Resolving Non-uniqueness in Maximal Lotteries

cfrank

Maximal lotteries are guaranteed to be unique when rankings are strict and there is an odd number of voters. Uniqueness can hold in other cases, and generally non-uniqueness only occurs when the majority margin matrix exhibits "pathological" and non-generic symmetries. It follows that small perturbations of the majority margin matrix almost certainly yields uniqueness.

This post is just a space to discuss how this non-uniqueness should be handled in a fair way when it arises. Here is an example of one idea (not novel):

If the maximal lottery is unique, great.
Otherwise, if we allowed non-strict rankings, induce strict rankings on each ballot's indifferences independently at random. (This raises questions about ballot format and implementation).
If the maximal lottery is still not unique, then the number of counted ballots is even. Produce one additional ballot, such as by randomly sampling one late ballot that was not yet counted (if one exists), by constructing a ballot from those already submitted (randomly sampled or a distributed construction), or by blatant authority (or even just a completely random ranking).

This will guarantee uniqueness of the maximal lottery. However, it may no longer satisfy the formal properties of maximal lotteries---as in, the achieved unique lottery may not actually be maximal for the ex-ante majority margin matrix (unless we use the latecomer ballot).

Alternatively, we can use a rule to select a maximal lottery from the admissible set, such as the maximum entropy maximal lottery, or sampling the maximal lottery from the Jeffrey's prior over maximal lotteries. My personal opinion is that the Jeffrey's prior makes the most sense.

As always, any thoughts are welcome.