[EM] Another lottery method

Simmons, Forest simmonfo at up.edu
Mon Sep 5 15:20:05 PDT 2005

The following lottery method is easier to explain in terms of ratings (range ballots), but can (and should) be adapted to rankings (ordinal ballots) by modifying the following definition.
Definition 1:   Lottery L1 beats lottery  L2  on a given range ballot
the L1 weighted average of the ratings (on the given ballot) is greater than the L2 weighted average of the ratings on that ballot.
Definition 2:  The lottery L1 pairwise beats the lottery L2 
the number of ballots on which L1 beats L2 is greater than the number of ballots on which L2 beats L1.
The method:
1.  Let alpha be the set of approval values of the alternatives (candidates).
2. For each x in alpha let L(x) be the random ballot lottery over the set of alternatives that have an approval of x or greater.
3. Let x be the smallest number in alpha such that for all  y greater than x, the lottery L(x) is not beaten pairwise by L(y).
4.  The winning lottery is L(x).
The idea is to have a random ballot lottery based on a top segment of the approval list, and to have that segment extend down the approval list as far as possible without having a pairwise preference for a shorter such segment.
By convention no lottery is pairwise beaten by an undefined lottery, so if no other lottery wins, then L(MaxApproval) wins, i.e. in that case the winner is chosen by random ballot from the alternatives tied for most approval.
I believe that this method is monotone, clone-proof, and Independent from Pareto Dominated Alternatives.
I'll post some examples later, when I get the time.
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