We vote, but do we elect whom we really want?
Don Saari
Institute for Mathematical Behavioral Sciences
University of California, Irvine, CA
dsaari@uci.edu
So much goes wrong in this area! So many mysteries!!
Aggregation rule
So, what goes wrong with voting indicates what goes
wrong elsewhere in the social sciences
in particular economics, business, engineering, etc.
What math can offer:
Beyond ad hoc approaches, goal should be to find
systematic approaches where ideas transfer to other
areas.
Business decisions
Party time!
Wash, Boston
Milk,Wine,Beer
6 Milw.
5
Boston,
Beer,Wash,
Wine,Milw
Milk
Plurality
Milk-6, Beer-5, Wine-4
Pairwise
Wine, Beer, Milk
Runoff election
Beer?
Wash, Boston,
Milw
4 Wine,
Beer, Milk
Milk
Wine
Milk
Beer
Beer
6
9
6
9
5
Wine
10
Why? That is
the basic issue
addressed
today
Rather than voter preferences, an election outcome
can
J C de Borda, 1770
Class ranking
Plurality: one point for first
place,
zero for all others
Weighted: Points to first, second,
third, ....
Borda: Number below, so for
three candidates 2, 1, 0
Beverage example:Seven different
election outcomes!
Problem: Which method is
best? i.e., respects voters
wishes
Recently solved
by Mathematics
C But, 7 outcomes? Procedure line
(1-s) Plurality + s Antiplurality
Plot election tallies
Actual elections
A
1/2
(2,
1,
0)
=
(1,
½,
0)
B
(1,
s,
0)
C
Converse
Positional rules
Normalize weights
Normalize election tally
A
Goal: find systematic
approach
A
P
B
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
Procedure hull
2ABCD
2CBDA
2 ADCB
3 D B CA
Vote for one (1, 0, 0,0): A wins
Vote for two (1, 1, 0, 0): B wins
Using different weights,
18 different strict (no ties)
elections rankings. With ties,
about 35 different election
outcomes!
Vote for three (1, 1, 1, 0): C wins
For about 85% of examples,
OK, so
something
goes with
wrong.
ranking changes
procedure
D wins
Borda, (3,2,1,0):
But how likely is all of
In general, for n candidates, can have (n-1)((n-1)!) strict
rankings! this?
Saari and Tataru, Economic Theory, 1999
How do we explain all positional differences?
Solved in 2000
Symmetry is
4
Wine>Beer>
Milk,
1
Milk>Wine>Beer
if ties
really
are ties!
(SystematicFind
rather
than
ad hoc)
the key!
5 Milk>Wine>Beer, 5 Beer>Wine>Milk
Bob:
20 votes,
Sue: 27
Here
we have
Z2votes
orbitsCancel votes in pairs: Sue wins
Me: A B C
Lillian: C B A
Candidate: A B C
Me:
1 0 0
Bias against Lillian:
0 0 1
Candidate: A B C
B!
Total:
1 0 1
Me:
1 1 0 Bias for B!
Lillian:
0 1 1
Candidate: A B C
Total:
1 2 1
Me:
2 1 0
Lillian:
0 1 2
Total:
2 2 2
A tie!!
Including the beverage
example
Only the Borda Count
Source of all
problems with
positional methods
Source of all cycles; voting, statistics, etc.
For a price .....
Mathematics?
Ranking Wheel
10 A>B>C>D>E>F Everyone
prefers C to D to E to F
10 B>C>D>E>F>A
10 C>D>E>F>A>B
Symmetry: Z6 orbit
Rotate -60 degrees
A
F
56
61
12
B
I will come to your group before your
next
election.
You
32
Reversal
+
45
D
E
C everyone in
43
tell me who
you want to win. After
talking
to
D
ranking
C
E
your group,
I will
design a “fair” election
rule, which
B
C
D
wheel:
A
B candidates. Your candidate will win!
includes all
A>B>C>D>E>F Explains all
A
F
F
B>C>D>E>F>A
three
No candidate is favored: each is in
C>D>E>F>A>B
candidate
first, second, ... once.
etc.
problems!
Yet, pairwise elections are cycles!
Fred wins by a landslide!!
lost information!! Consensus?
Example
3
x
A>C>D>B
2
C>B>D>A
6
A>D>C>B
x
5
C>D>B>A
3
B>C>D>A
2
X
D>B>C>A
5
B>D>C>A
4
X
D>C>B>A
Now: C>B>A
OUTCOME: A>B>C>D
Now: D>C>B
by
9: 8: 7:
2 3
4 6
6
Drop any one or any two candidates and outcome reverses!
Conclusion in general holds for
ALMOST ANY Weights -- except
Borda Count!
A mathematician’s take on all of this:
OK, some examples are given. Can we find
everything,
all possible
examples,
of what
could
Borda
is in variety;
minimizes
what can
go wrong
ever happen?
Chaos!
Symbolic
Extends
to almost
all otherDynamics
choices of weights
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.
There exists a profile whereby for each subset of
candidates, the specified ranking is the actual ranking!
Namely, there exists a proper algebraic variety of
weights so that if weights not in variety, then
anything can happen
Borda Count!
Seven candidates
50
10 Number Borda lists < Number plurality lists
More than a billion times the
Number of droplets of water in all oceans of the world
Problem resolved!
http://www.math.uci.edu/~dsaari
Using mathematical symmetry
Conclusion: The Borda Count is the
unique choice where outcome reflects voters
Only one example of where mathematics
plays crucial role in
views
understanding problems of our society
Thank you!
A>B, B>C implies A>C
Arrow
No voting rule is fair!
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
Voting rule:Pareto: Everyone has same
voters have transitive
ranking of a pair, then that is the societal ranking
preferences
Binary independence (IIA): The societal ranking of a
Borda
2,
1,
0
pair depends only on the voters’ relative ranking of
pair
Lost info: same as with
binary:And
cannot
see info like higher
transitivity
Modify!!e.g.,
of candidates between
symmetry
or #transitivity
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
Dictator = EX profile restriction
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
D
D
E
C
C
B
B
A
A
F
Everyone prefers
C, D, E, to F
F wins with 2/3 vote!!
A landslide victory!!
Decision by consensus:
Mathematician’s take
Why? What characterizes all problems?
A mathematician’s take on all of this
OK, so something goes wrong.
But how likely is all of
this?
Saari and Tataru, Economic Theory, 1999
Instead of the plurality vote, how about using other
weights to tally ballots?