2. Traveller's Dilemma

Introduction

. Consider this situation:

You and your partner are flying home with identical valuable mementos of your trip. Both collections are damaged in flight. The airport manager states that he/she is happy to compensate you but is handicapped by not knowing the value of your mementos. He/she thinks that by simply asking the price from the travellers, they will inflate it. Instead, he asks each of them to write down the value of the mementos from $2 to $100 without discussing it together. So, if both write the same number, he/she will take that to be the true price, and will pay each of them that amount. But if they write different numbers, he/she will assume that the lower is the actual price and that the person writing the higher price is cheating. If this happens, they will be paid the lower number along with a bonus and a penalty - the person who wrote the low number will get $2 more as a reward for honesty and one who wrote the high number will get $2 less as a punishment. For example, if one writes 46 and another 100, the first one gets $48 and the other $44.167

What numbers would they write?

Comments on the activity

. This activity establishes a realistic context in which one or more individuals have choices to make and will be rewarded according to those choices.

. The objectives of this activity are

"...to contest the narrow view of rational behaviour and cognitive processes taken by economists and many political scientists, to challenge the libertarian presumptions of traditional economics and to highlight a logical paradox of rationality..."

Kaushik Basu, 2007

. The rational choice dictates that $2 is the best option, yet most people pick $100 or a number close to 100. If people do not chose 2, they have not thought through the logic and are deviating markedly from the rational choice. Furthermore, players reap a greater reward by not adhering to reason in this way, ie by acting illogically, they end up reaping a greater reward. So this type of outcome demands a new kind of formal reasoning.

. The rationale for selecting 2 as the logical choice is based on a style of analysis called a backward induction, which is commonly used by game theorists. Consider a plausible line of thought that might be pursued: the first idea would be to write the largest possible number, ie 100 which will earn him/her $100 and similarly for his/her partner (if the partner is similarly greedy). Furthermore, if the memento cost considerably less than $100, he/she will be happily thinking about the stupidity of the proposed activity. Then he/she realizes if $99 is nominated, he/she could make a little more money, because in that case he/she will get $101. On the other hand, the partner could have the same thought, ie written $99. Thus they would both receive $99. If the partner wrote $99, then the other would do better by writing $98, in which case he/she would get $100. Yet the same logic would lead the partner to choose $98 as well. In that case, he/she would deviate to $97 and earn $99; and so on. Continuing on this line of reasoning, the travellers spiral down to the smallest permissible number, namely, $2. This means that the 2 players earn $98 less than they would if they each naively chose 100 without thinking through the advantages of picking a small number. It is highly implausible that this would happen, ie choosing 2; on other hand, this is where logic would lead us.

. The game theory approach is better understood by examining a payoff matrix - a square grid containing all the relevant information about potential choices and payoffs for each player. The matrix for the Traveller's Dilemma (TD)

Other partner's choice (dollars)

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. For TD, the payoff matrix is with the first partner's choice in the left most column; other partner's across the top row. The first number in the square at the intersection of the chosen row and columns is the first partner's payoff and the second number is the other partner's payoff. For example, the first partner chooses 98 and the second 99, then the first partner receives $100 and the other partner receives $96. The outcome in which both players choose to earn $2 (see shaded box) is called the Nash equilibrium (an outcome from which no player can do better by deviating unilaterally). Both players perform badly if they choose any other number than 2. The choice of 2 is called the dominant choice because it is the best thing to do - no matter what the other player does.

"...game theory insists that rationality could lead players to select 2, but most people pick an integer closer to 100. A new kind of reasoning is needed to gain a vigorous understanding of this rational choice not being rational. The results of Traveller's Dilemma contradict economists' assumption that standard game theory can predict how supposedly selfish rational people will behave. They also show how selfishness is not always good economics..."

Kaushik Basu, 2007

. TD helps us understand how competing firms may undercut each other's price to their own detriment.

. It has been suggested that the choice of 100 is a spontaneous emotional response; picking a number between 90 and 99 involves some strategic reasoning, eg some amount of backward induction, etc and takes the longest time in making a decision about a certain number, ie players give more thought to their selection; anything between 2 and 90 is regarded as a random choice.

. Remember: what is rational to one player is not necessarily to another, ie rationality is not necessarily common knowledge between players and is a source of conflict between logic and intuition. In TD, intuition wins.

. In trying to explain the apparently irrational behaviour in which most players choosing a high number and expecting other players to do the same, some have suggested

"...altruism is hardwired into our psyche alongside selfishness, and a behaviour results from a tussle between the two......altruism, socialization and faulty reasoning guide most individuals' choices. Yet I do not expect that many would select 2 if those three factors were all eliminated from the picture...... the idea of behaviour generated by rationally rejecting rational behaviour is a hard one to formalize..."

Kaushik Basu, 2007

(source: Kaushik Basu, 2007)

 

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