Tuesday, June 05, 2007

Game Theory, Selfishness, Economics, Science, and Beer

In that order!

Scientific American June 2007 issueThere was the a very interesting article in the June 2007 issue of Scientific American about game theory that got me thinking. The article was written by the inventor of Traveller's Dilemma, which is a simple game devised as an experiment to test some principles of game theory.

It works like this: you and another person return on a plane from a tropical island. Both of you pack a souvenir that you bought on the island in your checked baggage. Predictably, your baggage is handled roughly, and the souvenir is reduced to fragments. The man in charge of the baggage claim desk has to reimburse you for your loss, but he doesn't want to overpay. So to keep you honest, he offers the following deal: both you and the other passenger write the cost of the souvenir, which can be no less than $2 and no more than $100. If both of you write the same number, he will assume that you are both honest and pay you that amount. However, if one person writes a lower amount than the other, he will assume the lower amount is the real price and pay you both that amount, but with an honesty bonus of $2 to the person who wrote the lower amount and a dishonesty penalty of $2 to the person who wrote the higher number.

For example, if you write $55, but the other passenger writes $45, he would get $47 (the low price of $45 plus the $2 honesty bonus) and you would get $43 ($45 minus the dishonesty penalty of $2).

Got all that?

Now how would you play the game? Your first instinct might be to simply write $100, or some other very high number ($98, $95, etc.). But the optimal strategy is actually to write $2!

It may sound counter-intuitive until you run the logic a bit. Here is an example of the voices inside the heads of the two players:

You think: I'll write $100.
He thinks: I could write $100... but if he writes $100, I'll write $99, that way I'll get $101 instead of just $100!
You think: But what if that bastard writes $99, thinking I'll write $100? I'll show him! I'll write $98, and thus get $100 even though he tried to screw me by undercutting my bid.
He thinks: But what if he is thinking this way too? I'd better write $97, so if he writes $98 to try and beat me, I'll still get $99.
You think: I've got it! I'll write $2, the lowest allowable, and there is nothing he can do to screw me over!
He thinks: I have to write $2, or otherwise he can undercut me.

Congratulations, you have arrived at the optimal strategy: $2. And since so has he, you guys are mired in the Nash equilibrium (yes, named after that John Nash).

But the interesting thing that Dr. Basu--the author of the article and inventor of the game--points out, people tend not to play as dickeshly as that. His conclusion is that humans have an altruistic instinct that is in constant conflict with the selfish one, and that models that start with the assumption that people are going to make decisions rationally and selfishly tend to get hack-tacular when they are munged to reflect people's actual observed behaviour.

This is the old argument: must a theory be elegant to be correct? Does nature allow for ugly models?

Stephen Hawking is an interesting chap, and he has an interesting viewpoint on reality. He subscribes to the so-called positivist doctrine, saying that a good scientific theory must make predictions about a wide variety of phenomena that can then be observed experimentally. Following from this is the idea that reality is completely subjective, that different mathematical models can be equally valid and equally basic if their predictions agree with observations. So this would mean that ugly theories can be as valid as elegant ones, but the history of science has often showed us that mathematical ugliness hides a flaw in the theory or the maths.

(Seems like I read a book review in SciAm a few months back on a book that talked all about elegance in mathematical models... I think it may well have been Why Beauty Is Truth: A History of Symmetry.)

Anyway, returning to the Traveller's Dilemma, the observation that people do not play selfishly and reduce the TD to a zero-sum game is a very interesting result. Classical economic libertarianism, as espoused by Adam Smith et al., assumes that people do act selfishly, and the very act of everyone trying to serve his self-interest above all regulates the market. Hence the invisible hand.

So most studies of economics and game theory have assumed, until fairly recently, that people will be selfishly rational. Seeing as how this is not necessarily the case might complicate the question, but should simplify the maths.

However, these experiments have been carried out on people, not corporations. It seems to me that corporations as a whole probably do act in a rational, selfish way, controlled by the aggregate authority of their shareholders (or, in the case of private firms, stakeholders). So maybe classical economics is not dead yet. I need to read more about game theory and how it has developed in the latter half of the 20th century in order to understand its implications for global capitalism.

