# [EM] Siimple proof for 3 candidates

MIKE OSSIPOFF nkklrp at hotmail.com
Fri Mar 9 11:32:12 PST 2001

```I like to say "demonstration" instead of "proof", but "proof" is
briefer in the subject line.

The approach that's useful with many voters isn't useful with few
voters, because then the probability that voting for i will change the
winner from j to i depends on whether you're voting for j.

So let me define yet another meaning for Pij:

Pij is the probability that voting i over j will change the winner from
j to i.

With 0-info, Pij is the same no matter what 2 candidates are i & j.
That's true even with very few candidates.

Say there are 3 candidates, and you prefer them in the order A>B>C.

The gain from voting for B is Pbc(Ub-Uc)
The costs of voting for B is Pab(Ua-Ub)

You have postive net gain from voting for B if:

Pbc(Ub-Uc) - Pab(Ua-Ub) > 0

Pbc(Ub-Uc) > Pab(Ua-Ub)

Dividing both sides by the same number:

Ub-Uc > Ua-Ub

2Ub > Ua + Uc

Ub > (Ua + Uc)/2

When that's extended to more than 3 candidates, it will no doubt be
shown that the above-mean strategy for 0-info is valid even for very
few voters, with any number of candidates. That's suggested by the
fact that above-mean is the best strategy when there are many voters
and any number of candidates, and when there are very few voters and
3 candidates.

Mike Ossipoff

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