[EM] Methods based on sequential voting

[Simmons, Forest]

In some elections not all of the ballots are cast at the same time, and furthermore, the partial results (from exit polls, say) may be available to voters later in the sequence.
 
In some applications, like the US presidential election, geography roughly determines the order of the ballots.  The West coast voters have information about the East coast voters votes before voting.
 
In other, more local elections, a first come, first serve policy determines the order.
 
In small groups, the order of seating, may be deemed acceptable.
 
In other applications, a random permutation of the voters will be seen as sufficiently fair.
 
In ballot-by ballot Declared Strategy Voting it is essential to process the ballots one-by-one, and a different order of processing can yield a different outcome.
 
In any case, some voters  may prefer to vote near the beginning of the sequence, others may prefer to vote near the end, when the partial results make optimal strategy more obvious, and others may prefer to vote near the middle.
 
It may be that in some electorates all voters would rather vote last, but some may feel more strongly about this than others.
 
Here are some ways of allocating places in the voting sequence in such a way that every person thinks she has gotten her fair share of the voting power:
 
 
(1)  Have each voter specify exactly how much weight would have to be attached to the first vote in order to make it equal (in her opinion) in value to the average of the remaining votes.  The voter that bids the lowest, gets the first position. Apply this procedure repeatedly among the remaining voters for the remaining postitions until everyone has a place in line.
 
(2)  Use Sperner's lemma to find an envy  free allocation of positions with associated weights attached.  How to do this is beyond the scope of this short message.  But it means that each voter ends up with a position in line and a weight for her vote, such that according to her own measure (subjective or objective) she would not prefer any of the other voters' position/weight assignments above her own.
 
Method (1) is an example of "fair division."  Method (2) is both fair and "envy free."  Here's another method that is merely fair, but is likely to be nearly envy free:
 
(3)  Have each voter specify what percent of the total weight should go to voters in the first half of the line.  The half of the voters who bid the lowest will be recursively assigned to the first half of the line.  The other half will be recursively assigned to the rest of the line.   If the number of voters N is not even, then interpret "first half of N" as all of the positions less than  N/2 .
 
All of these methods violate "one-person, one vote, "  but only to compensate for the inequity of that dictum in the case of sequential voting.
 
Forest
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