[EM] More candidates, few voters, 0-info

[MIKE OSSIPOFF]

Say that, instead of 3 voters, there are 4: A, B, C, & D.

Say we want to determine whether or not to vote for B.

If we're voting for C, then voting for B just means that we aren't
voting C over B. And so Bbc is the probability that C's vote total
is equal to B's vote total, or one vote less.

If we're not voting for C, then Pbc is the probability that B's
vote total is equal to C's vote total, or one vote less.

Since it's zero info, both of those probabilities are equal,
and both are equal to the probability that B's vote total is
equal to D's vote total or one vote less (Pbd), and are both equal
to the probability that A's vote total is equal to B's vote total,
or one vote less (Pba).

So, no matter whether or not we vote for C, Pbc is the same, and
is equal to Pba & Pbc.

The Pij are defined and they're all equal in zero info elections,
even with very few voters.

And so my demonstration that 0-info strategy is to vote for all the
above mean candidates is valid even with very few candidates.

Sum, over all j, of Pij(Ui-Uj) =
Sum over all j of (Ui-Uj)  [dividing each term by the same number, Pij]

= Sum over all j of Ui - Sum over all j of Uj
= Ui(N-1) - Sum over all j of Uj

That must be greater than zero, to vote for i, and so,

Ui(N-1) > Sum over all j of Uj
Ui > (Sum over all j of Uj)/(N-1)

Mike Ossipoff


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