Mathematical utility expectation maximization in Approval

[Bart Ingles]

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.

Bart



Richard Moore wrote:
> 
> [...]  keep
> in mind Bayes' Theorem for conditional probabilities.
> 
> Ignoring three-way (and higher) ties...
> 
> Let Pt|ij be the probability that there is a tie given that i and j are the
> front runners.
> Let Pij be the probability that i and j are front runners (Mike's
> definition).
> Let Ptij be the probability of a tie between i and j.
> Let Pij|t be the probability that, given a tie, i and j are the front
> runners (Bart's).
> Let Pt be the probability of any tie.
> 
> Two conditional probability equations can be written:
> 
> Ptij = Pt|ij * Pij
> Ptij = Pij|t * Pt
> 
> Setting the two expressions for Ptij equal to each other, and rearranging
> the terms, you get:
> 
> Pij = (Pt * Pij|t) / Pt|ij
> 
> or
> 
> Mike = Bart * (Pt / Pt|ij)
> 
> Since Pt is constant for all candidate pairs, and Pij|t is different for
> each
> pair of candidates, then the relationship between Mike's and Bart's
> definition isn't a proportional one.
> 
> While Mike has matched the correct definition to the notation he chose,
> I think Bart's conditional probability is better for determining strategy.
> The probability that A and B are front runners isn't the probability that
> your vote will decide between them. I suspect Mike's strategy as
> modified by Bart is mathematically equivalent to mine, which I defined
> in terms of differential probabilities. I didn't give a formal equation in
> my earlier post but what I have in mind is something like
> 
> Ui = sum over j of (dPj|i * uj)
> 
> where Ui is the incremental utility of voting for i, uj is the utility of
> candidate j, and dPj|i is the change in probability of j winning caused
> by my vote for i (for monotonic voting methods, dPj|i will be
> non-negative when i = j).
> 
> For zero-info voting with N candidates, we can add that
> 
> dPj|i = -dPi|i / (N-1)
> 
> for all j not equal to i, which simplifies the solution, as I showed in
> my earlier post.
> 
> Hopefully my differential method and Bart's method will prove
> equivalent. I wonder which is harder to calculate for the non-zero
> info case, Pt|ij or dPj|i?
> 
>  -- Richard