Mathematical utility expectation maximization in Approval

[MIKE OSSIPOFF]

Bart wrote:

>The probability definition I used came directly from Merrill's book.
>The strategy presented there is to vote for candidate i if
>
>   u(i) > the sum of all (t(j) x u(j)) for each candidate j
>
>where u() is voter utility and t() is the probability of a candidate
>being in a tie given one exists.  In other words u(i) is being compared
>to a weighted average of all u(j), where each candidate utility in the
>average is weighted by his likelihood of being in a tie for first place.

The fact that Merrill said that doesn't mean that your definition
of Pij defines a different probability than does my definition of
Pij.

The probability that i & j will be the 2 frontrunners in the
election's votecount is the same as the probability that, if there's
a tie for 1st place (considering only 2-candidate ties), then
i & j will be the candidates in that tie.

That's true no matter how Merrill or anyone else defines Pij.
I don't deny that many authors define Pij the way you do. In fact
your definition is probably more widely used than mind. But
both definitions define the same probability. They're two wordings
of the same definition.

Mike Ossipoff


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