Wednesday, December 14, 2011

Game Theory X: Metagames: Extortion

Since it's finals week and I'm busy, I've arranged for the Topic to have a guest lecturer. Well, two of them. They're going to be giving this lecture through an audio feed, as due to circumstances almost entirely within their control, they are unable to be here. Say hello to the Piranha Brothers.
The following material is flagged Green Level. It is intended to reflect material that the author believes to be a matter of consensus among experts in the field. This belief may be incorrect, however; and as the author is not an expert and does not have an expert fact-checking the article, errors may creep in.
Okay, so our topic this week: As we have seen, communication can be helpful. But can it ever be used to cause harm to another player? Are there any conditions under which it would be beneficial to pay for communication to be forbidden?
Dinsdale, Doug, I hope you've read the notes I sent you.
No, I used them for a bogroll. Of course I read them, you daft person!
Of course. And I can see why you wanted us to teach this class. If I understand right, he wants us to talk about the "Other Other Operation".
Right. So, the Other Other Operation. What we would do is, we would pop by someone's place, and say that if they did not pay us, we would beat them up. So, when this sit-you-ay-shun existed, the game looked a bit like this:

Pay Do not pay
Break Face (A-B,-[A+C]) (-B,-C)
Do Not Break Face(A,-A)(0,0)
with all numbers positive, and A<C.
And no, don't ask how I pronounce a table like that.
So, what do those letters mean, Doug? I mean, I get that it's algebra, and that the letters represent numbers, but what do the numbers represent?
Well, A is obvious, innit? A is what we're asking our "client" to pay us. B is the cost to us to break a face, since it's such a pretty face and it pains us to ruin it. And C is, obviously, the cost of having a broken face. It should go without saying that the zeroes are there because if the person doesn't pay and we don't break any faces, nothing's changed.
Okay, I get that. But from what I've read, "Break Face" and "Pay" are both what he calls "dom-in-ay-ted stra-teh-jies". So why would we ever break a face, and why would anyone ever pay us?
That is a good point. Do not break face/do not pay is the Nash equilibrium. So what do you think, Dinsdale? Why would anyone pay us?
Might it have something to do with Tit-For-Tat? If they don't pay, we break their face next time?
No, Dins. Tit-For-Tat only works if there is a "next time". (Although punishment can play a role in it). Think carefully: what do we do before the game happens?
We... oh. So that's it? They pay because we threaten them? But why would we carry out the threat, since it's a dominated strategy?
Well, that gets into what we talk about. Let's say that we can make promises, and that those promises are binding. So we promise that, unless the other person promises to pay us, we will break their face.
That makes the metagame this, with the rational moves highlighted in green:
Promise to pay

Break Face (A-B,-[A+C])
Do Not Break Face(A,-A)
=(-A, A)
Do not promise to pay

Pay Do not pay
Break Face (A-B,-[A+C]) (-B,-C)
Wow, Doug, you really do talk pretty. But it's not like we can make promises we can't break.
Right. So let's say there's something that will punish us for breaking our word, by an amount D; and will punish the other person by an amount E. For instance, we'll say loss of reputation or a bunch of legbreakers coming around. The metagame now looks like this; if D>B and E>A, the rational actions are highlighted in green:

Response to Threat:Promise to payResponse to Threat: Do not promise to pay
Make threat

Pay Do not pay
Break Face (A-B,-[A+C]) (-B,-[C+E])
Do Not Break Face(A,-A)(0,-E)

Pay Do not pay
Break Face (A-B,-[A+C]) (-B,-C)
Do Not Break Face(A-D,-A)(-D,0)
Do not make threat

Pay Do not pay
Break Face (A-B,-[A+C]) (-B,-C)
Do Not Break Face(A,-A)(0,0)

Pay Do not pay
Break Face (A-B,-[A+C]) (-B,-C)
Do Not Break Face(A,-A)(0,0)
So, to us, the ability to communicate and make these kinds of promises is worth A, and worth -A to our "client". In other words, our "client" would pay any amount up to A to keep us from being able to make promises. And we would pay any amount up to A to be able to.
Doug, there was something in the notes about "side payments"...
Oh, right. See, in some games, it's possible for one player to make a payment to the other beyond the game itself. In this case, that isn't quite true, since we can change A to whatever we want. It's effectively the same thing, but not quite. The thing is, as long as A<C, we can make A whatever we want it to be. So, as long as the amount we're asking for is less than the harm caused by a broken face, we can ask whatever we like.
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