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

[Kristofer Munsterhjelm]

On 06/08/2013 10:16 PM, 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)

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

> 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.

It seems that this could be generalized to any top-two runoff method. 
Consider a base method X, that picks two candidates for the runoff. 
Then, even if X passes the FBC, if the situation is so that:

- Candidates A and B go to the runoff if voting is honest,
- some voters that have sincere preferences A>C>B can replace A with C 
by favorite-betrayal,
- in an {A,B} runoff, B will win; in a {B, C} runoff, C will win,

then there's an incentive for favorite betrayal. To completely protect 
against that, a method would have to pass a criterion where voters who 
prefer C to the honest runoff winner B can't replace either of the 
candidates by betraying their favorite (who might not be C). Call this 
passing "double FBC". But I don't see how a method could possibly do that.

Does that mean that we can't have both FBC and LIIA? The argument would 
go in this vein:
- assume X passes both FBC and LIIA, and that the same set of votes are 
used for both rounds.
- then the winner in round 2 is the winner in round 1, by LIIA.
- This means that we don't need to consider the runoff as such, only the 
base method X.
- And by FBC, for any strategy that involves favorite betrayal, there's 
another, non-favorite-betraying strategy that also works.
- So the runoff, being equivalent to just running base method X under 
the assumptions given, should pass the FBC.
- But it can't, by the argument above.
- Hence having both FBC and LIIA is impossible.

But something is strange here. Approval is said to pass both FBC and IIA 
(which is a superset of LIIA). So where's the flaw?

Thinking a bit further, it seems the flaw is in that the votes don't 
change. In the example above, if the runoff is Centre-Right vs. Right, 
the 21 CR > R > L voters aren't going to approve both Centre-Right and 
Right in the second round. Since Approval ballots are binary, it's 
impossible to express a rank preference over more than two levels, and 
so the assumption only holds if the voters' preferences are inherently 
dichotomous (in which case the voters who approved of both runoff 
candidates would just stay home on the second round).

The argument would still seem to hold for ranked voting, however - at 
least if you include the assumptions that voters who vote "A = B > C" 
would vote "A = B" in the runoff. To make the impossibility proof 
formal, one would just have to show that no ranked method can pass 
double FBC.

-

Finally, I'd like to say that I do understand that reality is a lot less 
neat. What Abd says about differences in turnout in the first and second 
rounds of a runoff means that criteria are not as useful as for 
single-round methods because the votes in the different rounds would 
change.

One could even argue that if they don't, there's no reason to add a 
runoff to an advanced method, and the only reason for Plurality to have 
a runoff is to patch problems in Plurality itself. I have seen reasoning 
of this sort from some IRV advocates who both say "top-two runoff is 
also nonmonotonic, so don't go around saying TTR is better than IRV" and 
"IRV is better than TTR in every way because it's clearly better than 
the contingent vote".