[EM] 0-info approval voting, repeated polling, and adjusting priors

[Simmons, Forest]

Jobst wrote ...

A point which troubles me is this: The justification of placing the approval cutoff at the expected or median utility according to the current priors is based upon the two assumptions that (1) given a top tie between two candidates, the events "x is among the two tied candidates" and "y is among the two tied candidates" are *independent* and that (2) the priors give the respective probabilities of these events. But neither do I believe that this independence can be assumed, nor do I believe that the probability of winning the poll is the same as the probability of being one of the two tied candidates given a top tie...

Does anybody know a solution to this?

I reply:

I don't know a solution, but perhaps if we gave definite meaning to the probabilities we could make some progress.  It might be easier to talk about winning probabilities if those probabilities have a more definite roll in picking the winner.

Suppose for example, that at each stage a marble is drawn (with replacement) at random from a bag containing one red and 999 green marbles, and that the first time the red marble is drawn, the winner is to be chosen by a lottery based on the current set of probabilities.  Suppose further that at each stage the only thing reported back to the voters is who won the previous stage.  If two candidates tie at some stage, then the tie is broken by coin toss before reporting the winner, so the voters don't know about the tie.

Under these circumstances and from the point of view of the voters how would the probabilities evolve from one stage to the next?

Forest

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