# Unscrupulous diner's dilemma

In game theory, the unscrupulous diner's dilemma (or just diner's dilemma) is an n-player prisoner's dilemma. The situation imagined is that several individuals go out to eat, and 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 presupposed that the expensive dish is better than the cheaper, but not by enough to warrant paying the difference when eating alone. Each individual reasons that the expense s/he adds to their bill by ordering the more expensive item is very small, and thus the improved dining experience is worth the money. However, having all reasoned thus, they all end up paying for the cost of the more expensive meal, which by assumption, is worse for everyone than having ordered and paid for the cheaper meal.

## Formal definition and equilibrium analysis

Let a represent the joy of eating the expensive meal, b the joy of eating the cheap meal, k 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 ${\displaystyle k-l>a-b}$. 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, ${\displaystyle a-{\frac {1}{n}}k>b-{\frac {1}{n}}l}$

Consider an arbitrary set of strategies by a player's opponent. Let the total cost of the other players' meals be x. The cost of ordering the cheap meal is ${\displaystyle {\frac {1}{n}}x+{\frac {1}{n}}l}$ and the cost of ordering the expensive meal is ${\displaystyle {\frac {1}{n}}x+{\frac {1}{n}}k}$. So the utilities for each meal are ${\displaystyle a-{\frac {1}{n}}x-{\frac {1}{n}}k}$ for the expensive meal and ${\displaystyle b-{\frac {1}{n}}x-{\frac {1}{n}}l}$ for the cheaper meal. By assumption, 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 k and the utility of every player is ${\displaystyle a-k}$. On the other hand, if all the individuals had ordered the cheap meal, the utility of every player would have been ${\displaystyle b-l}$. Since by assumption ${\displaystyle b-l>a-k}$, everyone would be better off. 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.[1]

## Experimental evidence

Gneezy, Haruvy, and Yafe (2004) tested these results in a field experiment. Groups of six diners faced different billing arrangements. In one arrangement the diners pay individually, in the second they split the bill evenly between themselves and in the third the meal is paid entirely by the experimenter. As predicted, the consumption is the smallest when the payment is individually made, the largest when the meal is free and in-between for the even split. In a fourth arrangement, each participant pays only one sixth of their individual meal and the experimenter pay the rest, to account for possible unselfishness and social considerations. 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.[2]

## References

1. ^ Glance, Natalie S.; Huberman, Bernardo A. (March 1994). "The dynamics of social dilemmas". Scientific American.
2. ^ Gneezy, Uri; Haruvy, Ernan; Yafe, Hadas (April 2004). "The inefficiency of splitting the bill" (PDF). The Economic Journal. 114 (495): 265–280. doi:10.1111/j.1468-0297.2004.00209.x. Archived (PDF) from the original on 2015-06-08. Retrieved 2015-06-08.