“Geometry of Departmental Discussions”

1. Why is it that no matter how hard we try, somebody will propose an improvement! “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.

2. Lost information!! Cannot see full symmetry Allproblems with pairwise comparisons due to Zn orbits For a price, I will come to your department .... Mathematics? Ranking Wheel 10 A>B>C>D>E>F 10 B>C>D>E>F>A 10 C>D>E>F>A>B A F B E C D 1 6 2 5 3 4 6 5 1 4 2 3 Rotate -60 degrees Symmetry: Z6 orbit D E C B A F D C B A A>B>C>D>E>F F B>C>D>E>F>A No candidate is favored: each is in first, second, ... once. C>D>E>F>A>B etc. Yet, pairwise elections are cycles! 5:1 Coordinate direction!

3. Pairwise majority voting 2 3 1 Core: Point that cannot be beaten by any other point Core is widely used; e.g., median voter theorem Resembles an attractor from dynamics In one-dimensional setting, core always exists Two issues or two dimensions? No matter what you propose, somebody wants to “improve it.”

4. {1, 3} Hours 3 2 1 Salary core does not exist McKelvey: Can start anywhere and end up anywhere Stronger rules? Actual examples: MAA, Iraq No matter what you propose, somebody wants to “improve it.” Monica Tataru: Holds for q-rules; i.e., where q of the n votes are needed to win Tataru has upper and lower bounds on number of steps needed to get from anywhere to anywhere else

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

6. Two natural questions Generically ˆ When does core exist? If not, what replaces the core? Always McKelvey Banks Plott diagram 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.) q=6, n = 11 5 on losing side 6-5=1 to change vote Number of voters who must change their minds to change the outcome Saari, Math Monthly, March 2004 q=41, n=60 19 on losing side, so need to persuade 41-19 = 22 voters to change their votes Proof by singularity theory So this core persists up to 22 different issues Added stability Answered question when core exists generically.

7. 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”

8. Replacing the core Predict what might happen? 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 Ellipse: sum of distances is fixed Define “d-finesse pt” in terms of ellipses

9. Minimizes what it takes to respond to any change -- d Practical politics: incumbent advantage d-finesse point is where all three d-ellipses meet Generalizes to any number of voters, any number of issues and any q-rule 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.

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

11. Arrow Inputs: Voter preferences are transitive No restrictions Conclusion: With three or more alternatives, rule is a dictatorship Output: Societal ranking is transitive cannot use info that voters have transitive preferences 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 Modify!! With Red wine, White wine, Beer, I prefer R>W. Are my preferences transitive? Cannot tell; need more information You need to know my {R, B} and {W, B} rankings! Determining societal ranking

12. Lost information!! Cannot see full symmetry Allproblems with pairwise comparisons due to Zn orbits For a price, I will come to your department .... Mathematics? Ranking Wheel 10 A>B>C>D>E>F 10 B>C>D>E>F>A 10 C>D>E>F>A>B A F B E C D 1 6 2 5 3 4 6 5 1 4 2 3 Rotate -60 degrees Symmetry: Z6 orbit D E C B A F D C B A A>B>C>D>E>F F B>C>D>E>F>A No candidate is favored: each is in first, second, ... once. Yet, pairwise elections are cycles! 5:1 C>D>E>F>A>B etc.

13. 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. D E C B A F 10 A>B>C>D>E>F 10 B>C>D>E>F>A 10 C>D>E>F>A>B D C B A F Everyone prefers C, D, E, to F F wins with 2/3 vote!! Election outcomes need not represent what the voters want! Why?? Consensus?