We vote, but do we elect whom we really want? Don Saari

Institute for Mathematical Behavioral SciencesUniversity of California, Irvine, CA

dsaari@uci.edu

What math can offer:Beyond ad hoc approaches, goal should be to

find systematic approaches where ideas transfer to other areas.

So much goes wrong in this area! So many mysteries!!

So, what goes wrong with voting indicates what goes wrong elsewhere in the social sciences

in particular economics, business, engineering, etc.

Aggregation rule

Party time!

6

5

4

Milk,Wine,Beer

Beer, Wine, Milk

Wine, Beer, Milk

PluralityMilk-6, Beer-5, Wine-4

Milk Wine

6 9

Milk Beer

6 9

Beer Wine

5 10

PairwiseWine, Beer, Milk

Beer?Runoff election

Rather than voter preferences, an election outcome can

reflect the choice of an election method!

Why? That is the basic

issue addressed

today

Milw. Wash, Boston

Boston, Wash, Milw

Wash, Boston, Milw

Business decisions

J C de Borda, 1770

Plurality: one point for first place,

zero for all othersWeighted: Points to first, second,

third, ....

Borda: Number below, so for three candidates 2, 1,

0Beverage example:Seven different election outcomes!

Problem: Which method is best? i.e., respects

voters wishes

Recently solved by Mathematics

Class ranking

A

C

B

A

B

C

Plot election tallies

Normalize election tally

Positional rulesNormalize weights

(1, s, 0)

(2, 1, 0)1/2 = (1, ½, 0)

(1-s) Plurality + s Antiplurality

P

A

Actual elections

Converse

But, 7 outcomes? Procedure line

Goal: find systematicapproach

Good news and bad, first: How bad can it get?Three

candidates:About 70% of the time, election

ranking can change with weights!More candidates, more severe problems

2 A B C D 2 A D C B 2 C B D A 3 D B C A

A wins

B wins

C wins

D wins

Using different weights,18 different strict (no ties) elections rankings. With ties,about 35 different election outcomes!

For about 85% of examples,ranking changes with procedure

Vote for one (1, 0, 0,0):

Vote for two (1, 1, 0, 0):

Vote for three (1, 1, 1, 0):Borda, (3,2,1,0):OK, so something goes wrong.

But how likely is all of this?

Saari and Tataru, Economic Theory, 1999

In general, for n candidates, can have (n-1)((n-1)!) strict rankings!

Procedure hull

How do we explain all positional differences? Solved in 2000

Bob: 20 votes, Sue: 27 votes Cancel votes in pairs: Sue wins

Me: A B C Lillian: C B A Candidate: A B CMe: 1 0 0Lillian: 0 0 1Total: 1 0 1

Candidate: A B C

Me: 1 1 0Lillian: 0 1 1Total: 1 2 1

Candidate: A B CMe: 2 1 0Lillian: 0 1 2

Total: 2 2 2

Find if ties really are ties!

A tie!!

Bias against B!

Bias for B!

Here we have Z2 orbits

Source of all problems with

positional methods

Only the Borda CountIncluding the beverage example

4 Wine>Beer> Milk, 1 Milk>Wine>Beer

5 Milk>Wine>Beer, 5 Beer>Wine>Milk

Symmetry is the key!

(Systematic rather than ad hoc)

I will come to your group before your next election. You tell me who you want to win. After talking to everyone in your group, I will design a “fair” election rule, which includes all candidates.

Your candidate will win!

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>A

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

Symmetry: Z6 orbit

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

Source of all cycles; voting, statistics, etc.For a price .....

Yet, pairwise elections are cycles!

lost information!!Fred wins by a landslide!!

Everyone prefers C to D to E to F

Consensus?

Reversal + ranking wheel:

Explains all three

candidate problems!

3 A>C>D>B 2 C>B>D>A

6 A>D>C>B 5 C>D>B>A

3 B>C>D>A 2 D>B>C>A

5 B>D>C>A 4 D>C>B>A

X

OUTCOME: A>B>C>D

by 9: 8: 7: 6

X

Now: C>B>A

x

Now: D>C>B

Drop any one or any two candidates and outcome reverses!

Conclusion in general holds forALMOST ANY Weights -- except

Example

2 4

x

3 6

Borda Count!

Borda is in variety; minimizes what can go wrong

Extends to almost all other choices of weights

A mathematician’s take on all of this:OK, some examples are given. Can we find everything, all possible examples, of what

could ever happen?Chaos! Symbolic Dynamics

Theorem: For n >2 candidates, anything you can imagine can happen with the plurality vote!Namely, for each set of candidates, the set of n,

the n sets of n-1, etc., etc., select a transitive ranking.

Namely, there exists a proper algebraic variety of weights so that if weights not in

variety, then anything can happen

There exists a profile whereby for each subset of candidates, the specified ranking is the actual

ranking!

Number of droplets of water in all oceans of the world

Borda Count! Seven candidates

Number plurality listsNumber Borda lists<1050

More than a billion times the

Problem resolved!

Using mathematical symmetryConclusion: The Borda Count is the

unique choice where outcome reflects voters viewsOnly one example of where mathematics plays crucial role in

understanding problems of our society

Thank you!

http://www.math.uci.edu/~dsaari

ArrowInputs: 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 thatvoters have transitive

preferences

Modify!!

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

A>B, B>C implies A>C No voting rule is fair!

Borda 2, 1, 0

And transitivity

Dictator = EX profile restriction

e.g., # of candidates betweenLost info: same as with binary: cannot see info

like higher symmetry or transitivity

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

Decision by consensus:

Everyone prefers C, D, E, to

F

D

E C B

A F

DC

BA

Mathematician’s take

F wins with 2/3 vote!!A landslide victory!!

Why? What characterizes all problems?

A mathematician’s take on all of thisOK, so something goes wrong.

But how likely is all of this?

Saari and Tataru, Economic Theory, 1999Instead of the plurality vote, how about using

other weights to tally ballots?