Wednesday, December 27, 2006

The game game-theorists should have played

Confession of the day: As a pre-teenager, I was into "Asteroids", which I think is from Atari. Nice game (if you're from the Half-life2 age you'll know how old I am and how sad this sounds). A bunch of large rocks on a 2D space, and some UFO's thrown in to shoot your spaceship.

Anyway, one day I was dazzled with this guy. Instead of finishing up the rocks and moving on to a harder stage, like real man do, the guy destroyed almost all rocks, and, subsequently sat and waited for the UFOs to come.

The point being? The points, of course: one large rock shot would give you 20 points, while a small UFO would bring 1000 points. Imperial powers don't have as many lifes to spare than that guy did--it was awesome. This is, perhaps, a game that game-theorists should have been playing all along.

There is a huge momentum on "behavioral game theory", or "cognitive game theory", or whatever you want to call it. For a review, see the book with the former title from Colin Camerer. The basic idea is that, rather strangely (for economists only), people do not "play" their best options. Real people do not play "rationally". They stubbornly deviate from the rational choice, even in simple settings.

Perhaps the best example is that of the problem known as 'ultimatum bargaining'. Suppose that an agent A makes a proposal and another agent B decides to accept or reject the offer.

A: do you want ice cream?
B: yes

B's reply should always be yes, as long as the offer holds "utility" (we'll only use the term under quotes, ladies and gents, for reasons yet to be brought up). But here's the catch: A has some amount of money, say, $100, and proposes a split between A and B.

A: I get $60 and you get $40?
B: (having no option but to accept or reject the $40) Ok...

Here economic theory goes nicely with psychology. The "rational" choice for B is, always, to accept the offer. But...

A: I get $99 and you get $1?
B: (having no option but to accept or reject the $1) Rot in hell.

This scenario is an established psychological fact, and a departure from "economic rationality". After all, $1 is more than $0, and $0 is what our friend decided to get. So behavioral game theorists (a psychological type of game-theoretician (a mathematical type of theoretician)) have invented the label that humans have, quote, "negative reciprocity". But here lies the problem with inventing yet one more heuristic or bias:

Recent theories attempt to explain rejections using social preference functions which balance a person's desire to have more money with their desire to reciprocate those who have treated them fairly or unfairly, or to achieve equality. [...] Economists have resisted [such functions] because it seems to be too easy to introduce a new factor in the utility function for each game [in which behavior diverges from EUT]. (from Camerer 2003, p.11)


This is the problem with the heuristics and biases program today. As the number of biases increases with each new paper, how can economics manage to have an integrated mathematical model underlying decision-making? The flurry of distinct functions make it seem like the enormous number of incompatible unix and linux distributions. Are we going to have "utility", plus or minus a laundry list of exclusions? Where is the dismal science going?

Perhaps it is high time for an overhaul of the basic concept, centered in human intuition; a mathematization of Klein's model (this is almost given by Hofstadter's FCCA)--but we're not there yet.

In the case of ultimatum bargaining, why do people reject? Perhaps because they do not play the game at all, or even see that there is a game (and a gain) in it. The reasoning is simple: consider for instance Hofstadter's variations of "the same phrase":

* He didn’t give a flying f*** .
* He didn’t give a good God damn.
* He didn’t give a tinker’s damn.
* He didn’t give a damn.
* He didn’t give a darn.
* He didn’t give a hoot.
* He didn’t care at all.
* He didn’t mind.
* He was indifferent.

These phrases have the same meaning. A choice of one, however, tells a lot about who you are and where you are and who you are talking to. Have you heard of the smallest conversation ever? Here it goes:

"?"
"!"

Ok. Now here's the context in which this occurred. Victor Hugo had just finished "Les miserables", and went for a vacation of sorts. Anxiety, however, made him restless and made him write to his publisher the following letter: "?", to which his publisher, knowing that sales were good, and not to be outdone by V.H., responded: "!".

A conversation takes place in a context. Content and context fight to create meaning. If you don't care about something and want to tell someone that fact, it makes a big difference which phrase you will pick from Hofstadter's list: it means a lot about who you are and who you think your listener is.

And so does a game. If you, as a proposer, offer someone 1/100th of what you have, you are, in effect, by the subtle connotations of the act, telling people how much worth you view them. Their rejection has absolutely nothing to do with money, rationality, or retribution--they never framed the situation as an economic game.

If people were playing "nature", or an inanimate object, they would not mind to accept humble offers; it means nothing about who they are. Miners do not scream out: "I've had it with you, you greedy mountain; I know you have a million dollars in gold, stop offering me these miserable bargains". Instead, they accept the bargains, and hope for better ones in the future. People will shoot the large asteroids for a mere 20 points. Yet, given the chance, they might rather choose to get a better bargain, as in the case of waiting for those small, juicy, 1000-points, UFOs.




Game theory and practice

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