[EM] Chris: Approval vs IRV

[Adam Tarr]

Stephane Rouillon wrote:

>Maybe I need more mathematical support on this but, even if I agree with 
>Mike, I evaluate the number of time I would have to bury my favourite in 
>order to get it elected with IRV far less than the number of time I could 
>lose him by not setting properly my approval cut-off with approval. It is 
>a matter of odds (probability).  Could someone evalute those, even just 
>for a small case?

OK, a small case follows.

The names of the candidates are with respect to your preferences.  The 
breakdown of voters is uncertain, but it is EITHER:

35% Worst
5% Worst>Compromise>Favorite
12% Compromise>Worst>Favorite
13% Compromise>Favorite>Worst
35% Favorite>Compromise>Worst

OR,

30% Worst
5% Worst>Compromise>Favorite
12% Compromise>Worst>Favorite
13% Compromise>Favorite>Worst
40% Favorite>Compromise>Worst

The difference, of course, is that in the first case, you lose the second 
round runoff in IRV, and in the second case, you win it.

In IRV, if you really hate Worst, you have good reason to sell out Favorite 
and put Compromise in first.  This causes Favorite to lose in the first 
round, so that Compromise wins the runoff.  This is the classic "lesser of 
two evils" scenario.

In Approval, I can always vote for favorite, but the question is whether I 
approve compromise as well.  It's a very similar dilemma, although I don't 
have to actually sell out my favorite.

Now, here's the twist.  Say the second situation is the 
reality.  Furthermore, suppose that 8% of the 40% in my faction decide to 
sell out favorite (and a similar 1% of the 5% in the W>C>F faction).  So 
the votes, in IRV, look like:

30% W
4% W>C>F
13% C>W>F (including 1% insincere)
21% C>F>W (including 8% insincere)
32% F>C>W

So, favorite gets eliminated in the first round, and compromise wins, and 
that 8% of the electorate that sold out Favorite spends the next four years 
feeling like idiots.  Now, suppose we have the same number of "double 
approvers" in the Approval election.  I'll assume a fifth of the 
compromisers (3% on each end) also approve their preferred wing, just like 
for the other factions.  So, with an equal amount of "compromising" going 
on, the votes are:

34% W
4% WC
19% C
11% FC
32% F

Now, despite the same amount of "favorite betrayal" as in the IRV example, 
Favorite still wins.  So here we have a pretty good argument that approval 
is MORE forgiving in this case than IRV.  In general, it takes twice as 
much mistaken compromising to sink the stronger wing candidate in approval 
versus IRV.

And one last note: this whole strategy problem is trivialized in 
Condorcet.  Just vote sincerely, and go home.  As such, this example makes 
a good case for my Condorcet>approval>IRV>plurality preferences.

-Adam