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

[Jobst Heitzig]

Dear Forest,

you wrote:
> Here's a two line description of the simplest method that Jobst's proof applies to:
> 1. Ballots are ordinal with approval cutoffs.
> 2. In each round (and on each ballot) the approval cutoff is adjusted by moving it next to 
> (without moving past) the approval winner from the previous round.
> This method converges because the approval winner of the previous round is on the same side 
> of the approval cutoff as in the previous round, so his approval is the same in the new round, 
> so the new approval winner (if different from the old) must have greater approval, which can only 
> happen finitely many times.

This is true and could possibly be considered a good replacement for strategy A with guaranteed convergence. However, the underlying philosophy of it, just as in strategy A, is to place maximal trust in the last poll, and use earlier information only to decide whether to put the cutoff above or below the previous winner.

My original thoughts went in the opposite direction: What is a reasonable and realistic way for voters to make use of the information contained in *all* polls so far. This seems important *especially* since the last poll may be stronlgy influenced by short-term strategic considerations.

So let me propose to study also strategies which adjust the priors not only slightly but also less and less, for example by putting

(i)   p(x,i,t+1) = (t-1)/t * p(x,i,t) + 1/t * delta(x,w(t))

or, more generally, by putting

(ii)  p(x,i,t+1) = (t-alpha)/t * p(x,i,t) + alpha/t * delta(x,w(t)).

Both ways will still lead to convergence (same proof as before). In the first situation, the prior for  x  will ultimately converge to the relative frequency with which  x  won the polls, which seems to be a reasonable requirement when we want to estimate the probability that  x  will win the next poll. In the second situation, the more recent polls have more influence on the priors than the older polls, but unlike in a pure exponential weighing scheme the trust in older polls grows over time. 

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?

Jobst


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