# Unscrupulous diner's dilemma

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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.

Game theory is the study of mathematical models of strategic interaction in between rational decision-makers. It has applications in all fields of social science, as well as in logic and computer science. Originally, it addressed zero-sum games, in which each participant's gains or losses are exactly balanced by those of the other participants. Today, game theory applies to a wide range of behavioral relations, and is now an umbrella term for the science of logical decision making in humans, animals, and computers.

In game theory, an n-player game is a game which is well defined for any number of players. This is usually used in contrast to standard 2-player games that are only specified for two players. In defining n-player games, game theorists usually provide a definition that allow for any (finite) number of players.

The prisoner's dilemma is a standard example of a game analyzed in game theory that shows why two completely rational individuals might not cooperate, even if it appears that it is in their best interests to do so. It was originally framed by Merrill Flood and Melvin Dresher while working at RAND in 1950. Albert W. Tucker formalized the game with prison sentence rewards and named it "prisoner's dilemma", presenting it as follows:

Two members of a criminal gang are arrested and imprisoned. Each prisoner is in solitary confinement with no means of communicating with the other. The prosecutors lack sufficient evidence to convict the pair on the principal charge, but they have enough to convict both on a lesser charge. Simultaneously, the prosecutors offer each prisoner a bargain. Each prisoner is given the opportunity either to betray the other by testifying that the other committed the crime, or to cooperate with the other by remaining silent. The offer is:

## 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.

In game theory, the Nash equilibrium, named after the mathematician John Forbes Nash Jr., is a proposed solution of a non-cooperative game involving two or more players in which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only their own strategy.

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]

The tragedy of the commons is a situation in a shared-resource system where individual users, acting independently according to their own self-interest, behave contrary to the common good of all users, by depleting or spoiling that resource through their collective action. The theory originated in an essay written in 1833 by the British economist William Forster Lloyd, who used a hypothetical example of the effects of unregulated grazing on common land in Great Britain and Ireland. The concept became widely known as the "tragedy of the commons" over a century later due to an article written by the American biologist and philosopher Garrett Hardin in 1968. In this modern economic context, "commons" is taken to mean any shared and unregulated resource such as atmosphere, oceans, rivers, fish stocks, roads and highways, or even an office refrigerator.

In the Abilene paradox, a group of people collectively decide on a course of action that is counter to the preferences of many or all of the individuals in the group. It involves a common breakdown of group communication in which each member mistakenly believes that their own preferences are counter to the group's and, therefore, does not raise objections. A common phrase relating to the Abilene paradox is a desire not to "rock the boat". This differs from groupthink in that the Abilene paradox is characterized by an inability to manage agreement.

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Ayelet Gneezy is an associate professor of marketing at the Rady School of Management, UC San Diego.

## 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 2016-02-05. Retrieved 2015-06-08.Cite uses deprecated parameter |dead-url= (help)