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## The Prisoner's Dilemma

##### (http://faculty.washington.edu/krumme/450/prisoner.html)
This remarkable innovation did not come out in a research paper, but in a classroom. As S. J. Hagenmayer wrote in the Philadelphia Inquirer ("Albert W. Tucker, 89, Famed Mathematician," Thursday, Feb. 2, 1995, p.. B7) " In 1950, while addressing an audience of psychologists at Stanford University, where he was a visiting professor, Mr. Tucker created the Prisoners' Dilemma to illustrate the difficulty of analyzing" certain kinds of games. "Mr. Tucker's simple explanation has since given rise to a vast body of literature in subjects as diverse as philosophy, ethics, biology, sociology, political science, economics, and, of course, game theory." [Source]

Exercise:

Formulate a theoretical scenario with the general payoff structure for a "locational dilemma" involving the decision of a store owner to either join a shopping center [e.g. with (mutual) agglomeration economies, but longer distances to travel], or to select a decentralized location (or stay at the present location). Be sure you specify your model so that it complies with the Prisoner's Dilemma conditions and, please, thoroughly interpret your results.

Internet Sources:

Prisoners' Dilemma

A decision situation which illustrates the benefits of cooperation or collective action but also the difficulty of arriving at such an outcome. The decision payoffs are structured so that it is individually beneficial not to collaborate (with the fellow prisoner) even though collaboration by both would yield acceptable outcomes (and clearly better than if both defect). Each prisoner feels that she has to defect due to the uncertainty about the "partner's" action.

More specifically:

 PRISONER II's Collaboration Defection I's WorstOutcome Actions PRISONER I'sACTIONS Collaboration +1, +1 -2, +2 -2 Defection +2, -2 -1, -1 -1

The General Case:

 PRISONER II's Collaboration Defection I's WorstOutcome Actions PRISONER I'sACTIONS Collaboration R, R S, T S Defection T, S P, P P

A Prisoners' Dilemma situation requires that the following conditions be met:
 S < P < R < T and 2R > S + T

where: R = Reward (for collaboration)
T = Temptation (to defect and get away with it)
S = Sucker's payoff (was taken in)
P = Punishment (both defect)

Geographic application: (Hotelling model, importantly with some distance elasticity of demand; initial configuration: Both duopolists are located at center)
R = Private benefits from moving to Hotelling's quartile locations (which also happens to be socially more desirable)
T = Benefits from not moving from Center while partner moves (expansion of market share)
S = Loss from having moved while partner stays in center. Partner gains a permanent competitive advantage.
P = Both stay in the center and have to share the distance-elastic demand

The problem can be reformulated so that "collaboration" might imply benefits from togetherness due to a variety of substantial agglomeration economies. However, if only one moves to this potential agglomeration location (e.g. shopping center), she will lose her former clients (and is now stuck at an individually inferior location) without gaining from the expected external benefits. The non-moving competitor is not just saving relocation costs but also gains a competitive advantage by expanding her market into the vacated market area.