Traveler’s Dilemma

I read an article in the June 2007 issue of Scientific America that discussed the traveler’s dilemma. The dilemma is sort of like the famous prisoner’s dilemma and is another decision theory model. One important thing to keep in mind about these models: the people who make them and run them always insist that it is unsporting to bring in real world factors or to act in anyway outside of the rules of their cleverly constructed scenarios. In any case, here is the scenario (roughly put):

Bill and Jane bought the same sort of antique while on a trip and both items are damaged during the flight. Ted, the person in charge of making restitution for the broken item has no idea of its value (and presumably both lost their receipts and there is no mention of the item’s value anywhere else in the world). But, being part of a decision theory scenario, he will make a deal. Bill and Jane will write down the value of the item, from $2 to $100. If one value is lower than the other, then Ted will take that as the actual value of the item and pay Bill and Jane that amount. There is, however, one major catch. The person who puts the lower amount will be rewarded for honesty and given $2 extra while the person who puts down the higher number will be punished with a $2 penalty. If both numbers are the same, there is no reward or penalty.

Obviously this does not happen in the actual world, but here is how it plays out in the land of decision theory. If Bill and Jane are both rational (yet not smart enough to contact Ted’s supervisor and report his obvious insanity) they will reason as follows: If I write down $100 and the other person writes down $99, they will get $101 and I’ll get $98. Similarly, if I write down $99 and they write down $98, they will get $100 and I’ll get $97. So, if I write down X and they write down X-1, then I’ll end up with $2 less and they will end up with $2 more. Laying aside honestly writing down the value of the item, the allegedly most rational strategy is to write down the lowest amount, namely $2. If the other person writes down a higher number, then s/he will get nothing and I will get $4. Naturally, the other person realizes this as well, so s/he will write down $2 also. So, Ted ends up giving Bill and Jane $2 each.

This is a very good deal for Ted-he just hands over $4 instead of having to pay up to $200. This shows that it is smart to get people like Bill and Jane into this sort of dilemma.

But, is it smart for Bill and Jane to play this way? Given the reasoning above, the initial answer would seem to be that $2 is their best option. However, this seems intuitively incorrect. This is because they could both walk away with $100. If Bill and Jane see getting more money as better than getting less money, then the $2 option is inferior to the one that yields $100.

If Bill and Jane could communicate (presumably Ted has the power to prevent any interaction, even knowing winks, nods or lip reading between the two) the best strategy would be for them to both write down $100. Each of them goes home with $100 and that is a lot more than $2.

Of course, if Bill and Jane are smart, they would know that this is a better outcome for them-better by $98 each. But, of course, they need to worry that the other person will write down $99 instead of $100 and thus get the extra $2-which takes them back to the situation described above.

In a slightly more realistic world, Bill and Jane would no doubt look at each other and think “is this person such a tool that s/he would throw away $98 just to try to come out $2 ahead of me?” They might also think: “Ted is trying to screw us over with this decision theory crap. If we both play the game the ‘rational’ way, we both get screwed and Ted wins. But, if we both put down $100, we both win and that fruit bat Ted is out $200.” The game becomes more interesting when Bill and Jane see themselves as playing against Ted and not playing against each other.

If Bill and Jane are clever, they will both write down $100, take the money and beat Ted at his silly game. After all, while $101 is better than $100, $100 is much, much better than $2.

If Bill and Jane are vindictive and motivated to simply making sure that the other person does not get more, then the $2 is the best choice for them. And, for being vindictive weasels, they deserve to go home with $2 and not $100.

If I was in this scenario with Jane, I’d put down the actual price of the item. I generally value honesty more than money. But, if Ted insists on being a fruit bat and forcing this game, then I’d put down $100 to try to beat him. If Jane put down $100, I’d high five her and we’d mock Ted for being a loser. If Jane put down $2, I’d happen to mention to airport security that Jane seemed to be muttering something about “death to America” the whole flight and that she was having trouble sitting still, no doubt because she has a bomb hidden in her butt. Sure, having her subjected to a cavity search might be morally questionable, but those are the breaks in the land of decision theory. Naturally, I’d donate my $2 to charity to offset this. But wait, I couldn’t because I’d have to pay the penalty and I’d be walking away with…nothing. Enjoy the cavity search, Jane. 😉