Tuesday, January 31, 2006

I'm game if you are



You're not going to believe what I learnt in class today.


You know how we're always arguing over what to do over the weekends? Like last week, when I wanted to catch that special Antonioni retrospective and you wanted me to take you shopping?

I DID NOT argue. You promised that we would go to the sale, and then backed out at the last minute. I mean, you don't even like Antonioni for god's sake, and....

Ya, ya, okay. It was a once in a lifetime event and I didn't know about it when I promised, but okay, so it was my fault. The point is you know how we always have trouble deciding? Well, I discovered in class today that there's this game called the Battle of the Sexes which looks at exactly that problem. It's really cool.

You were sitting around in class playing games?

No, no, it's Game Theory. It's like this uber-cool branch of economics. So listen. It turns out that there's something called a mixed strategy equilibrium that provides a better answer to the problem than just having one person give in to the other.

Really? That's interesting. How does it work?

Well, for starters, you take a dice.

A dice? What kind of dice?

A normal dice. Like the one that comes with our snakes and ladders game.

So this mastermind solution has something to do with snakes and ladders?

No, no. Look, it could be any dice. It just has to be unbiased, that's all.

Hmmm...I'm not sure that snakes and ladders thing is unbiased actually. The way you kept getting all those sixes last time was really suspicious. If we're going to need a dice for this method of yours I think we should get a new one.

Yes, yes, all right, we'll get a new dice. But listen. So once you have this dice, one of us, say I, throws it, and if it comes up 1 or 2 then I do what you want, otherwise I do what I want.

Huh? Wait. So basically you're saying there's twice as high a chance of you getting what you want than of me getting what I want. That's convenient, isn't it? Why don't we just split the bloody dice equally - if it comes up 1, 2 or 3 we do what I want, and we do what you want otherwise. I'm okay with that. I don't see why I should be at a disadvantage just because it's your game.

Ah, but you won't be at a disadvantage. You see, after I throw the dice, you throw it too and the same rules apply. If it comes up 1 or 2 then you do what I want otherwise you do what you want. So you see, there's a two-in-three chance that you'll end up doing what you want as well.

So basically you're saying there's a two-thirds chance that you'll do what you want AND there's a two-thirds chance that I'll do what I want?

Yes, exactly.

Not very strong on fractions, are they, these game theorists of yours. It feels like one of us has a third of a chance too many.

No, no, obviously we don't always do the same thing you see. We can separate and each do our own thing.

So now you don't want us to spend weekends together? You want us to separate and do things by ourselves? Why don't you just come out and say that you don't want to spend time with me, instead of trying to blame it on some vague model that you made up?

No, look, it's not that I don't want to spend time with you. You know I do. It's just that that's the way the rules are set up. You do what the dice tells you to do and I do what the dice tells me to do. The rest is just up to chance. That's the whole point of it. Besides, it's not like we would be spending all our weekends separately. In fact, there's a 4/9th chance that we'd be together - either doing what you want or what I want (2/9th chance of each). It all depends on what the dice comes up with.

So basically we spend more than half our weekends apart. What joy. And wait. What happens if we both throw a 1? Then you'd have to do what I want and I'd have to do what you want, wouldn't we?

Yes, exactly.

What? You mean I'd have to go to an Antonioni retrospective while you went shopping for dresses? That doesn't make any sense. Why wouldn't one of us just go with the other?

Because you have to stick with the rules. Start switching decisions opportunistically and the whole thing will break down. Look, anyway, it's not like that's going to happen all the time. The probability of getting that outcome is just 1/9.

So basically your master strategy to keep us from arguing is to say that 5 times a year we each go do something that we don't want to, without the person we're doing it for?

Well, it wouldn't be 5 times a year exactly, of course. It could be more. It could be less. It's all probabilistic, you see.

I see. So we could end up doing this every weekend for an entire year.

We could. But it'd be really, really unlikely. A total freak chance.

But still, it could happen. If we were going to do this at all (and I'm NOT saying I agree, mind you) why wouldn't we just schedule it? Set aside the weekends that we were both going to be miserable? That way at least we wouldn't run the risk of it happening more often.

No, no, then it wouldn't be random anymore would it? That doesn't work.

Right. So not only should we be doing something neither of us wants to do, but we should also do it completely at random. Sort of like a weekly surprise.

Yes, exactly. Look, I know it sounds a little daft when you put it like that, but it really works. Honest. You can show mathematically that there's no way that either of us could do better by ourselves.

Really? We couldn't just talk about it like grown-ups, for instance? Instead of playing games with dice like we were five years old.

No, no, that's where the arguing starts. It's precisely to avoid having to talk about it that you need the algorithm in the first place.

So you're basically admitting that you'd rather leave your weekend plans up to chance than talk to me about them. Do you really hate talking to me that much? Has it really come to this?

