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Diner’s dilemma

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In game theory, the Diner’s dilemma is an n-player Prisoner’s dilemma. The situation imagined is several individuals go out to eat, prior to ordering they agree to split the check equally between all of them. Each individual must now choose whether to order the expensive or inexpensive dish. It is supposed that the expensive dish is better than the cheaper, but not better enough to warrant paying the difference if you were eating alone. Each individual reasons that the added expense she adds to her bill by ordering the more expensive item is very small, and thus the improved gustatory experience is worth the money. However, every individual reasons this way and they all end up paying for the cost of the more expensive meal, which, by hypothesis, is worse for everyone than ordering and paying for the cheaper meal.

Formal definition and equilibrium analysis

Let g represent the joy of eating the expensive meal, b the joy of eating the cheap meal, h is the cost of the expensive meal, l the cost of the cheap meal, and n the number of players. From the description above we have the following ordering <math>h>g>b>l</math>. Also, in order to make the game sufficiently similar to the Prisoner’s dilemma we presume that one would prefer to order the expensive meal given others will help defray the cost, <math>g - \frac{1}{n}h > b - \frac{1}{n}l</math>

Consider an arbitrary set of strategies by a player’s opponent. Let the total cost of the other player’s meals be x. The cost of ordering the cheap meal is <math>\frac{1}{n}x + \frac{1}{n}l</math> and the cost of ordering the expensive meal is <math>\frac{1}{n}x + \frac{1}{n}h</math>. So the utilities for each meal are <math>g - \frac{1}{n}x - \frac{1}{n}h</math> for the expensive meal and <math>b - \frac{1}{n}x - \frac{1}{n}l</math> for the cheaper meal. By hypothesis the utility of ordering the expensive meal is higher. Remember that the choice of opponents’ strategies was arbitrary and that the situation is symmetric. This proves that the expensive meal is strictly dominant and thus the unique Nash equilibrium.

If everyone orders the expensive meal all of the diners pay h and their total utility is <math>g-h<0</math>. On the other hand suppose that all the individuals had ordered the cheap meal, their utility would have been <math>b-l>0</math>. This demonstrates the similarity between the Diner’s dilemma and the Prisoner’s dilemma. Like the Prisoner’s dilemma, everyone is worse off by playing the unique equilibrium than they would have been if they collectively pursued another strategy.

Experimental Evidence

Gneezy, Haruvy and Yafe (2004) tested these results in a field experiment. Groups of six diners faced different billing arrangements. As predicted, subjects consume more when the bill is split equally than when they have to pay individually. Consumption is highest when the meal is free. Finally, members of some groups had to pay only one sixth of their individual costs. There was no difference between the amount consumed by these groups and those splitting the total cost of the meal equally. As the private cost of increased consumption is the same for both treatments but splitting the cost imposes a burden on other group members, this indicates that participants did not take the welfare of others into account when making their choices. This contrasts to a large number of laboratory experiments where subjects face analytically similar choices but the context is more abstract. It could be that laboratory experiments are misleading in predicting behavior in real-world settings.

References

Glance, Huberman. (1994) “The dynamics of social dilemmas.” Scientific American.

Gneezy, U., E. Haruvy, and H. Yafe (2004) “The inefficiency of splitting the bill: A lesson in institution design” The Economic Journal 114, 265-280.