we vote, but do we elect whom we really want? don saari institute for mathematical behavioral...
Post on 21-Dec-2015
221 Views
Preview:
TRANSCRIPT
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?
top related