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

[Simmons, Forest]

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.
 
For practical purposes (i.e. not worrying about tied ranks or tied approvals at some stage) this method is summable in an (N+1)^3 array, where N is the number of real candidates, and the "plus one" represents the virtual approval cutoff candidate (whether or not the ballot is presented that way). 
 
The contribution of a ballot to the (i,j,k) entry of this array is unity if candidate k is ranked ahead of the j side of candidate i on the ballot, else the contribution is zero.
 
This is the same array that is used in batch style Approval DSV (Declared Strategy Voting) with strategy A, but it is used differently.
 
In both versions the first index i represents the current approval winner.  In DSV the second index j represents the current runner-up.  But in this new application, the second index represents the approval winner of the previous round.
 
That's why we need entries for the virtual approval cutoff candidate:  this candidate is taken as the previous winner in the first round.
 
This adds a new feature to the method:
 
If the methods converges on the virtual candidate, then the winner should be picked by random ballot from among the set of candidates that won at least one round.
 
If no real candidate wins even one round, then .... ?
 
Another thought: why not use the "all winners lottery" even if the limit of the sequence of winners is not the virtual candidate?
 
Is monotonicity too much to hope for?
 
It seems like Rob LeGrand, Kevin Venzke, and I briefly considered this method a few years back, but then we didn't have the benefit of Jobst's proof of convergence.  Rob had a whole list of strategies A, B, C, D, E, ... for DSV Approval, and I think that this was one of them.
 
This method is a limiting case of Jobst's idea where (in a mixture of Jobst's and my formulation and notation) lambda is infinitesimal in the weighted average
 
   L_(k+1) =  lambda*L_k + (1-lambda)*L'_k  .
 
The other limiting case, where (1-lambda) is infinitesimal might also be interesting to study, but I don't think it will be summable.  Furthermore, it will have to deal with ties in probabilities (and therefore more frequent ties in ranks and approvals).
 
Thanks to Jobst for the great proof insight under this subject heading!
 
Forest
 
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