May 22, 2007
GAME THEORY: The Traveler's Dilemma
Kaushik Basu in Scientific American:
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?
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.
More here.
Posted by Azra Raza at 04:35 AM | Permalink
Comments
Surely it is at least as logical as the proposed solution for A to think: I will write $100, then no matter what B writes I will get B's choice, less $2 (or $100, if B should write that). The proposed solution nets B's choice plus $2 (or $2, if B should write that). The $4 difference seems minimal weighed against the chances that the other person will write either $100 or the actual value (surely more than $2 or why fuss?). And in any case, honest (as opposed to "economically rational"!) people would just write the real value.
The problem here is that logic does not apply to humans the same way it does to numbers.
Posted by: Bill | May 22, 2007 10:20:03 PM
The actual problem is that Basu's proposed solution is not, in fact, logical and hence no "logical paradox of rationality" is demonstrated.
To see why, let me first state Basu's reasoning:
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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
...
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.
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Let me now state the objective of this exercise: to obtain the greatest absolute amount of money possible.
Given the rules, there are three ways of computing the obtained sum
1)equal to the claimed amount (in case of identical claims). Range: $2-$100; Possible cases: 99 (from 2 to 100)
2)$2 more than the claimed amount (in case of one's claim being lower).
Range: $4-$101; Possible cases: 98 (from 2 to 99)
3)$2 less than the other's claimed amount (in case of one's claim being higher).
Range: $0-$97; Possible cases: 98 (from 3 to 100)
Just to rephrase, the goal is not to secure more money than one's opponent, but to secure as close as possible to the global maxima, which is $101.
$101 is obtainable only in 1 scenario - you claim 99, your opponent claims 100. Next maxima, 100 is obtainable in exactly 2 scenarios - both claim 100 or you claim 98 and your opponent claims 100 or 99. From here, it's all downhill.
Now the game iterations:
(assumption: both players know of and understand the objective)
1)I should put 100.
2)But my opponent will have thought the same and thus if I put 99, I would get 101.
3a)Wait a minute, so will my opponent, so I could put 98 and still get at least 100.
3b)Wait a minute, so will my opponent, so I should just put 100 after all (or if optimistic, 99) and still get 100. Hopefully, my opponent thinks like me. [Ed: unstated assumption of Basu as well]
Basu seems to take (3a) as the only 'logical' contingency and then proceeds to iterate further
4)Wait a minute, so will my opponent, so I could put 97 and get 99. [The problem here is that one is now limiting compensation by one's own agency, which is contrary to the objective and hence, in fact, illogical]
Posted by: Gyan | May 23, 2007 1:13:53 AM
This is a lot like the scenarios where you have to meet up with a random person in some large area (like New York city.) No matter what you think the optimal "logic" is, the idea is to be right at or below the other person's guess. Humans naturally look for landmarks. In New York, you think "The Empire State" or "Statue of Liberty" and hope they think similar. In this game, the only two "landmarks" are 2 and 100. Since you want a lot of money, 100 is vastly preferable. The risk of +/-$2 (on that $100) is peanuts compared to the "logical" play plan of $2 to start. Combine that with even mild altruism, and most people would rather behave cooperatively for $100, than compete for $2. As Gyan pointed out, once your logical play strategy takes you more than $2 from optimal, it's no longer optimal. You're trying to maximize gain, not just beat your opponent. If you wanted the $2 scenario to really work, play the game where the lower price bidder gets the everything, and the higher guy gets nothing.
Posted by: Xepher | May 23, 2007 10:22:19 PM
These comments are wonderful -- truly illuminating. I never know what to make of problems like TD because the logic always seems to force some absurdity on the players. I don't really understand why TD is significantly different from a story problem in math. What do you achieve with TD that you couldn't achieve just by doing the math? Someone please tell me how this illuminates probable human behavior -- I would like to get it.
Posted by: Elatia Harris | May 23, 2007 11:02:47 PM
I'm not sure I DO understand the "big deal" with this problem. It seemed to me like the article's author just ignored a whole lot of potential motivations and/or assumptions about human nature to make $2 the "logical" outcome. It really seems to be as if he was conflating the goal of "beat your opponent" with "maximize personal gain." You almost always beat the other player at $2, but if the goal is to make the most money, $99 or $100 are the only two logical guesses that make money. Guessing any lower is only useful if you're worried about a $2 difference. True maximum means guessing either a) $1 below your opponent, or b) the same as your opponent. ANY other guess reduces your potential income. Basically, you need to hit one of two moving targets out of 98... but if you assume your chances are (reasonably) bad for that, it doesn't matter (more than $2 worth) how wrong/above that you guess. Guess one over is just as bad as guessing 90 over (if you lose.) On the other hand, if you "win" (your opponent guesses at or above your guess) you win AT LEAST what you pick. Thus, if you bet on the 48/49 odds, rather than the 1/49 odds, $100 or $99 are the best choices. In this sense, you say "screw the $2" and then there's only two outcomes. Guessing right at or below what your opponent picks (odds: 2/98) or "anything else" (odds: 96/98.) If you assume/bet on the obvious 96/98 odds, then there's two outcomes. You can either bet high, which results in winning your opponents guess... which you have no control over... -$2, or you can bet low, which results in winning LESS than a high bet would, since your opponent's uncontrollable guess is nominally "the same" no matter what you pick. The end function of the game is to figure out if people value a $2 penalty more than the chance at $100 (or $101) dollars. Ask your state's lottery commission for that answer, and stop spending money on pointless "research" into this scenario. :-)
Posted by: Xepher | May 24, 2007 12:41:42 AM
It seemed to me like the article's author just ignored a whole lot of potential motivations and/or assumptions about human nature to make $2 the "logical" outcome.
Exactly so. Logical to whom? Mathematicians maybe. Anyone else is going to pick $100 -- and, as demonstrated in the article and the comments above, they will be right to do so.
This is not a "paradox" or "contradiction", it's a problem with the underlying model of how humans behave. (See also economic theories, unremitting failure of.) What Basu calls the "rational" choice is in fact only rational from one (rather odd) point of view; the model is too simple. Basu writes:
Argh. To me, this view has things exactly backwards: it is implausible, and that does matter -- in fact that, not your beloved formal logic, is the "whole point".Someone please tell me how this illuminates probable human behavior -- I would like to get it.
I don't think it does. Humans do not behave as rational machines driven by formal logic.
Posted by: Bill | May 24, 2007 12:37:26 PM
I'm especially struck by the fact that the designers of the study didn't think it important to tell the subjects how much the antique was actually worth. In other words, being honest with the airline was not an option.
Might it not be both "rational" and "sensible" not to commit fraud?
Posted by: Chris Schoen | May 24, 2007 6:48:58 PM
Humans do not behave as rational machines driven by formal logic.
Apparently I have failed. Bill, as I tried to show above, even formal logic does not lead to $2 as the best answer. The "logical" solution is itself wrong.
Posted by: Gyan | May 25, 2007 12:04:34 AM
Good comments here, and some echo of my own thoughts. Note that studies have shown that human players do NOT pick $100. My own micro-survey suggested $70-$80 was more likely.
On the one hand the principle of maximizing gain was completely ignored in favor of "beating" the other player.
On the other hand, there is an assumed risk that doesn't exist in reality. There was no pre-existing guarantee (in the scenario) that you would get ANY money back. So the "antiques" broken or not can be considered a sunk cost. Any gain made off the airline transaction is money you never had, and weren't expecting. Therefore you should try to get as much as possible, ignoring the apparent risk of "loss".
Posted by: Jonas | Jun 26, 2007 1:23:43 PM
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