Transcript More Voting Theory: Classifying Election Outcomes

More Voting Theory:
Possible Election
Outcomes in Positional
and Approval Voting
&
Classifying Election
Manipulations
Equilateral Triangle & Ranking Regions
• Refer to the handout for the different ranking regions
in an equilateral triangle for a three-candidate election
– this was discussed on Wednesday.
• A voter profile is a vector-type quantity
p = (p1, p2, p3, p4, p5, p6)
where each pi represents the number of voters with
the i-th type ranking.
• For example, p2 is the number of voters with the Type
2 ranking of “A > C > B.”
Weightings of Positional Elections
• Remember the Borda Count from Wednesday’s
presentation. This is a type of election procedure that
falls under the general category of positional
elections.
• Positional elections assign points to each candidate
based on their rankings. For example, we can assign
3 points to the top-ranked candidate, 2 points to the
second-ranked, and 1 point to the last-ranked.
• We can begin to normalize these positional
weightings by always assigning 0 points to the thirdranked candidate. In the example above, this would
be the same as using the weighting (2, 1, 0). Note
that this does not change the final outcome.
• To fully normalize these positional rankings and
weightings, we can make another vector-type quantity
ws = (1, s, 0)
where s corresponds to the positional method used.
In the (2, 1, 0) procedure, we would use s = 1/2;
In the (6, 5, 0) procedure, we would use s = 5/6.
• Note that the plurality tally is just a special case of
this ws weighting with s = 0.
• Another type of tally is known as anti-plurality,
which uses s = 1. This is the same thing as “voting
against” a candidate.
Calculating Outcomes Using ws
• We can very quickly and easily calculate election
outcomes for different procedures using the value of s
in the ws weighting vector.
• A ws weighting only takes into account the profiles
where a particular candidate is first or second ranked.
• So, to calculate a particular candidate’s tally, use the
general formula
a + bs
Where a is the number of voters who rank a
candidate first, and b is the number of voters who
rank that candidate second.
Classifying All Outcomes in Positional
Three Candidate Elections
• A normalized tally qs=(q1, q2, q3) can also be
calculated for any positional election procedure with
three candidates so that q1+q2+q3 = 1. Just divide
each candidate’s tally by the total number of voters.
• This qs is a point in the plane x+y+z =1 bounded by
the positive axes of R3, but we can also think of this
as a point in the equilateral triangle representation
discussed on Wednesday. We simply suppress the
axes and “flatten” the plane to get the equilateral
triangle.
• It is possible to classify all possible positional
election outcomes for a particular profile p by
constructing what is known as the procedure line.
• After a bit of algebra, it can be shown that a tally of
votes qs for a particular positional procedure s always
obeys the following equation:
qs = (1-2t)q0 + 2tq1
where q0 is the plurality tally and q1 is the
antiplurality tally for the profile p, and t = s/(1+s).
This equation is the procedure line of a particular
profile p.
Geometric Intuition of Procedure Line
• The procedure line can be thought of as the line in the
equilateral triangle representation of voter preferences
for a particular profile p that stretches from q0 to q1.
• Any ranking regions that this line crosses are possible
election outcomes!
• Thus, for varying procedures, we can get very different
election outcomes. For instance, it is entirely possible
that in some voter profiles, the commonly-used Borda
Count of (2, 1, 0) will lead to a different winner than
simple plurality vote.
Example: 1992 Election
• An economic researcher (Tabarrok) wrote a paper
pertaining to the possible outcomes in the 1992
Clinton-Bush-Perot election.
• Using polling data, he calculated the profile
p = (.2085, .2210, .135, .0615, .1635, .2105)
p1 is C > B > P, p2 is C > P > B, p3 is P > C > B,
p4 is P > B > C, p5 is B > P > C, p6 is B > C > P.
• He then constructed the procedure line for the 1992
election using the points q0 and q1 and found …
Procedure Line for the 1992 Election
The procedure line is
completely within the
“Clinton > Bush > Perot”
region!
Interesting Paradoxes
• Recall procedure line: qs = (1-2t)q0 + 2tq1
• Suppose q0 = q1, then plurality and anti-plurality
tallies are the same. In this case, the procedure line is
just a point, meaning all procedures give the same
result.
• Suppose the procedure line passes through the
intersection of the three perpendicular bisectors in the
equilateral triangle. Then some procedure allows for
an exact tie of A ≈ B ≈ C, and the only other two
possible outcomes have opposite election rankings.
Geometric Classification of Approval
Voting Outcomes
• Recall that in Approval Voting, you can cast a vote
for each candidate of whom you “approve.” Note
that casting an approval for each candidate would be
like not voting.
