[EM] Approval's worst problem isn't a problem

[Michael Ossipoff]

The example that Rob posted …

45: A>B>C
20: B>C>A
35: C>B>A

...Is similar to the worst problem example frequently brought up about 
Approval. The difference is that usually B and C are very close in numbers, 
and the A voters don’t have a preference between B and C.

So it might be morelike:

40: A
31: B>C>A
29: C>B>A

Again B is the CW, but this time just barely. And maybe no one knows who is 
CW, because it’s so close. What’s known is that B and C add up to a majority 
that could easily defeat A--if they co-operate.

But, regrettably, co-operation between B and C is a Nash disequilibrium.
If the vote is:

A: 40
B: 31+29 = 60
C: 31+29 = 60

….then the C voters could improve the outcome for themselves by withdrawing 
their vote for B, and thereby stealing the election from the CW.

The first comment that could be made is that this second example, the more 
difficult one, is also more unlikely. Something more like Rob’s example is 
much more probable. And there, there is a clear and obvious CW.

Let me suggest reasons why it isn’t such a problem:

As I said before Approval always has a Nash equilibrium in which the CW wins 
and no one reverses a preference.

Though co-operation isn’t a Nash equilibrium, there are Nash equilibria. If 
B or C wins because his own supporters vote only for him, and the other 
faction vote for him too, that’s a Nash equilibrium. Then, either faction 
would get worse outcome by changing their vote.

If, by one of the justifications listed below, one faction publicly 
announces its refusal to co-operate, that faction is doing a strategy for 
the purpose of imposing a certain Nash equilibrium.

That’s common in military confrontation (e.g. mutual assured destruction) 
and in the animal kingdom, and maybe in legal systems too. Often it’s a 
dominated strategy, as in the example of mutually assured destruction. But 
they’re still used, and they still work. Such strategies of deterrence are 
widely used, and so they must have some success. For instance, later I 
mention the example of the cat defending its own territory against another 
cat.

1. Principle:

Maybe either B or C is more principled (or their voters are). If so, and if 
it’s made clear before the election, the principled faction has a great 
advantage in threat credibility. Like the cat that’s defending its own yard 
against an intruding cat.

If it’s obvious that there’s going to be the co-operation/defection dilemma, 
and that bilateral co-operation isn’t going to happen, then it will be 
obvious that the more principled faction isn’t going to co-operate, and that 
the other faction should co-operate.

2. Condorcet Winner:

Usually it will be pretty obvious that either B or C is very likely to be 
the CW. Therefore it will be clear that someone needs that CW. And if, as in 
the case of Rob’s example, A prefers B to C, then it’s also likely that A, 
being closer to B than to C, wouldn’t be as bad for B voters as for C 
voters. All the more reason why the B voters don’t need C as much as the C 
voters need B.
It’s sometimes pointed out that, when there are 3 candidates, the supporters 
of the middle candidate have no reason to vote for anyone else.

3. What if Principle and CW conflict. I suggest that principle trumps CW or 
apparent CW:

“Maybe your Senator Gladhand is “middle” in the sense of being between the 
other candidates, and has been called middle by the media, but that doesn’t 
mean that he is near the voter median. Even you don’t really like his 
policies, if you’re honest with yourself. We aren’t voting for Gladhand, 
because he’s a slimeball. He therefore isn’t someone you can count on to 
win, and that’s because of his own lack of merit.

"Don’t pin your hopes on him by voting only for him. Our candidate is 
someone whom you can actually trust and respect.”

4. Strategies such as BF-Uncertain, BF-Certain, BF-Probability and 
Better-Than-Expectation:

Some of us have pointed out that usually you won’t need a mathematical or 
numerical strategy method to decide for whom to vote--it will be obvious to 
you whom to vote for. But maybe this classic Approval worst problem, the 
co-operation/defection dilemma, is a situation where sometimes the strategy 
methods could help. For instance, in Rob’s example, the BF strategies made 
short work of the problem. Of course the B voters could have merely pointed 
out that B is the clear CW, and that would ordinarily be enough.

Of course, in this kind of a situation, it isn’t just a question of what 
strategy for one faction to use--it’s a matter of what _overall_ vote 
configuration makes the most sense in terms of strategy for all the 
factions.

Maybe the consistent vote-totals-based strategy equilibrium could sometimes 
have a role in making it more obvious which of {B,C} is the rightful winner.
By the way, “DSV equilibrium” would be a briefer name for consistent 
vote-totals-based strategy equilibrium.

In any case, the strategy methods are there for when they’re needed.

My point so far is just that there’s a whole toolbox of solutions for 
Approval’s worst problem.

5. But the existence of the problem requires that B & C be somewhat similar, 
and so how bad would it really be for one faction if the other faction’s 
candidate won?

I’d often be inclined to vote for the other faction, because the important 
thing is to keep A from winning. If the C voters want it that bad, enough to 
defect and betray, then let them have it, in the interest of defeating A.

6. But if the C voters betray, how good is their chance that they count on 
the help of the B voters ever again? In fact, one or both factions could 
make that explicit before the election:

“We’re on the same side. Our candidates are nearly the same. Together we can 
easily defeat A. The choice between B and C can be made by other voters 
outside the B and C factions (such as maybe the A voters, or various 
others). We’re voting for your candidate, but we want to make it quite clear 
that if you steal the election from us by lowdown betrayal, you can forget 
about ever having any support from our faction.”

7. So, sometimes, for any of the above reasons, you might rightfully give 
in, if the indications such as CW, principle, strategies, DSV equilibrium, 
etc., indicate that to be right. Or sometimes you might rightfully insist, 
taking a stand based on one or more of the justifications given above.

Approval’s worst problem isn’t really a problem.

Mike Ossipoff