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 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 $h-l>g-b$ . 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, $g-{\frac {1}{n}}h>b-{\frac {1}{n}}l$

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 ${\frac {1}{n}}x+{\frac {1}{n}}l$ and the cost of ordering the expensive meal is ${\frac {1}{n}}x+{\frac {1}{n}}h$ . So the utilities for each meal are $g-{\frac {1}{n}}x-{\frac {1}{n}}h$ for the expensive meal and $b-{\frac {1}{n}}x-{\frac {1}{n}}l$ 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 h and their total utility is $g-h$ . On the other hand, if all the individuals had ordered the cheap meal, their utility would have been $b-l$ . Since by assumption $b-l>g-h$ , 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

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

External links

If You're Paying, I'll Have Top Sirloin by Russell Roberts

Topics in game theory

Definitions	Normal-form game Extensive-form game Escalation of commitment Graphical game Cooperative game Succinct game Information set Hierarchy of beliefs Preference

Equilibrium concepts	Nash equilibrium Subgame perfection Mertens-stable equilibrium Bayesian Nash equilibrium Perfect Bayesian equilibrium Trembling hand Proper equilibrium Epsilon-equilibrium Correlated equilibrium Sequential equilibrium Quasi-perfect equilibrium Evolutionarily stable strategy Risk dominance Core Shapley value Pareto efficiency Gibbs equilibrium Quantal response equilibrium Self-confirming equilibrium Strong Nash equilibrium Markov perfect equilibrium

Strategies	Dominant strategies Pure strategy Mixed strategy Tit for tat Grim trigger Collusion Backward induction Forward induction Markov strategy

Classes of games	Symmetric game Perfect information Simultaneous game Sequential game Repeated game Signaling game Screening game Cheap talk Zero-sum game Mechanism design Bargaining problem Stochastic game n-player game Large Poisson game Nontransitive game Global games Strictly determined game Potential game

Games	Prisoner's dilemma Traveler's dilemma Coordination game Chicken Centipede game Volunteer's dilemma Dollar auction Battle of the sexes Stag hunt Matching pennies Ultimatum game Rock-paper-scissors Pirate game Dictator game Public goods game Blotto games War of attrition El Farol Bar problem Fair division Fair cake-cutting Cournot game Deadlock Diner's dilemma Guess 2/3 of the average Kuhn poker Nash bargaining game Prisoners and hats puzzle Trust game Princess and monster game Monty Hall problem Rendezvous problem

Theorems	Minimax theorem Nash's theorem Purification theorem Folk theorem Revelation principle Arrow's impossibility theorem

Key figures	Albert W. Tucker Amos Tversky Ariel Rubinstein Daniel Kahneman David K. Levine David M. Kreps Donald B. Gillies Drew Fudenberg Eric Maskin Harold W. Kuhn Herbert Simon Hervé Moulin Jean Tirole Jean-François Mertens John Harsanyi John Maynard Smith John Nash John von Neumann Kenneth Arrow Kenneth Binmore Leonid Hurwicz Lloyd Shapley Melvin Dresher Merrill M. Flood Oskar Morgenstern Paul Milgrom Peyton Young Reinhard Selten Robert Aumann Robert B. Wilson Roger Myerson Samuel Bowles Thomas Schelling William Vickrey

See also	All-pay auction Alpha–beta pruning Bertrand paradox Bounded rationality Combinatorial game theory Confrontation analysis Coopetition List of game theorists List of games in game theory No-win situation Topological game Tragedy of the commons Tyranny of small decisions

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Unscrupulous diner's dilemma

Formal definition and equilibrium analysis

Experimental evidence

See also

References

External links