Friday, May 25, 2007

The Traveller's Dilemma

Two people independently pick a dollar value between 2-100. If they pick the same number, then both are paid this amount. Otherwise, the lowest value is paid out to each, with the greedier person paying a $2 penalty to the other player in addition. If you want to maximize your earnings, what number should you pick? Cue the game theorists:
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.

Jason Kuznicki has an interesting response:
The smallest figure only becomes a reasonable strategy when neither Pete nor Lucy have any guidance whatsoever about how the other traveler might respond. In the world of actual prices, this never happens. In other words, the $2 solution is only plausible when it is entirely divorced from economics, and when neither player has any cues at all for giving an answer.

I don't think market cues are relevant here. There's a perfectly salient "default option" even in the abstract case, namely, the maximum value of $100. Suppose the market price is $50. The recursive "race-to-the-bottom" reasoning will apply just as disastrously from 50-49 as it originally did from 100-99. Changing our starting point doesn't really change anything. It's the reasoning that's the problem.

The real solution, then, is to affirm norms of global rationality: look at the big picture, and reason according to a decision-procedure that will predictably yield better results. That means ditching the economist's "local rationality" of backwards induction and its race to the $2 bottom. The SciAm writer has it right:
If I were to play this game, I would say to myself: "Forget game-theoretic logic. I will play a large number (perhaps 95), and I know my opponent will play something similar and both of us will ignore the rational argument that the next smaller number would be better than whatever number we choose." What is interesting is that this rejection of formal rationality and logic has a kind of meta-rationality attached to it. If both players follow this meta-rational course, both will do well. The idea of behavior generated by rationally rejecting rational behavior is a hard one to formalize. But in it lies the step that will have to be taken in the future to solve the paradoxes of rationality that plague game theory and are codified in Traveler's Dilemma.

6 comments:

  1. I don’t see how that game theoretic argument is even rational since it fails to address the stated goal of maximizing earnings. If you pose the question, do I ever get a large return by being sure to underbid my partner, you can use the game theoretic argument to answer it no. Thus there is no return on an underbidding strategy. So then, if you are not going to underbid, the best strategy is to just bid a hundred and accept that your return will be whatever your partner bids less two. We all know enough about humans to know that the typical person will not bid two even if we don’t know how to predict how they actually will bid. Or another way of putting this is that if you grant that there is any chance that your partner might not have used the game theoretic argument, then you shouldn’t either: doubting the rationality of the argument shows that it is not, in fact, rational or at least depends on an unbelievable assumption.

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  2. What matters here is that people tend to have strategis as opposed to loking at each incident individually - part of this is because you cant tell that yu wont met the other guy again and you cant be bothered rethinking your strategy every time.

    You can reasonably assume that the other player plays the game a certain way (for example one of the generous methods) and that he is not entirely neutral to your wellbeing.

    So I'd probably pick the maximum ($99). Unless I had reason to believe you were a bad person....

    GNZ

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  3. ($100, $100) is also a perfectly logical outcome, if the assumption is that both players look to maximize the total return of the game (acting social). One problem with the stated version of the game is that there is little difference between the 'individually maxing bet ($99)', and the 'socially maxing bet ($100)'. A better setup for the game might be an exponential distribution, at least when applied in a real-world experiment - in other words: at what point would people starting 'feeling' the cost of acting social, and act individualistically'?
    WMM

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  4. btw to explain why I picked 99,
    "between 2-100" - 100 isn't an option because it is not between those two numbers.

    I wonder why no-one spotted that?

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  5. I've just came across this topic when I was reading Scientific America today.

    Although the topic is pretty interesting, I think this 'game' should be treated as a practical case.

    Originally, it was described as a 'game' offered by an airline representative, to compensate for two travelers' antiques, that were identical and broken during the same flight. He didn't want to ask the owners individually, as he believed they would make up some huge numbers, and thus he offered this game to test for honesty and so on...

    In the article that I've read, they've described the ($2,$2) solution is the Nash Equilibrium. Although I don't see any problems with their 'logic', I think there are problems because two important points have been overlooked.

    1) There is an actual price for the antique, and thus it wouldn't make sense for one to ask for $2; it just won't even pay off!

    2) Although one would hope to maximize his/her own earnings, I don't see why there should any incentive at all to 'earn more than the other person'. If I were to play this game, I would pick $100 (or occasionally %99).

    The reason for this decision is that even if the other party saw through my mind and chose $99. I would still be able to go home with 97. However, if they played the safest choice ($2), I would go home with nothing. On the other hand, if I were logical and chose $2, then I would go home with $2 or $4 at best. Comparing these outcomes, I would rather 'risk' getting nothing hoping to earn a large number, than to play safe and get $2 for sure.

    3) Having said these, it is now clear that the choices people tend to make are dependent (and quite heavily too IMHO) on the game parameters.

    a) Reward/Penalty: At one extreme, if the reward/penalty is heavy and greatly exceed the minimum value (e.g. it is possible for a traveler to be 'fined'), then one would have more incentive to be cautious and choose the smallest allowable number. One the other hand, if the reward is infinisimal, then one is negligibly less worse off even when the other other person is the one who've written a smaller number, and thus it's only 'natural' to choose the biggest number possible.

    b) the upper and lower limits: similar arguments can be presented to affect choices that one would make.

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  6. "Originally, it was described as a 'game' offered by an airline representative..."

    I'm pretty sure the "practical" details were merely filled in for the sake of exposition.

    "two important points have been overlooked"

    I'm dubious...

    1) Your past payment is a "sunk cost" -- it makes no difference to how you can maximize your profits now.

    2) While you're right to reject the idea of "any incentive at all to 'earn more than the other person'", I don't think anyone else was making that mistake either. (It's not assumed in any of the game theoretic reasoning, for instance.)

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