[EM] More results about computer simulations of elections
Kevin Venzke
stepjak at yahoo.fr
Tue Dec 6 13:46:07 PST 2005
Warren,
--- Warren Smith <wds at math.temple.edu> a écrit :
> Consider the following cheapo model of simulating an election. Each
> candidate to
> each voter has a utility which is an independent uniform random in the
> interval [0,1].
> There are some fixed number C of candidates and some number V of voters
> where we shall assume V is LARGE:
>
> Thm 1.
> Suppose all the voters magically know the identity X of the
> max-summed-utility candidate.
> Suppose each voter votes approval-style by approving of all candidates
> with more utility
> than f*U_X, where U_X is X's utility (to that voter) and f is a constant
> (for example f=10% or f=90%) with 0<f<1.
> Then: with 100% probability in the V=large limnit, X will win the
> election.
>
> Proof: X gets 100% approval. Any other candidate gets approval 0.5*f
> approximately. Q.E.D.
>
> Note 1a. Actually the above proof works for ANY pre-fixed candidate X,
> who instead could be the Schulze winner, the worst-summed-utility
> candidate, or whoever.
> X still will be elected.
Congratulations. You just proved that if all the voters can guess which
candidate I'm thinking of, this candidate will get 100% approval, no
matter how likely they think it is that this candidate will win.
I don't understand the "proof" section at all. How do you determine that
X will get 100% approval and everyone else will get 0.5*f approval?
> Thm 2.
> Suppose each voter votes approval-style by approving of all candidates
> with more utility
> than U_X*f+U_Y*(1-f),
I guess you wanted to divide by two here.
> where U_X is X's utility (to that voter) and X is
> the best
> among {A,B} and Y the worst (in that voter's eyes) and A and B are two
> randomly selected candidates (but fixed once and for all after their
> choice, which happens
> immediately before the election; think of A and B as the two
> "major-party" candidates)
> and f is a constant (for example f=90%) with 0.5<f<1.
>
> Then: the most popular among {A,B} gets elected with 100% probability in
> the V=large limit,
> i.e the same winner strategic plurality voting would elect.
>
> Proof: the most popular among {A,B} gets over 50% approval. Each other
> candidate gets (2-f)/3<0.5 approval in expectation. Q.E.D.
Sorry, you're wrong. Let's say f=50% and 50 voters have utilities 100 and 0
for A and B, and 50 voters have utilities 0 and 100 for A and B. Say
there is a third candidate C with utility 51 for every voter. Then C
wins with 100% approval.
> Note 3.
> This suggests that if f=50% exactly, then we should expect approval
> voting
> results better than strategic plurality voting.
I totally agree.
> Note 6.
> Observe that Approval voting in note 3 therefore will be expected to do
> better
> social-utility-wise than any Condorcet or IRV method (with this model and
> these voter strategies).
Yes, *with this model and these voter strategies*.
> Summary:
> These facts make some of Kevin Venzke's simulations look, in retrospect,
> pretty stupid.
I don't agree, since I didn't assume Schulze voters would use (extreme)
favorite betrayal. Also, I came up with useful numbers regarding Approval.
> If comparing voting system A versus B by using social
> utility,
> I would tend to want to make ther honesty:strategy fraction among the
> voters be the SAME
> for both A and B, to compar apples with apples. This is not required,
> but it seems more fair. If you make Schulze voters alwys be honest and
thus
> never susceptible to DH3, then a comparison with some other voting system
with 100% or 50%
> strategic voters, will of course give a bias in favor of Schulze.
I don't think it would be more fair to assume Schulze voters use DH3
strategy.
It is almost obvious that approval voters will use some kind of "better
than expectation" strategy, but it is not obvious that Schulze voters
will always try to bury the other frontrunner.
> My "honest" approval voters simply approved the above-average candidates.
Average among all candidates? That corresponds to "Zero-Info" in my
simulations.
I'll respond to your other message a bit later.
Kevin Venzke
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