Challenging problem?

Question

This one is called the "Traveler's Dilemma"

You and another passenger lose identical souvenirs on an airline. The manager cuts you a deal - you can both write a dollar amount on a piece of paper, between 2 and 100. If you both write the same amount, you both get that amount. However, if one person writes y and the other writes x, and y < x, then the person who wrote y gets y + 2, and the person who wrote x gets y - 2.

What moves will players rationally play?

Answer 1

I can't tell you what move they'd "rationally" make, but I can tell you what their statistical best move would be. Assuming that they are not allowed to collaborate, each player's best play would be 96 or 97.

Let's suppose I play n and the other player's guess is completely random.
Then if he chooses n, I win n.
If he chooses m < n, I win m-2.
If he chooses m > n, I win n+2.

X = my winnings, given that I played n
P(X = m-2) = 1/99 for 2 ≤ m < n [n-2 such cases]
P(X = n) = 1/99 [1 such case]
P(X = n+2) = (100-n)/99

My expected winnings,
E(X | n) = 1/99 * [(0+1+2+...+n-3) + n + (n+2)(100-n)]
= (406 + 193n - n^2) / 198

This is maximum when dE/dn = 0
or when n = 96.5
So if I bet 96 or 97, my expected winnings would be maximum at 49.08.

Of course the curve is not very steep at the high values of n, for example E(X | n=100) = 49.02. So generally it's better to write as high a number down as possible.

*EDIT*
But putting aside statistics, it makes logical sense to pick as large a number as possible. If you pick a small number, you doom yourself to small winnings, independent of what the other person picks. If you pick a large number, at least you have a chance of a large winning. If the other guy chooses a small number, then your pick doesn't matter much anyway. So the only way you have a chance of a large winning is to pick a large number.

Answer 2

First, this dilemma has to assume that both x and y are given this offer, and not able to collaborate afterwards, because otherwise obviously both x and y should put down $100.

The flaw to this question is the term "rationally". Rational according to what or whom? If I merely want to ensure that I'll get the same or more than the other person, I should put down $2. However, if I want to maximize reimbursment by the airline, I should put down $100. Then I would get $2 less whatever the other peson had the brains to put down, and it shouldn't matter to me if he actually ended up getting $4 than I did. This is one of the problems with game theory, because assumptions about "most rational course of action" could be off target or unrealistic.

Addendum: Dr D's analysis is interesting, because to him, rationally playing this is to assume that the other will pick a number at random. Indeed? Yes, I agree with his analysis if the other number is picked randomly, but is that really "playing it rationally"? As I said, this is what's wrong with game theory.

Answer 3

Both will play 100. Reasoning:

If other player plays 100, I get 100, if other player plays <100, I get 102. Therefore my worst case scenario is 100.