“Geometry of Departmental Discussions”Donald G. Saari

Institute for Mathematical Behavioral SciencesUniversity 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 complexand 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 symmetryFor a price, I will come to your department ....

10 A>B>C>D>E>F10 B>C>D>E>F>A10 C>D>E>F>A>B

D

E C B

A F

DC

BA

F

Mathematics?

16 2

5 3 4

A

F B

E C

D

Ranking Wheel

A>B>C>D>E>F

65 1

4 2 3

Rotate -60 degrees

B>C>D>E>F>AC>D>E>F>A>B etc.

Symmetry: Z6 orbit

No candidate is favored: each is infirst, second, ... once.

All problems with pairwise comparisons due to Zn orbits

Coordinate direction!Yet, pairwise elections are cycles! 5:1

Pairwise majority voting

1 2 3

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 existMcKelvey: 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 numberof 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 McKelveyTheorem: (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 outcomeq=41, n=6019 on losing side, so need to persuade41-19 = 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.

Plottdiagram

Added stability

BanksAlways

q=6, n = 115 on losing side6-5=1 to change

vote

Proof by singularity

theory

Consequences of my theorem(All in book associated with lectures)

Single peaked conditionsfor majority rule

Essentially a single dimensional issue spaceGeneralization for q rules

Ideas of proofSingularity theory

Algebra: Number of equations, number of unknownsExtend to generalized inverse function theorem

Extend to “first order conditions”

Replacing the coreCore: 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 samelens 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 meetGeneralizes to any number of voters, any number of issues and

any q-rule

Minimizes what it takes torespond 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 practicalway to handle these problems

Most surely there are other, maybemuch 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

ArrowInputs: Voter preferences are transitive

No restrictionsOutput: 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 thatvoters have transitive

preferences

Modify!!

You need to know my {R, B} and {W, B} rankings!

Lost information!! Cannot see full symmetryFor a price, I will come to your department ....

10 A>B>C>D>E>F10 B>C>D>E>F>A10 C>D>E>F>A>B

D

E C B

A F

DC

BA

F

Mathematics?

16 2

5 3 4

A

F B

E C

D

Ranking Wheel

A>B>C>D>E>F

65 1

4 2 3

Rotate -60 degrees

B>C>D>E>F>AC>D>E>F>A>B etc.

Symmetry: Z6 orbit

No candidate is favored: each is infirst, second, ... once. Yet, pairwiseelections are cycles! 5:1

All problems with pairwise comparisons due to Zn 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>F10 B>C>D>E>F>A10 C>D>E>F>A>B

Why??

Everyone prefers C, D, E, to

F

D

E C B

A F

DC

BA

F

F wins with 2/3 vote!!

Consensus?

Election outcomes need not represent what the voters want!