E > F > G > H > A > B • Here’s a challenge: can we find a procedure to pick H? Picking a candidate like C would be easy – just use the Borda Count. But what about H?

22. How to Elect H …(Moral Disclaimer: Don’t Try This at Home!) • Suppose the particularly manipulative and sly chair of this committee has decided on a procedure whereby candidates are eliminated iteratively. He or she tries to direct the order of comparisons so that … G against F => F moves on, G out of the running F against E => E moves on, F out E against D => D moves on, E out D against C => C moves on, D out C against B => B moves on, C out B against A => A moves on, B out Finally, A against H => H WINS … huh?

23. “Well, I Never …” • Saari writes, “The particularly smug reader may feel confident that his or her organization is immune to this strategic behavior, if only because their decisions are made in an open manner with full discussions and with consensus. Think again, this bothersome phenomenon can fully arise even during a friendly discussion” (Chaotic Elections, 101).

24. Conclusion • Basically, Wednesday’s and Today’s presentations carry the message of “voter beware.” • Especially in close elections, a small number of votes can make all the difference. Vote wisely!