[EM] Comparative Effectiveness of Approval and Condorcet in the case of a three candidate cycle.

[Gervase Lam]

> Date: Wed, 02 Feb 2005 18:45:15 -0800
> From: Russ Paielli
> Subject: Re: [EM] Comparative Effectiveness of Approval and Condorcet
> 	in the	case of a three candidate cycle.

> I've already brought up the issue of inaccurate polling data, and I
> think the effect of such uncertainty needs to be addressed before the
> effectiveness of Approval can really be evaluated. Someone somewhere has
> probably addressed this issue, but I am personally unaware of it.

In an e-mail sent to me 1 year ago today(!), Mike Ossipoff suggested doing 
a 2 round (or more strictly a 1.5 round) Approval election.  In the 1st 
round, the candidate with the most Approval votes above 50% of the votes 
is the winner.  If no candidate gets more than 50% of the votes, then a 
2nd round takes place, with the same candidates as in the 1st round.

The idea is that the 1st round acts as the accurate poll with which voters 
in the 2nd round would use.  In order to make the 1st round as accurate as 
possible, you "threaten" the voters with the fact that there can be a 
winner in the 1st round.

However, I don't think the 50% of votes level is right.  What really needs 
to be done is to calculate a level where there is a 0.5 probability (i.e. 
50% chance) of there being a winner after the 1st round.  I had a go at 
doing this a couple of weeks a go.

I had to assume that on one ballot, each candidate had a 50-50 chance of 
being approved.  This is because the rules to set the level shouldn't use 
poll information.  As a result, for the purposes of calculating the level, 
it is assumed that the election is a 0-info one.

What I did was to set the probability of all the candidate being below the 
certain level at 0.5 (i.e. 50% chance).  This means that the other 0.5 
chance is of one or more candidates being above the level.  This gave me 
the equation 0.5 = p^n, where p is the probability of each candidate being 
below the level and n is the number of candidates in the election.  With n 
= 3, p is 0.79.

There is a 50-50 chance of a candidate being approved on each ballot, with 
the total number of votes for each candidate following the Binomial 
Distribution.  However, the Binomial Distribution is practically the same 
as the Normal/Guassian Distribution (i.e. the classic bell curve).

Therefore, all that needs to be done is to find 0.79 in the Normal 
Distribution table that statisticians have.  On finding 0.79, the "z 
value" for 0.79 can be found.  It is then a simple matter of using the z 
"transformation formula" to find the approval level that should be set for 
the 1st round.

I don't have a Normal Distriubtion table in front of me.  But, if a 
remember rightly, the level is about 60%.

Sorry if there are any mistakes.  But I need to rush.

Thanks,
Gervase.