# “Geometry of Departmental Discussions” Donald G. Saari Institute for Mathematical Behavioral Sciences University of California, Irvine Voting.

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“Geometry of Departmental Discussions” Donald G. Saari Institute for Mathematical Behavioral Sciences University of California, Irvine dsaari@uci.edu Voting is very complex, with lessons that extend: we do not always elect who the voters want! Societal problems are surprisingly complex and annoying, such as when some group wants to “improve” your proposal. Why is it that no matter how hard we try, somebody will propose an improvement!

Lost information!! Cannot see full symmetry For a price, I will come to your department.... 10 A>B>C>D>E>F 10 B>C>D>E>F>A 10 C>D>E>F>A>B D E C B A F D C B A F Mathematics? 1 6 2 5 3 4 A F B E C D Ranking Wheel A>B>C>D>E>F 6 5 1 4 2 3 Rotate -60 degrees B>C>D>E>F>A C>D>E>F>A>B etc. Symmetry: Z 6 orbit No candidate is favored: each is in first, second,... once. All problems with pairwise comparisons due to Z n orbits Coordinate direction! Yet, pairwise elections are cycles! 5:1

Pairwise majority voting 1 23 Core: Point that cannot be beaten by any other point Core is widely used; e.g., median voter theorem In one-dimensional setting, core always exists Two issues or two dimensions? Resembles an attractor from dynamics No matter what you propose, somebody wants to “improve it.”

1 2 3 core does not exist McKelvey: Can start anywhere and end up anywhere Monica Tataru: Holds for q-rules; i.e., where q of the n votes are needed to win Actual examples: MAA, Iraq Salary Hours Tataru has upper and lower bounds on number of steps needed to get from anywhere to anywhere else Stronger rules? No matter what you propose, somebody wants to “improve it.” {1, 3}

Some Consequences: campaigning negative campaigning: changing voters’ perception of opponent 1 2 3 Positive With McKelvey and Tataru, everything extends to any number of voters

When does core exist? Two natural questions If not, what replaces the core? Generically ˆ McKelvey Theorem: (Saari) A core exists generically for a q-rule if there are no more than 2q-n issues. (Actually, more general result with utility functions, but this will suffice for today.) Number of voters who must change their minds to change the outcome q=41, n=60 19 on losing side, so need to persuade 41-19 = 22 voters to change their votes So this core persists up to 22 different issues Saari, Math Monthly, March 2004 Answered question when core exists generically. Plott diagram Added stability Banks Always q=6, n = 11 5 on losing side 6-5=1 to change vote Proof by singularity theory

Consequences of my theorem (All in book associated with lectures) Single peaked conditions for majority rule Essentially a single dimensional issue space Generalization for q rules Ideas of proof Singularity theory Algebra: Number of equations, number of unknowns Extend to generalized inverse function theorem Extend to “first order conditions”

Replacing the core Core: point that cannot be beaten Finesse point: point that minimizes what it takes to avoid being beaten lens width, 2d, is sum of two radii minus distance between ideal points All points on ellipse have same lens width of 2d Define “d-finesse pt” in terms of ellipses Ellipse: sum of distances is fixed Predict what might happen?

d-finesse point is where all three d-ellipses meet Generalizes to any number of voters, any number of issues and any q-rule Minimizes what it takes to respond to any change -- d For minimal winning coalition C, let C(d) be the Pareto Set for C and all d-ellipses for each pair of ideal points Finesse point is a point in all C(d) sets, and d is the smallest value for which this is true. Practical politics: incumbent advantage

The finesse point provides one practical way to handle these problems Most surely there are other, maybe much better approaches And, they are left for you to discover But, the real message is the centrality of mathematics to understand crucial issues from society

Arrow Inputs: Voter preferences are transitive No restrictions Output: Societal ranking is transitive Voting rule: Pareto: Everyone has same ranking of a pair, then that is the societal ranking Binary independence (IIA): The societal ranking of a pair depends only on the voters’ relative ranking of pair Conclusion: With three or more alternatives, rule is a dictatorship With Red wine, White wine, Beer, I prefer R>W. Are my preferences transitive? Cannot tell; need more information Determining societal ranking cannot use info that voters have transitive preferences Modify!! You need to know my {R, B} and {W, B} rankings!

Lost information!! Cannot see full symmetry For a price, I will come to your department.... 10 A>B>C>D>E>F 10 B>C>D>E>F>A 10 C>D>E>F>A>B D E C B A F D C B A F Mathematics? 1 6 2 5 3 4 A F B E C D Ranking Wheel A>B>C>D>E>F 6 5 1 4 2 3 Rotate -60 degrees B>C>D>E>F>A C>D>E>F>A>B etc. Symmetry: Z 6 orbit No candidate is favored: each is in first, second,... once. Yet, pairwise elections are cycles! 5:1 All problems with pairwise comparisons due to Z n orbits

For a price... I will come to your organization for your next election. You tell me who you want to win. I will talk with everyone, and then design a “fair” election procedure. Your candidate will win. 10 A>B>C>D>E>F 10 B>C>D>E>F>A 10 C>D>E>F>A>B Why?? Everyone prefers C, D, E, to F D E C B A F D C B A F F wins with 2/3 vote!! Consensus? Election outcomes need not represent what the voters want!

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