[EM] Why Approval is strongly nonfalsifying

[MIKE OSSIPOFF]

There's a CW. That means that for any candidate, X, more people
prefer that CW to X than prefer X to that CW.

Say, in Approval, everyone extends their Approval set down to, but
not past, the sincere CW. In other words, everyone votes for the CW,
and for everyone whom they like more, but not for anyone whom they
like less.

That means that everyone is voting for the CW. In order for another
candidate to voted for by everyone, he'd have to be considered equal
to or better than the CW by everyone. But then the CW wouldn't be the
CW. If anyone likes him less than the CW, as someone must, then he
gets fewer votes than the CW. The CW wins.

Say the voters who prefer X to the CW want to try to make X win,
and so they vote for X, but not for the CW.

The number of people who prefer X to the CW is less than the number
of people preferring the CW to X. The number of people voting for X
and not for the CW is less than the number of people voting for the
CW and not for X. The CW still gets more votes than X.

Shortening one's Approval set, then, brings no benefit. Obviously,
extending one's Approval set past the winner, to candidates whom one
likes less, can't benefit a voter.

So no one benefits by changing their strategy. No group of same-utilities, 
same strategy voters gains by changing their strategy
unilaterally.

Though it was assumed that there's a CW (that's part of the
premise of the definitions in the classification), no assumption was
made about there being no indifference between the CW & other
candidates.

Approval is strongly nonfalsifying.

Mike Ossipoff


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