93. The Traveller's Dilemma

'There is something rational about choosing not to be rational when playing Traveller’s Dilemma' (Kaushik Basu 2007).

The Traveller’s Dilemma (TD) game was crafted in 1994 by Kaushik Basu. Lucy and Pete go to a remote island in the Pacific, where they purchase two identical antiques. The antiques get damaged during their return journey, and the airline manager offers to compensate, except that he has no idea about the actual price. To make sure that the airline does not have to pay more than what is necessary, he comes up with a game plan. He asks each of them to write down simultaneously and without consulting each other the price of each antique as any integral number of dollars between 2 and 100, with the following proviso: If both write the same number, he will pay that amount to each of them. But if they write different numbers, he would conclude that the lower price is the correct one, and that the other person is trying to cheat. Therefore, the person writing the lower price will be rewarded for honesty with an extra two dollars, and the other person will be punished with a payment that is two dollars less than the lower of the two prices submitted by the travellers. For example, if Lucy writes $54 and Pete writes $100, then Lucy will be paid $56, and Pete will get $52. What will the two travellers write to maximize their gains? That is the Traveller’s Dilemma.

http://mybanyantree.wordpress.com/2007/05/23/the-travelers-dilemma/

Although most people will choose a number 100 or one close to 100, logic demands that both players should choose 2. The actual price of each antique was considerably lower than $100, but let us first suppose that Lucy decides to write $100, hoping that Pete would be similarly greedy. Lucy’s logic is that if they both wrote $100, they would both get $100. But she soon realizes that she can get a little more money by writing 99, because she would then get $101 if Pete has written $100. But the same thought can occur to Pete also, in which case Lucy would be better off by writing 98. She would then get $100, if Pete wrote 99 or 100. But once again, if this idea occurs to Pete also, Lucy should be writing $97. This kind of logic (called backward induction) can go on till both should write $2. So if both players in the TD game are rational, their joint payoff should be (2, 2). The table below (from Basu 2007) gives the complete payoff matrix for this two-player strategic game.

	2	3	4	…	98	99	100
2	2  2	4  0	4  0	…	4  0	4  0	4  0
3	0  4	3  3	5  1	…	5  1	5  1	5  1
4	0  4	1  5	4  4	…	6  2	6  2	6  2
…	…	…	…	…	…	…	…
98	0  4	1  5	2  6	…	98   98	100   96	100   96
99	0  4	1  5	2  6	…	96  100	99   99	101   97
100	0  4	1  5	2  6	…	96  100	97  101	100  100

The leftmost column in this table gives Lucy’s choices in dollars, and the top row those of Pete. The square where a chosen row and a chosen column intersect shows the payoffs of Lucy and Pete respectively. For example if Lucy chooses 4 and Pete chooses 98, the payoff to Lucy is $6 and that to Pete is $2.

The payoffs (2, 2) when both players choose 2 represent Nash equilibrium: Any unilateral deviation from the Nash choices gives a lower payoff. For example, if Lucy stays at the equilibrium by choosing 2, then Pete does worse by choosing any number other than 2. To give an example of an outcome that is not a Nash equilibrium, consider (100, 100). Lucy will be better off by choosing 99, because then the outcome will be (101, 97), implying that she would do better by deviating from (100, 100).

http://www.quora.com/Game-Theory/How-does-Nash-equilibrium-explain-the-concept-of-selfish-gene

We note that if each player in the TD has a choice of only 2 or 3, instead of every integer from 2 to 100, we get a payoff table similar to the table for the Prisoner's Dilemma (PD) I gave in Part 92. However, unlike in PD, there is no dominant choice (cf. Part 92) in the full version of the TD depicted in the above table. If Pete chooses any number from 4 to 100, Lucy would do well to choose a number greater than 2.

The existence of the Nash equilibrium in the TD game presents a logical paradox. Although it is a rationally correct solution, most of us would choose a number much higher than 2. It appears that our intuition contradicts the tenets of game theory.

Classical economic theory was based on the axiom that people make choices that classical game theory can predict, and classical game theory assumes that people are selfish and rational. But in practice, as demonstrated by several experiments in which real money was used for participating persons, and in which a variety of magnitudes of rewards and punishments were tested (Basu 2007), people do not play the Nash strategy on average. For low rewards, the average choice of numbers was high, and it fell somewhat when the rewards were increased. Repeated playing of the TD threw up some surprises as well. For high rewards, the play converged over time down to the Nash-equilibrium value. But for low rewards, the repeated play went in the other direction, i.e. more and more away from Nash equilibrium. Such experiments come under the relatively new field of research called experimental economics.

Why do people behave in such an illogical irrational manner? Why do they not follow the rational path to Nash equilibrium? Several reasons have been advanced (Basu 2007):

1. Many people are not capable of deductive reasoning, and end up making illogical, impulsive, irrational moves. But then why do even game-theorists, who presumably do not belong to this category, and who know how to reason deductively, end up making irrational choices? It turned out in real experiments that they chose high numbers because they expected other participating game-theorists to choose high numbers. Why?

2. Perhaps there is a tussle between altruism and selfishness in our brains, with simple-minded Darwinism favouring selfishness.

3. Perhaps many of us do not like to ‘let down’ our fellow traveller in the TD game just for the sake of getting rewarded an additional dollar. So we may choose 100, knowing fully well that choosing 99 can give us $101.

This is really an unsolved problem. Even if we eliminate factors such as faulty reasoning, altruism, and socialization, it is still doubtful if people would play logically and ruthlessly. People don't always make selfish rational choices (Ball 1999).