Tuesday, June 05, 2007

The Irrationality Of Rationality (Or Is It The Rationality Of Meta-Rationality?)

You've all heard of the Prisoner's Dilemma, but have you heard of the Traveller's Dilemma? I'll let the creator explain it.

Lucy and Pete, returning from a remote Pacific island, find that the airline has damaged the identical antiques that each had purchased. An airline manager says that he is happy to compensate them but is handicapped by being clueless about the value of these strange objects. Simply asking the travelers for the price is hopeless, he figures, for they will inflate it.

Instead he devises a more complicated scheme. He asks each of them to write down the price of the antique as any dollar integer between 2 and 100 without conferring together. If both write the same number, he will take that to be the true price, and he will pay each of them that amount. But if they write different numbers, he will assume that the lower one is the actual price and that the person writing the higher number is cheating. In that case, he will pay both of them the lower number along with a bonus and a penalty--the person who wrote the lower number will get $2 more as a reward for honesty and the one who wrote the higher number will get $2 less as a punishment. For instance, if Lucy writes 46 and Pete writes 100, Lucy will get $48 and Pete will get $44.

What numbers will Lucy and Pete write? What number would you write?

Scenarios of this kind, in which one or more individuals have choices to make and will be rewarded according to those choices, are known as games by the people who study them (game theorists). I crafted this game, "Traveler's Dilemma, in 1994 with several objectives in mind: to contest the narrow view of rational behavior and cognitive processes taken by economists and many political scientists, to challenge the libertarian presumptions of traditional economics and to highlight a logical paradox of rationality.

Traveler's Dilemma (TD) achieves those goals because the game's logic dictates that 2 is the best option, yet most people pick 100 or a number close to 100--both those who have not thought through the logic and those who fully understand that they are deviating markedly from the "rational choice. Furthermore, players reap a greater reward by not adhering to reason in this way. Thus, there is something rational about choosing not to be rational when playing Traveler's Dilemma.

In the years since I devised the game, TD has taken on a life of its own, with researchers extending it and reporting findings from laboratory experiments. These studies have produced insights into human decision making. Nevertheless, open questions remain about how logic and reasoning can be applied to TD.

Common Sense and Nash
To see why 2 is the logical choice, consider a plausible line of thought that Lucy might pursue: her first idea is that she should write the largest possible number, 100, which will earn her $100 if Pete is similarly greedy. (If the antique actually cost her much less than $100, she would now be happily thinking about the foolishness of the airline manager's scheme.)

Soon, however, it strikes her that if she wrote 99 instead, she would make a little more money, because in that case she would get $101. But surely this insight will also occur to Pete, and if both wrote 99, Lucy would get $99. If Pete wrote 99, then she could do better by writing 98, in which case she would get $100. Yet the same logic would lead Pete to choose 98 as well. In that case, she could deviate to 97 and earn $99. And so on. Continuing with this line of reasoning would take the travelers spiraling down to the smallest permissible number, namely, 2. It may seem highly implausible that Lucy would really go all the way down to 2 in this fashion. That does not matter (and is, in fact, the whole point)--this is where the logic leads us.

What is the point of all this? As Basu says, there are two broad lessons to be taken from this. First, the assumption of classical economics that a society of selfish people acting in their self-interest will produce the greatest economic good needs to be challenged. In this case, the greatest economic good would come from both players choosing 100. The deductive logic, however, suggests both players should choose 2, leaving them considerably worse off than if they had behaved irrationally.

Second, because repeated experiments have shown that players don't choose 2, and that these choices aren't a reflection of stupidity or not understanding the deductive logic behind the game - even a group of game theorists, when playing this game, by and large chose numbers other than 2 - we must recognize the limits of game theoretic or logical reasoning to explain social scientific outcomes. According to Basu, we must somehow account for what he calls "meta-rationality," which contains an explicit rejection of formal rationality. Anyways, it's a really interesting piece and I highly recommend going and reading the whole thing.

Link courtesy Freakonomics


NB said...

That really doesnt make any sense to me in logical terms. Ultimatley if she chooses 90 and pete chooses 80 sure she incurs X% chance of losing 2 dollars more than pete, ie getting 78 dollars. But risking two dollars is still better then complelty forfieting about 88 bucks, with a probabllilty of 100%. In terms of expected values, she would opt for a reasonably high figure because shes increasing her odds of receiving a higher figure, ie in case pete places higher than her, at the very least they receive the amount shes stated. Pete would probably do the same.

I dont think you can say a choice other than two is illogical, ie i dont think it leads us to explicitly reject formal rationality (because what is demonstrated to be formal rationalism in this example isnt really rational at all)

(btw i didnt read the article, i just read the excerpt)

Ahsan said...

in a way, you are right. you are predicting "real world" behavior which goes counter to what deductive rationalist logic would predict. in other words, you are merely proving one of basu's points for him.

basu's point is that "real" people would never act the way rationalist logic would say they would. one of the core tenets of rat choice is the assumption that ALL players will be rational. so in this case, using your numbers, lucy would choose 90 for an instant, and then realize that it is pete's best move to say less than 90, say 80. but upon realizing that, lucy's best move IMMEDIATELY becomes 79, because if she chooses 90, and pete 80, she ends up with 78 and pete with 82. on the other hand, if she chooses 79 and pete chooses 80, she ends up with 81 and pete 78. since she ends up with 78 in the first outcome and 79 in the second, it is in her best interest to give a number lower than pete's. the point to note is that solved mathematically (including, as basu points out, strong equilibrium, nash equilibrium, weak equilirbium, and almost any other equlibrium you can think of), this game ends with both players choosing 2.

this then is one of the two central points of the article. one, as you have noted, in the real world, people would never play like that. even game theorists, when experimented upon with this game, did not choose 2. what does this tell us? that rational choice and deductive logic need to be taken with a grain of salt, especially when it comes to social science.

the second point is that classicial economics says everyone will be best off (collectively) if everyone follows his/her self-interest. this game shows that not to be the case, because if everyone DID rationally act in their self interest, the outcome would be a decidedly inferior one for the collective and for the individuals.

Ahsan said...

oh and i suggest please read the article. the excerpt is about 20% of it, and much of what i just said will be much better explained by an actual game theorist who, you know, actually invented the damn game in the first place.