[EM] Nash Lottery question

[Forest Simmons]

*Given score style ballots, the Nash Lottery is the lottery (i.e.
probability vector on the alternatives) that maximizes the product P of the
ballot expectations. *

*Let’s argue that when an alternative X is in the support of the Nash
Lottery L, raising the rating of that alternative on a single ballot
without changing any other rating on that ballot (or any other ballot) will
not bump that candidate from the support of the lottery:*

*First of all note that the new winning lottery L’ (if there is a change)
must yield a greater product P’ of ballot expectations than the old product
P, since even the old winning lottery L will now have a greater product of
ballot expectations due to the increased rating of X on one ballot.*

*Now suppose that the support of the new winning lottery L’ does not
include X.  Then this lottery L’ yields the same product of ballot
expectations before and after the change in the rating of X .  So we have
P’ > P even before the change in the single ballot that raised X.  In other
words the lottery L’ was better than L even before the change, so L could
not have been the Nash Lottery winner, after all.*

*Now another question arises:  can raising the rating of only one
alternative X (already in the support of the winning Nash Lottery L)
increase the cardinality of the support of the winning lottery?*

*If not, then Random Ballot on the support of the Nash Lottery winner is a
monotonic method.*

*Any insight into this question?*
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