No. Look, it's just a game, all right.

Our relationship is just a game to you?

No. This decision rule thing is just a game I learnt at school. I just thought it might be fun to try it.

I don't think so. Why are you spending your time doing stuff like this anyway? I thought your PhD was in Corporate Strategy. What are you doing trying to solve husband-wife conflicts?

No, no. I mean, the whole husband-wife thing is a metaphor, obviously. This stuff has tons of applications to other business problems.

Really? Like what exactly?

I don't know. Tons. Mergers and Acquisitions stuff. Systems integration. Competitive Dynamics. Lots of great stuff like that. You wouldn't understand.

I'm sure. So now you want our lives to be determined by a metaphor? What else can this Game Theory of yours do for us? Is there anything in it about sharing the TV remote, for example? I think I could use some innovative solutions there.

Well, actually, there is this game that kind of has the same structure. The way it works is...

I DON'T CARE!! Don't think I can't see what you're trying to do with this. Trying to weasel your way out of things you don't want to do. Well it won't work. That sale you didn't take me to last week is still on this weekend, and we're going. And no amount of manipulation with dice is going to get you out of that. So there.

Um...right. Okay. No need to get all hot and bothered about it, darling. I was just saying...


I sometimes wonder why game theorists don't get divorced more. Don't get me wrong. I love the stuff. I watched A Beautiful Mind three times. Once I even cried. I'm just unconvinced that Game Theory has anything to do with real life.

Before people who love Game Theory start jumping down my neck, though - let me say two things. First, I'm well aware that the 'husband''s understanding of the Battle of the Sexes game above is woefully inadequate. There's a lot more nuance to the solution (and to Nash equilibria more generally), which he completely misses. And second, that I'm not saying that my criticism of being unconnected to the real world is true only of Game Theory. I actually think it holds for much of academia, I just think the Game Theory disconnect is the most amusing, and the description of the Battle of Sexes game simply lends itself to being made fun of.

Categories: ,


... said...

Woah! This conversation is so "us"!! Very funny though! It is weird how alike most husband-wives are :-) Loved your post!

Heh Heh said...

"You can show mathematically that there's no way that either of us could do better by ourselves."

not true, once you involve pre-commitment (which drawing up a schedule does). also it would make it a repeated game, and interpretations of mixed strategy equilibria as averages over a large number of trials of a single game are oxymoronic.

as to your statement about game theory vs. the real world, i agree. too much promise, too little applicability. there are other branches of economics that certainly have more applicability in the real world. most recent work in game theory is little more than intellectual masturbation.

just showing off :)

ozymandiaz said...

I met Nash once. Not long ago there was a trial going on here, a big business, big money trial. He was an expert witness. So I met him briefly. I wonder though, in his state of mind, if you only meet someone once do you suspect if they are real or not? I mean i suspect reality all of the time anyway and my mind is a raisin in the road compared to his...

Falstaff said...

...: Thanks.

heh: See point 1 in the second paragraph of the footnote (I knew I should have become a lawyer). Part of what I was trying to do with the post was make the 'husband' deliberately obtuse (though you may be one of the few people reading the post who actually see the subtleties of this) - there are obviously a lot of reasons why applying the single game mixed equilibrium solution to the actual situation makes no sense. Even if you ignore the repeated games problem, there's the more basic point that nothing in the Nash equilibrium says you could't do better if you could credibly cooperate.

What's interesting to me is the fact that game theory would even try to spin the story they do around the game - I mean one would think (or hope, at least) repeated games and the possibility of collusion would be pretty much ubiquitous in marriages, no? Not to mention the difficulty of assuming rationality on the part of people who are, after all, married.

Oz: Wait, you're telling me that there are actually people out there who are handing out convictions based on Game Theory?? That's seriously scary.

ozymandiaz said...

Honestly I don't recall the content of his testimony. It was a case between Maris distributing and Anheuser-Busch. So there was no "conviction" per se, just a $50 million verdict.

:-) said...

Game theory is more effective when used to primitive instincts. You cant apply game theory to intellectual conversations. :-)

Also Games are always generalizaions. Obviously individual responses vary.

Nice post. :-)

Prakriti said...

This is just too bloody brilliant :)!

I looooved your post! If I start on a similar mood, I would leave it up halfway through. You carried till the very end which made it so beautiful!

Did this conversation actually happen? :)

iyer education said...

dropped in thru desipundit...

fundooo post... if john gray reads this, he's gonna kill himself... and if stephen covey reads this, he cud easily substantiate his "win-win" strategy mathematically ;)

good post again :)

tangled said...

This post was utterly wonderful.

However, what does it say about me that I mentally replaced dice with die wherever it was a singular? *sigh* I'm incurably uptight, that's what.

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