Traveler's dilemma

The holiday is a dilemma in 1994 by Kaushik Basu thought up, game theoretical thought experiment in which the participants can earn more profit by game theory wrong action than the "correct" solution. The original English title of " traveler's dilemma" is not to be confused with the "traveling salesman problem", ie the problem of the traveling salesman. The dilemma is not a zero -sum game, because there are always positive values, ie profits, paid, even if the advantage of a player's disadvantage of the other player is the same.

Framework for action

The background story exists in several versions, as Basu 's dilemma repeatedly published the while further embellished. The version shown here is derived from an article from the magazine " Spektrum der Wissenschaft ", probably the first German declaration of the dilemma.

Tanja and Mark have indeed made ​​at the same time on the same remote Pacific island holiday; but they learn only after the flight back to his native airport know - in the compensation department office. The airline has zerdeppert namely the antique vases, each of which of the two had bought a copy locally. The agent recognizes their claim without further notice, however, with the best will not assess the value of works of art. From a survey of travelers he promises, apart from large exaggerations, precious little. Upon reflection, therefore, he decides to trick a richer approach. He asks both to write independently the value of the vase in Euro on a piece of paper, as an integer between 2 and 100 Any prior agreement is of course forbidden. But what he is known in advance, the payment procedures: Give both the same value, it will consider this to be the actual purchase price and pay it to each of them. The data differ, however, he will keep the lower quotation for true and the higher for a fraud attempt. In this case, both receive a refund of the lower amount - but with a difference: The one of two who has written the lower value, gets 2 euro more than a reward for honesty, the other is a penalty fee of 2 euros deducted. Selects Tanja So, for example 46, but Markus 100, so she gets 48 € and he only 44

The paradox

The amazing thing about this game is that the game theory predicts, would rationally by players to choose a value of 2 €. This answer of course contradicts common sense, but to understand by some logical considerations.

Tanja and Mark - or abstract A ​​and B - will consider how the act is each other. The first choice is logically 100, since then the most profit can be achieved. However, Player A 's payoff even increase to 101 by specifying 99 and occupies the bonus. Since Player B thinks the same as Player A - this is one of the properties, the "rational" summarizes the game theory, the term - he will come to the same conclusion, so now select both 99. A knows that B feels the same way, and tries to increase again in the same way his payoff: it selects the next lower value of 98, giving him (selects B still 99) the bonus payment and after all, still a payout of 100 brings. B will now follow suit again be undercut by the same conclusions of A, etc. The result is that there will be any number is a better, and, although the respective lower. So is the logical choice for both players 2 through the deviation by one unit (ie, 3 ) you can only cause a deterioration, regardless of what the other player chooses that option 2 is more favorable. So here is the so-called Nash equilibrium of the game. The choice of the equilibrium strategy 2 by both players is ultimately anything but advantageous since only minimum payments can be achieved.

The mistake

We have to distinguish at least three possible goals of the people involved. The choice of 2 euros for player A under the objective as possible not less than winning player B correct and understandable. Pursued by a player, the goal is to achieve the highest possible total payoff amount of the insurance, he will choose 100 euros. More difficult is the decision for player A, when it comes to an overall profit maximization him. Only if it is assumed that players with negligible probability selects a higher amount than 3 € B, he will choose even 2 euros. Player B will be more likely themselves strive for a Gesamtgewinnmaxinierung and call a high amount.

It is true that players can not perform worse A when changing from 100 to 99 euros. Player B has chosen 100 euros, so Player A wins 101 euros, 99 euros chose Player B, Player A receives as well as 99 euros, but compared to 97 euros in the original choice of 100 euros. A change from 100 to 98 euros is also useful. A change from 99 to 98 euros but it is not in every case. Assuming that player B chooses amounts in the upper area with approximately the same probability, a change from 97 to 96 euros with no advantage would be more connected.

Mixed strategies as a possible explanation

One way to approach the human behavior is based on the probability theory rather than game theory. Players choose a specific value ( 2-100 ), but each value with a certain probability. Since Player A does not know how B chooses his chances, he can take, for example, a uniform distribution. For each choice of A can now calculate its expected value. Assuming the probability that A selects a certain value, is proportionate to the payout that he can expect on average in the choice of this value when B holds its own distribution ( the expected value ), you can see the distribution calculate the probabilities of A. The result can now be used instead of the uniform distribution for B. If you repeat the process with the new output distribution results in a different distribution which in turn can enter as initial distribution. By repeatedly performing the distribution to a limit distribution with the maximum converges in 97

Real behavior of people in the tourist dilemma

Over time, several experiments were conducted to determine the behavior of "real" people in the tourist dilemma. Almost always stated ( at low bonuses ) the vast majority of the maximum ( in the original version 100), the rest is distributed to approximately equally among the three alternatives: Nash equilibrium values ​​just below the maximum and random values ​​in between. In any case, the average of these values ​​was relatively high.

A real player will not simply accept the above calculated Nash equilibrium, but represent parts of the logical chain of conclusions into question. Maybe he will consider the insurance as another antagonist or set the potential gains in relation to an uninvolved fictional other players.

One must note that the change is returned only excluded from 99 to 100 euros for a strictly logical player when he sees the game as a pure battle between Tanya and Mark. By changing back the other player would have the lower number is specified and the changer would get the deduction. Even a change of both players is impossible, since each considered the game only from his perspective. The original 100 x 100 table so to speak, has been reduced to a 99 x 99 table. By backward induction, only one cell with the value 2 remains under this assumption at the end.

Contrary to theoretical considerations, a person will judge in the described situation, the attention to personal profit maximization. The comparison with the other Insured is of secondary importance to him. He will prefer to choose the highest possible amount in order to preserve the opportunity for a high profit. Nonsensical would be to choose the smallest possible amount to only cut off 2 euros more than the opponent of this view. The fact that the other player also chooses a correspondingly high number that action pays off only. Basu calls this an " overarching rationality ".

Parallels to other problems

The holiday dilemma is basically a generalization of the well-known prisoner problem. This corresponds to a case of the holiday dilemma with the lower limit and the upper limit of 2 3, which is the upper left four cells of the payoff matrix. Therefore, the prisoner's dilemma raises similar difficulties as the vacationers dilemma; the difference between human choice and the prediction of game theory, however, occurs when vacationers dilemma much more to light.