[EM] How Approval locks on to the CW in two or three moves.

[Forest Simmons]

Let's suppose that there are three candidates, and that one of them C 
is preferred over the other two by fifty percent plus majorities (not 
just by more for than against).

Suppose that candidate X (not equal to C) is the perceived front runner 
going into the first election.  Then strategic voters will approve all 
candidates that they prefer to X.  

Since more than half of the voters prefer C to X, more than half of the 
voters will approve C.

This is enough to ensure that C will be one of the two front runners in 
the next election (since the loser of the frontrunner battle will get 
less than fifty percent approval).

Assume that preferences haven't changed so that C and (say) Y are the 
two frontrunners in the next election.

[If Y=X, as in the likely case that X (the previous top front runner) 
did not come in last, then C would be the top front runner in the 
second election. But we are not assuming that here.]

Then on every strategic voter ballot precisely one of these two (C or 
Y)will be approved (whichever the voter prefers) along with others 
approved by the voter of the ballot.

Since C is preferred pairwise over Y by more than fifty percent of the 
voters, C will get more than fifty percent approval, and Y will get 
less than fifty percent approval.

This is enough to make C the top frontrunner in the third election, 
being the only candidate to get more than fifty percent approval in 
both previous elections.

In this election all strategic voters will put their approval cutoff 
next to C on the side of the other frontrunner (say) Z.

So C's margin of approval over Z will be the same as Z's margin of 
pairwise defeat by C. And C will get more than 50% approval (by the 
same reasoning as for the previous election).

The only question is could the other candidate (say) B get more 
approval than C in this election?

In other words, can B be above the approval cutoff (which is right next 
to C) more than C is?

C is above the approval cutoff more than fifty percent of the time. 
Could B be above the cutoff more than fifty percent of the time?

Well, B cannot be above the cutoff without also being above C, no 
matter which side of C the approval cutoff is placed.

So the question becomes, can B be above C more than fifty percent of 
the time?

The answer is no, because C beats B pairwise.

So no candidate B can have more approval than C in this election.

This makes C the election winner of the third approval election.

So once C makes it to top frontrunner status, C will get more approval 
than either of the other two candidates.

This argument shows that approval winner status (once attained) is 
stable for CW's.

In summary, within three elections (and probably sooner than three) the 
CW will reach and keep approval winner status (until voter preferences 
change).

I'm sure that this crude analysis can be improved upon, including 
relaxing the fifty percent plus condition and the three candidate 
condition.


Forest