Posted by: bridget | 31 May 2007

Travelers, Game Theory, and Philosophy

Scientific American’s latest issue has an article about the benefits of irrationality.  It begins with a “Traveler’s Dilemma,” which is similar to the famous prisoner’s dilemma.  The scenario: an airline loses the identical antiques for two separate, unrelated travelers.  It is difficult to determine the value of the antique, so the airline administrator devises a plan.  The two people will be asked to write the value of the antique on a piece of paper, without conferring with each other.  If the numbers are the same, the travelers will get the value of the antique.  If the numbers are different, he will assume that the person with the higher number is cheating, pay him $2 less than the stated value, and pay the other traveler $2 more than the value as a reward for honesty.  The travelers may write any number between $2 and $100.

Game theorists believe that the Nash equilibrium is $2 – the point at which the traveler will not be penalised for his choice and will likely be rewarded with $4.  A player, for example, who states that the antique is worth $100 will likely “lose:” the other person will write a lower number and he will be given that lower number, minus $2.  Moving backwards, the safest strategy is to bet $2.

When players give higher numbers, they are presumed to be acting irrationally.   This is inaccurate.  Within the context of the game, the travelers paid a certain price for the antique – let’s say $40.  It is irrational to write anything less than $38 – the point at which the $2 reward for writing down the lower price does not exceed the value of the antique.  (A player who writes down $39 could win $41, if the other player writes down $40; however, the chance that both players will write down $39 reduces the expectation value of the $39 bet.)

Moreover, there is no difference between writing down $4 and matching the other player’s prediction of $4, writing down $2 and winning the $2 bonus.  A person who writes down $4 has four outcomes: earn either $1 or $0 if the other player writes down $2, earn $4 if the other player also writes down $4, and earning $6 if the other player writes down a higher number.  The expectation value (really, the most rational way to examine the behaviour) depends on the likelihood of the other person choosing those options.

Dr. Basu places a high emphasis on competitive behaviour: to him, the reward of winning the $2 is very high and he associates high costs with losing $2.  In reality, the expectation value should govern, not whether or not the original bet was reduced or increased.  If a player writes down $100, believing that there is a 90% chance that the other player will write down a lesser number, he is better off than if he wrote $2.  Assuming, for the sake of argument, that he is opposing one of ten people, one of whom will write down $100 and nine of whom will write down $2.  He should still write $100: his expectation value is $10, whereas it would be $2.40 if he wrote down $2.   Granted, a person could write down $99 and win an extra $2 if his opponent wrote $100, but, beyond $98, the potential gain ($2) of being the “right” traveler is tempered by the fact that one could have more money by guessing the highest possible number.

Ultimately, people are not acting irrationally by calculating expectation value.  They need not maximise their worst-case scenario (which would cause them to both bet $2 every time) so much as maximise the expectation value of their guess.

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