[EM] Does Top Two Approval fail the Favorite Betrayal Criterion [?]

[Abd ul-Rahman Lomax]

Thanks to Chris for attempting this. This is a partisan election, 
apparently, which is an issue in terms of whether or not scenarios 
are realistic. But I'll set that aside for the moment.

At 03:16 PM 6/8/2013, Chris Benham wrote:
>Yes.
>Say there are three candidates: Right, Centre-Right and Left, and 
>the approval votes cast are
>49: Right
>21: Centre-Right (all prefer Right to Left)
>23: Left
>07: Left, Centre-Right (sincere favourite is Left)

It's really irritating that "Center" is misspelled, and "favorite," too. :-)

Okay, this is an election where Left is no-hope. The winner is going 
to come, with a sane method, from Right or Centre-Right. 51% of 
voters prefer Centre-Right to Right, reasonable compromise. The votes 
are reasonable, except that 23 Left bullet voters are unreasonable, 
in fact. I'd assume they know their position. Not too unreasonable, 
though. After all, Left is in second place as to first preferences. 
This is a center squeeze election.

>Approval votes: Right 49, Left 30, Centre-Right 28.
>
>The top-2 runoff is between Right and Left and Right wins
>70-30.

I've generally pointed to the problem of assuming the same electorate 
for runoffs, but let's, again, leave that aside. This is reasonable.

What is somewhat unreasonable about this scenario is the Right bullet 
voters. It is *very strange* that none of them also approve Centre-Right.

>All the voters who approved Left prefer Centre-Right to Right. The 7 
>voters who approved both Left and Centre-Right can change the winner 
>to Centre-Right by dumping Left (their sincere favourite) in the first round.
>49: Right
>28: Centre-Right
>23: Left
>
>Now the top-2 runoff is between Right and Centre-Right and 
>Centre-Right wins 51-49.
>
>Seven voters have succeeded with a Compromise strategy.

Looks correct to me. They can do that. The strategy works, in fact, 
because Centre-Right is the best winner, though only by a small 
margin. The strategy is *necessary* because of the 23 Left voters who 
don't approve of Centre-Right.

But there is something else going on here. In a real runoff election, 
those 23 Left bullet voters are not likely to show up to vote. 
Motivating them will be difficult. Remember that they didn't vote for 
Centre-Right in the primary; this represents low preference strength 
between R and CR. Why will they suddenly have high enough preference 
strength to show up and vote? In the U.S., that is.

If the turnout of those original Left voters is even slightly 
depressed, Right will win the runoff. Right has high preference 
strength, that's shown by the lack of additional approvals for Centre-Right.

I think the strategy could easily fail. Would *probably* fail. The 
low Left vote count in the primary would damage the Left party. They 
might *possibly* get a better result from this election, but the 
next, they are dead. Left will become increasingly irrelevant.

I agree that the example shows FBC violation on the face, and thank 
Chris for this example.

I also see that this example is (1) implausible, ultimately, and (2) 
would not lead to voter backlash. I.e., the Left voters would vote as 
they voted, and the Left voters would be kicking themselves for not 
voting for Center-Right, those that didn't, not for *failing* to 
betray their favourite. The problem that led to Right winning was not 
that they voted for their Favourite, but that they failed to vote for 
second-best, leaving that to the runoff. Just barely, they got their 
favourite into the runoff, wasting everyone's time, and their own 
campaign funds.

Approval has this problem, we know that. To fix it, Bucklin.

Let's take the same apparent voting strengths

49: Right translates to
40: R
  9: R,-,CR
21: Centre-Right (all prefer Right to Left) translates to
21: CR,-,R
23: Left translates to
15: L
8:  L,-,CR
07: Left, Centre-Right (sincere favourite is Left) translates to
  7: L,CR

I just guessed at some voting patterns, I did not tweak them to 
produce a desired result. In real Bucklin elections, we saw *lots* of 
additional preferences added. I assumed the voters here are 
sophisticated enough to know that skipping middle rank expresses 
higher preference strength, that third rank is *minimal approval*.

1st rank:
49: R
21: CR
30: L
shows sincere first preferences. That's an advantage in itself. A 
strategy might win, but has other costs.

2nd rank pulled in
49: R
28: CR
30: L
both L and R are failing to get additional preferences.

3rd rank pulled in
70: R
73: CR
30: L

Either CR wins or there is a runoff between R and CR, so the L voters 
do not need to betray their favorite.

I used all the preferences mentioned, but I added some level of 
delayed second-choice for the R voters. That's not necessary. If 
those voters continued to bullet vote, the result would be the same 
with a mandatory runoff.

What makes the election work with Bucklin is the addition of second 
preferences for supporters of a no-hope candidate. The only reason 
the 7 voters need to betray their favorite is from the intransigence 
of their ostensible partners. If they all voted a more complete 
ballot, Centre-Right would win. But this election is *extremely 
close* between R and CR, as is shown in the projected runoff.

As I pointed out above, if the L voters really do have a Tweedle-dum 
and Tweedle-dee opinion about Right vs Centre-Right, they will *not* 
vote for Centre-Right as expected. Holding that edge, then, could be 
quite difficult, and the strategy could utterly fail, with the 
humiliation of not having voted sincerely heaped on top of it.

I vastly prefer that voters may specify their favourite without 
significant harm to their voting power.

However, if it's Approval, there is an interesting possibility:

The election is not close. L is a no-hope candidate. Now, we could 
look at a collaborative strategy between CR and L voters. CR needs to 
do something to motivate L voters to vote for her. With L and CR 
together, they could win the election, it's very shaky without that. 
This collaboration would want to get L and CR into the runoff. So, 
with nothing to lose -- they have no R votes anyway, back in the 
primery, L and CR campaign together, urging *all of their followers 
to vote for both of them, that this is necessary to defeat R.*

If they do that, L and CR go into the runoff. The R voters will turn 
out. CR will win, very likely, by 70:30, if the R voters aren't too 
pissed at CR. A new coalition will have been formed. The politics of 
this jurisdiction will never be the same again....

Just a fantasy? Maybe. We really don't know how public approval 
elections will function.