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
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.