## Saturday, May 26, 2007

### Annals of game theory: Traveller's dilemma and libertarian presumptions

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?

Cornell economist Kaushik Basu has a fascinating piece in the Scientific American on the game of traveller's dilemma. Not surprisingly, the way people answer the above question goes very strongly against the 'rational' solution. As Basu says:

[Traveller's dilemma] undermines both the libertarian idea that unrestrained selfishness is good for the economy and the game-theoretic tenet that people will be selfish and rational.

While on game theory, here's an earlier SciAm article -- The Economics of Fair Play (pdf) -- on Ultimatum and Public Goods games which show a strong evidence for people's sense of fair play, generosity and altruism. Charmingly irrational! The authors -- Karl Sigmund, Ernst Fehr and Martin Nowak -- combine these results with evolutionary psychology to conclude that "in social interactions, our preferences often turn out to be far from selfish.

1. Rahul Siddharthan said...

Hm... I have a rather obvious comment that Prof Basu doesn't address at all: how do the results change if the price is between \$2,000 and \$100,000, rather than between \$2 and \$100, and the penalty is \$2,000 rather than \$2?

After all, in Prof Basu's version, if you guess higher than your partner, you get only \$2 less than what your partner guessed. And if you had guessed lower than your partner you aren't exactly ahead of the game. Ideally you need to guess \$1 less than your partner, and then you make \$1 more than his guess, which leaves you \$3 ahead of what you would have if you guessed too high.

Most people who play Prof Basu's version may figure that a gain of \$1 or \$3 is not worth fighting for: more to the point, they may figure that their partner will think the same way. At the same time, they may not shoot for the full \$100. Something like \$98 may be what I would do, and I would certainly not call it irrational.

However, if a wrong guess costs you \$2000 rather than \$2, then it gets much more interesting.

The experiments Prof Basu cites, conducted by others, were played for much lower rewards than \$2-\$100. They say it's because they wanted to keep the budget manageable. But an experiment with fake money would not be uninformative.

2. Blue said...

The problem with this situation is that it presupposes that people will always choose personal gain over group gain. The link you provided says that the logical mind will always work its way down to \$2, because only \$2 guarantees a "win."

But wouldn't another kind of logical mind bet something that might work out towards a larger group gain? Instead of being assured of a win (even if it is only \$4), why not be assured that both parties will benefit?

Or, for that matter, why not bid the actual price of the item? Because you don't trust your friend to do the same? Because you think your friend is going to try to screw you over? Not much of a friendship, then... ^__^

And finally, why wouldn't the logical mind bid \$100, knowing that if the other party also bid \$100, both parties would take home \$100? Just because bidding \$100 means that you will never get \$101?

3. Krish said...

Do I even have to say that this post got me smiling :-)

4. Unknown said...

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.

5. Unknown said...

Most the comments written before are suffering from a common mistake. the game of "traveller's dilemma" was intentionally devised so as to show that if we follow the 'definition' of
best response and that of Nash Equilibrium then we would reach a situation that is not exactly commonly seen. one can be assured of the fact that bidding anything above \$2 is not "rationalizable" and cannot be supported in the game provided we agree on the basic axioms of rationality. hence bidding anything above \$2 is certainly not "rational" even though it looks like so and bidding anything above \$2 cannot constitute a Nash Equilibrium.
I agree with rahul's point.