[EM] Another PR method

[Simmons, Forest]

Disclaimer: this method is for theoretical purposes only.  Those who don't believe in theoretical purposes should delete it immediately.
 
Somtimes PR STV is introduced by asking the reader to imagine a PR election in which voters vote sequentially with knowledge of the current running subtotal of votes before casting their ballots.  Once a candidate reaches the quota, no voter will waste any more votes on her.  Even if she were your first choice you would vote for someone else, once she is already elected.
 
The STV expositor is not suggesting that STV faithfully mimics this sequential method, nor even that it would be a good idea to use it.  He only uses it to illustrate the idea of vote transfer in STV.  Similarly, I am not advocating its use, but I would like to consider it and some refinements of it to illustrate some ideas.
 
It has a couple of obvious problems.  (1) It requires a choice of the order of voting.  Here in the USA, we vote East to West, since that's how the time zones are allocated.  Who has the advantage?  The early voter?  The late voter?  If we could figure that out, could we compensate by weighing the votes heavier near one coast or the other?  (2) The second obvious problem is that of voting strategy.  For example, the very first voter has nothing to go on other than the usual outside polls.  It's the plurality strategy problem all over again. 
 
Let's consider this problem in more detail.  If the first voter knew how the rest of the voters were going to vote, then she could easily figure out how to vote without wasting her vote, even if her vote could not be pivotal; in that case she would simply give moral support to her favorite candidate by casting her vote for her.
 
Since, it is impossible for the first voter to know how all of the rest of the voters are going to vote, why not let her cast a "contingent vote" in the form, I vote X if all the other voters vote thusly ... but I vote Y if their distribution of votes is this ... etc."
 
Each voter in turn specifies a contingent or conditional vote, until the last voter, who has no later contingencies to worry about.
 
Now for the nitty gritty of how to do this.
 
Let's say that there are  N voters and 26 candidates to fill seven seats, by "sequential PR."
 
Let L be a list of all N letter code words made up from the 26 letters of the English alphabet in alphabetical order.
 
Replace each code word by a set of seven letters representing who would win the election if the number of votes each candidate ended up with were the number of times that candidate's letter appeared in the code word.  Break any ties by some appropriate method before proceeding further.
 
Now group these sets in (order solid) groups of 26, and have the first voter choose one set from each group of 26, and through out the other 25 from each group.
 
Now group the remaining sets in groups of 26 and have the second voter select one from each group to be kept.
 
ETC.
 
The last voter chooses the winning set from the remaining group of twenty-six.
 
If you think in terms of trees with twenty-six branches branching out from each node, then this is a trimmig process which picks out one code word, so that in the end everbody knows which candidate their vote supported.
 
It seems to me that this is a way to pick out a NASH equilibrium from the set of all code words.
 
Forest
 
 
 
 
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