[EM] Modeling elections

[Blake Cretney]

  Adam Tarr wrote:

>>> Here's my problem with your model: you assume that a voter can 
>>> identify where they, or a candidate, lies on a "rightness" scale.  
>>> In reality, the voters only know where they and the candidates lie 
>>> on the issue space.  So in order to model this idea accurately, you 
>>> need to assign a "rightness" function that maps any point on the 
>>> issue space to the interval [0,1].  (or (-infinity, 1] as you have 
>>> it modeled)
>>
>>
>> Condorcet's method is often accused of favouring the middle 
>> candidate. An example that assumed truth would lie in the middle 
>> could be accused of making too many centrist assumptions in favour of 
>> Condorcet.  My first model takes the opposite position, and puts 
>> truth at the far extreme.
>
>
> I don't mean to imply that the truth is necessarily in the middle, 
> only that it is not necessarily on the extreme.  Consider the fact 
> that the issue space is not really one-dimensional, but has dozens of 
> dimensions.  If the "correct" position is on the extreme in the 1-D 
> approximation, this implies that the "correct" position is on an 
> extreme in every dimension of the issue space.  Maybe that seems 
> realistic to you, but it does not seem realistic to me.  I'd imagine 
> that the correct position is on the extreme on some issues, but 
> somewhat moderate on others.

What I'm saying is that I intentionally developed models that were at 
possible extremes.  If I think a particular method does best when the 
winner is in the middle or at one of the two extremes, then I am 
building a case that it is in fact best for all 1-d scenarios.

>> Plurality, Borda, Approval, and Random Candidate are all strongly 
>> affected by candidate distribution.  Plurality tends to favour a 
>> region that isn't represented by as many candidates.  The others tend 
>> to favour a region that has more candidates representing it, at least 
>> in the model I used.
>
> This is consistent with our general perceptions of these election 
> methods; I'd take this as a sign that your model is not total junk.  
> Well actually, I'm not sure approval would really have this effect, 
> but given your rough model of approval strategy it's not too surprising.

My strategy is the one that is generally understood to be best in the 
absence of information about who is likely to win.  I would argue that 
it is roughly equivalent  to sincere voting, if the term has any meaning 
for Approval.

There seem to be two major threads in approval thought.  Some people 
like Approval because they expect that voters will vote more or less for 
the candidates that they think are above average.  These people reason 
that this will result in higher "utility" outcomes and avoid "bad" 
Condorcet winners.  The other group believes that voters will be highly 
strategic and in fact bring about Condorcet winners, which is what they 
want.  It seems that because we don't really know how people will vote 
in Approval, people are free to imagine it will give whatever outcomes 
they want.

Anyway, I created another simulation with quite different assumptions. 
 It works as follows.  Voters and candidates have opinions on 31 
true/false questions.  1 is set as the right answer.  Voters and 
candidates are assumed to vary in their tendency toward right answers. 
 So, each voter and candidate is assigned a number between 0 and 1, 
equal to that person's chance of being right on each issue.  These 
probabilities come from a normal distribution, but the distribution for 
voters has median >.5, so that people are to some degree drawn toward 
better answers.  Once a candidate or voter is assigned its individual 
correctness probability, it is assigned an answer on the 31 questions 
based on this probability.

The distance between a voter's opinion and that of a candidate is equal 
to the number of questions on which they disagree.  Rank ballots have 
voters voting for the nearer candidates first.  In approval, voters vote 
for candidates that are closer than the average for that voter.

The issue space is in effect 31-dimensional, although there are only 2 
points in each issue space.  There isn't always a CW, so I use various 
Condorcet completion methods.  Here are some results, using 
normalvariate(.6, .4) for both voters and candidates.  That is, a normal 
curve with mean of .6 and s.d. of .4.  10,000 iterations.  Only the 
first two digits are likely significant.  The CW's shows the fraction of 
results that had at least one Condorcet winner.

CWs 0.9392
plurality 0.714225806452
approval 0.800816129032
borda 0.899893548387
random 0.561138709677
schulze 0.908383870968
rp 0.908709677419
minmax 0.908051612903

Next, I changed the candidate distribution to normalvariate(.4,.4)

CWs 0.9478
plurality 0.753303225806
approval 0.707303225807
borda 0.832167741936
random 0.418335483871
schulze 0.853212903226
rp 0.853061290323
minmax 0.853238709677

Next, I used a candidate distribution from an even, random distribution 
between 0 and 1.

CWs 0.9406
plurality 0.806016129032
approval 0.768338709677
borda 0.874848387097
random 0.496990322581
schulze 0.889070967742
rp 0.888941935484
minmax 0.889370967742

Next, I used a distribution of 1-normalvariate(.6,.4)

CWs 0.9487
plurality 0.758364516129
approval 0.708670967742
borda 0.833464516129
random 0.410993548387
schulze 0.854887096774
rp 0.854770967742
minmax 0.854535483871

The program is in the same archive as my previous model.  That is, 
http://vote.sourceforge.net/sim/sim.zip
This model is in the vsim2 folder.

---
Blake Cretney


----
For more information about this list (subscribe, unsubscribe, FAQ, etc), 
please see http://www.eskimo.com/~robla/em