Unscrupulous diner's dilemma
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 people go out to eat, and before ordering, they agree to split the cost equally between them. Each diner must now choose whether to order the costly or cheap dish. It is presupposed that the costlier dish is better than the cheaper, but not by enough to warrant paying the difference when eating alone. Each diner reasons that, by ordering the costlier dish, the extra cost to their own bill will be small, and thus the better dinner is worth the money. However, all diners having reasoned thus, they each end up paying for the costlier dish, which by assumption, is worse than had they each ordered the cheaper.
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 <math>k-l>a-b</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>a - \frac{1}{n}k > b - \frac{1}{n}l</math>
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 <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}k</math>. So the utilities for each meal are <math>a - \frac{1}{n}x - \frac{1}{n}k</math> for the expensive meal and <math>b - \frac{1}{n}x - \frac{1}{n}l</math> 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 <math>a-k</math>. On the other hand, if all the individuals had ordered the cheap meal, the utility of every player would have been <math>b-l</math>. Since by assumption <math>b-l>a-k</math>, 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.