What value should each traveller write down?
The solution given by game theory
According to wikipedia the rational solution is $2. The justification given is as follows:
A) If Alice only wants to maximise her own payoff, choosing $99 trumps choosing $100. If Bob chooses any value between $2 and $98, then $99 and $100 give the same payoff for Alice (ie $2 less than Bob's choice). If Bob chooses $99, then choosing $99 nets Alice $99, and choosing $100 nets her $97. If Bob chooses $100, then choosing $99 nets Alice $101, and choosing $100 nets her $100. So in all cases she gets more by choosing $99 rather than $100.
B) Similarly, choosing $98 beats $99. If Bob chooses any value between $2 and $97, then $98 and $99 give the same payoff for Alice. If Bob chooses $98, then choosing $98 nets Alice $98, and choosing $99 nets her $96. If Bob chooses $99, then choosing $98 nets Alice $100, and choosing $99 nets her $99. Finally, if Bob chooses $100 then Alice will get $100 by choosing $98 and $101 if she chose $99. So the only case where Alice gets more by choosing $99 rather than $98 is if Bob chooses $100. But Alice believes that Bob is also rational and that he uses the logic of paragraph A to decide that $99 is preferable to $100.
C) The same reasoning can be applied to all of Alice's options until she finally reaches $2, the lowest price.
My solution
The rational argument given above seems to prove that Alice and Bob should both choose $2, yielding only $2. Yet this solution looks wrong, indeed it is wrong. It is wrong because if both Alice and Bob choose $100 then this is what they will get. So how come the rational argument leads to the worst possible outcome? The reason for this paradox is that the above proof treats the decisions of Alice and Bob as independent events. Their choices are not independent because both use the same logic. Because they use the same logic Alice knows that Bob will make the same choice. Knowing this makes $100 the obvious choice. Ask any kid!
Wikipedia and other reputable sources that feature the Travellers' Dilemma omit the last sentence of the first paragraph, ie "Alice and Bob each believe that the other will make a rational choice." The assumption that the other person is rational must be included in the problem statement, because if we don't believe that the other person is going to make a rational choice then we cannot reason logically about it. When the game is played between real people, not hypothetical rational agents, the usual result is that both players choose a value at or close to $100, and so do much better than using game theory. In fact, a research team ran the experiment using professional game theorists playing for real money. Even among game theorists, game theory failed! Nearly a fifth chose $100, and two-thirds chose $95 or higher. So even game theorists don’t believe their own theory.
Nor should they. The game theory analysis is faulty because it does not take account of the hidden feedback loop, ie that Alice and Bob use the same logic, and are aware of doing so.
If the assumption of rationality is removed from the problem statement then $100 is still the best choice, as it is the natural one to make using that much disparaged faculty called common sense. Of course, if you are playing with a die-hard game theorist, or someone who thinks you are a die-hard game theorist, then you may wind up getting zilch. Playing the game without the assumption of rationality is much like playing rock-paper-scissors, ie a game of psychology and chance.
Finally, one can argue that even if Alice and Bob are both rational it could be that this problem has no rational solution. If so, Alice and Bob may not arrive at the same choice, undermining my argument. However, it is enough if they both believe there is a rational solution.