22 May, 2007

mathematicians are idiots

i haven't read through this whole article yet. i probably won't, because i find the premise ridiculous and, frankly, irritating. however, it is useful (in the opening paragraphs) as an illustration of how brilliant people can be remarkably stupid. particularly game theorists.

here's the setup:

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?
if you've any sense, you're choosing $100. in know i am.

now, the startling conclusion!
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.
amazing! irrationality wins! er, not really. see, what these guys don't realize is they are basing their logic on the assumption that people are machines. they aren't. sure, mathematically, it may be logical to pick 2, because then you can't lose. but the most you are going to get is $4. yippie fuckin' skippy. a formula, or super math geek, might think in these terms, but generally, humans don't. hell, i don't even think vulcans would. it is far more rational to assume the other person is going to try to maximize their reward, not their odds of winning. sure, if you pick 100, instead of 2, you get nothing. but the other person only gets 4 bucks. you aren't out a hundred dollars. you're out four. in terms of risk, that's pretty damn low. so why bother to low ball it? i'd venture to guess that no statistically significant portion of people choose less than 50. why? it's in the premise for choosing this reward/punishment scenerio: people are going to inflate the value. if you are setting up a problem because you think the subjects are going to try to reap a higher monetary reward, what possible reason would you have to believe someone would choose 2 as an answer to the problem? well, that's covered:
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.

Game theorists commonly use this style of analysis, called backward induction. Backward induction predicts that each player will write 2 and that they will end up getting $2 each (a result that might explain why the airline manager has done so well in his corporate career). Virtually all models used by game theorists predict this outcome for TD--the two players earn $98 less than they would if they each naively chose 100 without thinking through the advantages of picking a smaller number.
read that last sentence again. now, think about what this man is saying. the logical conclusion leads to you netting $2, $98 less than naively choosing $100. this is most decidedly not an advantage. and yes, it seems highly implausible that someone would choose 2 because it is highly implausible. the disadvantages of low reward more than outweigh any advantage of choosing low.

anyway, i couldn't really read past that point. the model is flawed, because it is based upon the least important variable in the equation. you don't need a game like this to realize that human decision making is not guided by strict logic. what this says to me is not that it is irrational to defy strict logic, but that it is irrational to stick to it.

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