Mathematical utility expectation maximization in Approval

[MIKE OSSIPOFF]

>> > Pij is the probability that i & j will be the 2 frontrunners.
>
>
>Slight correction/clarification:  The precise meaning of Pij is usually
>taken to mean the probability, given a tie exists for first place, that
>i and j will be involved in that tie.

Different wording, same thing. If i & j are the 2 highest votegetters,
and there's a tie for 1st place, then it's between them. If they
aren't the 2 highest votegeters, and there's a tie for 1st place, then
it isn't between them.

>
>Thus in the thumbnail description of mathematical Approval strategy,
>frontrunners are candidates those with the best chance of being involved
>in a tie, not necessarily the best chance of winning.

The 2 frontrunners are the only candidates with any chance of being
in a tie for 1st place. I'm talking about the 2 frontrunners in
the final election count.

But, as I said, there's a relation between the probability of being
a frontrunner and the probability of winning.

>
>Example:
>- You have candidates A, B, C, D, and E,
>- A has an 90% chance of winning,
>- B and C each have a 4% chance of winning,
>- D and E each have a 1% chance of winning,
>then A, B, and C are considered the frontrunners
, since they are equally
>likely to be involved in a tie for first place (either 2-way or 3-way).

But what I meant by the 2 frontrunners is: the 2 candidates who,
in the final election count, have the highest vote totals.

Candidate A is more likely than B or C to be in a tie for 1st place,
because he has a greater win probability. Numbers proportional to
these candidates' individual frontrunner probabilities are:
A: .95, B: .2, C: .2   There's a .04 probability that B & C will be
the 2 frontrunners, but there's a .19 probability of A & B being the 2
frontrunners in the final election count. Those are estimates based
on Tideman's square root suggestion.

Mike Ossipoff


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