There was plenty of other good stuff in this month's issue of SciAm (and every month's issue, for that matter): an article on network coding; another on an alternate hypothesis to the RNA world one to explain the origin of life (isn't it great that Wikipedia has an "Origin of life" category!?); a bold new idea in the depressing field of conservation; etc.

Which leads me to my next point, that popular science is a wonderful and important thing. Today's scientific fields are so specialised and technical that it is very difficult for a layman to understand the exciting work going on. This simply means that scientists must work harder, because a public that is enthusiastic about science is one that is more likely to push for their elected representatives to fund science (other than extremely dubious weapons research).

Some scientists and laymen alike are rising to the task, and here are three fantastic books I've read about science in the last year, in order of ascending excellence:

  1. A Short History of Nearly Everything, by Bill Bryson - a fantastic history of scientific thought, theory, and of the characters who did the thinking and theorising. This really is the best general introduction to the foundations of modern science that I have seen. It covers everything from how our solar system formed to how human civilisation arose to the structure of an atom, and much more.

  2. The Universe in a Nutshell, by the great Stephen Hawking - Though not as good as the next book I'll talk about, TUiaN is a wonderful look at the basics of string theory, supergravity, M-theory, parallel universes, black holes that ain't so black, quarks, massless force-carrying virtual particles, and my all-time favourite, imaginary time. Simply brilliant. But wait, it gets better, with...

  3. The Illustrated Brief History of Time, also by mah homey S-dot. This book is beautiful in every way, from the lavish illustrations to the thick, creamy paper to the lively prose to the mind-blowing science. Stephen Hawking makes science more exciting than anyone, and the most amazing thing about ABoT is how accessible it is without being dumbed down. Dr. Hawking leaves all of the ideas in and only omits the maths, which are frankly over our heads anyway. :) Just buy this book, or check it out from the library or something. But read it! Now!

Ah yes, beer. I have not forgotten you, my constant companion! I've been having tremendous fun with ratebeer.com. I started using it just as a way to keep track of the interesting beers I've tried, and what I thought of them, but I now find myself addicted to beer tasting. I swore that I would never be one of those haughty-taughty types that drank their beer only from a glass and remarked on the palate of the brew, etc. But damnit, it is just so much fun!

So yes, my hobby is now beer tasting. Unlike wine tasting, however, you get to actually swallow the stuff. If you care to, you can probably keep up with my adventures on my beer rating page.


Dave said...

I read that article too, and it seems to me that the whole theory underestimates the intelligence of the participants.

If the goal is to get more money than ("beat") the other player, then $2 is optimal. But if the goal is to maximize my own profit, then presumably I don't give a damn if the other guy might make $3 more than me. So I would reason that I should say $100, in which case I have a chance of making $100 and some chance of making less - possibly as little as 0 if the other guy is a total dick (but is it rational to assume he's a total dick?).

If I say $2 then my max take is $4 and my min is $2. I'd reason that my best chance for making the most money is to assume that the other player can see the downside of trying to "beat" me, and the upside of cooperating. Therefore I'd expect a sort of non-coordinated cooperation.

I think the reason real people tend to pick higher numbers rather than the predicted low number is not altruism, but that they can see the benefit of cooperation and trust that the other player can see it too.

Considering who I'm accusing of being more or less totally wrong here, I would kind of be surprised if there isn't a huge hole in my argument, but what the hey. Lemme know if you can see it...

Josh Glover said...

I agree with your assessment with individuals, mostly. Looking at the game dispassionately, it seems rather obvious that playing a high number is better.

It is just that once you are in the game, you tend to get sucked into the downward spiral toward the Nash Equilibrium. You start thinking, "I can do better, but I wonder if he is thinking that too?"

This seems to happen in real world "games", too. Look at nuclear brinksmanship; isn't that basically the Nash Equilibrium in the Cold War?

I think the main point the author was trying to make is that game theory is not as simple as most current models predict. And that is an excellent take-away.