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 . 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,
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 and the cost of ordering the expensive meal is . So the utilities for each meal are for the expensive meal and 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 h and their total utility is . On the other hand if all the individuals had ordered the cheap meal, their utility would have been . Since by assumption, 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.
Read more about this topic: Unscrupulous Diner's Dilemma
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