• Using similar geometric ideas to the construction of
the procedure line, we can classify all possible
outcomes for approval voting.
• In a three-candidate election, we first look at the
highest possible tally for each candidate (s = 1,
antiplurality tally) and the lowest possible tally for
each candidate (s = 0, plurality tally).
• This makes sense because nobody would reasonably
choose to vote for all three candidates in an AV
scenario with three candidates.
• Now, look at the “extreme points” based on
combinations of these plurality and antiplurality
tallies. There will be up to 8 possible choices for
these extreme points.
• Normalizing these extreme points, we can again plot
them in the equilateral triangle.
• The set of all possible AV outcomes lies within the
convex hull defined by these points.
• This gives us a much wider range of possible
outcomes than with positional procedures!
Back to the 1992 Election
• “For any profile, any undesired or troubling election
outcome that results from the use of some positional
method must be an admissible AV outcome.
Moreover, AV allows many other election outcomes
that never can occur with a positional method” (Saari,
Chaotic Elections 56-57).
• The same researcher (Tabarrok) who calculated the
procedure line for the 1992 election also figured out
all possible AV outcomes.
• AV outcomes can vary widely, but just how widely?
Possible AV Outcomes in 1992
Yes, both Bush and Perot
could have been elected using
AV! While highly unlikely, it
is still theoretically possible.
Using Multivariable Calculus to
Classify Manipulation
As this political cartoon alludes
to, close elections can be
altered by a small number of
voters who choose to vote
strategically. Ideas from
multivariable calculus can help
us to classify the effects of this
type of strategic behavior.
Altering Voter Profiles
• Suppose a certain three-candidate election has a given
profile p, but some of the voters choose to alter their
preferences for strategic means.
• For example, if a voter chooses to change from a
Type II voter to a Type IV voter, the change on the
profile can be given by the vector quantity
v = (0, -1, 0, +1, 0, 0)
• The resulting altered profile will then be p + v.
Effect on Election Outcomes
• Define f (p) to be election procedure which
determines the winner or election ranking. This is a
function mapping profiles to candidates/rankings.
• The change that a particular voter (or voters) can
effect on a election by switching types can be given
by the expression
f (p+v) – f (p)
• Suppose f (p+v) elects Gore, and f (p) elects Bush.
But what the heck does {Gore} – {Bush} mean?
Making Sense of {Gore} – {Bush}
• Notice that this equation resembles the definition of
the gradient and directional derivative in that
f (p+v) – f (p) ≈ grad f (p) • v
• If the gradient grad f existed, it would define a vector
orthogonal to a level set of f (e.g. f = {Bush} or f =
{Gore} ) in the direction of greatest change.
• The direction of greatest change is given by a normal
vector N between the boundary of two f level sets
(e.g. those profiles which elect Bush and those
profiles which elect Gore).
Determining the Effect of v
• Note that the boundary between two level sets of f
(e.g. {Gore} and {Bush}) is essentially where the two
candidates are tied.
• Suppose N points into the set that elects Gore. The
effect of v can be loosely classified as follows:
N • v > 0 if the change helps Gore
N • v < 0 if the change helps Bush
N • v = 0 if the change is neutral.
A Particularly Manipulative Scenario
• Suppose a selection committee of 30 members is split
evenly (10 each) among the following profiles for a
new candidate:
A>B>C>D>E>F>G> H
B>C>D>E>F>G> H>A
C>D>E>F >G> H>A>B
• Here’s a challenge: can we find a procedure to pick
H? Picking a candidate like C would be easy – just
use the Borda Count. But what about H?
How to Elect H …
(Moral Disclaimer: Don’t Try This at Home!)
• Suppose the particularly manipulative and sly chair of
this committee has decided on a procedure whereby
candidates are eliminated iteratively. He or she tries
to direct the order of comparisons so that …
G against F => F moves on, G out of the running
F against E => E moves on, F out
E against D => D moves on, E out
D against C => C moves on, D out
C against B => B moves on, C out
B against A => A moves on, B out
Finally, A against H => H WINS … huh?
“Well, I Never …”
• Saari writes, “The particularly
smug reader may feel
confident that his or her
organization is immune to this
strategic behavior, if only
because their decisions are
made in an open manner with
full discussions and with
consensus. Think again, this
bothersome phenomenon can
fully arise even during a
friendly discussion” (Chaotic
Elections, 101).
Conclusion
• Basically, Wednesday’s
and Today’s
presentations carry the
message of “voter
beware.”
• Especially in close
elections, a small
number of votes can
make all the difference.
Vote wisely!