What Is the Prisoner’s Dilemma?
Game of Strategy in Social Science. Prisoner dilemma: a study in conflict and cooperation.
Game theory is a branch of applied mathematics that is used in the social sciences, most notably in economics, as well as in biology, engineering, political science, international relations, computer science, and philosophy.
Game theory attempts to mathematically capture behavior in strategic situations, in which an individual’s success in making choices depends on the choices of others. While initially developed to analyze competitions in which one individual does better at another expense (zero sum games), it has been expanded to treat a wide class of interactions, which are classified according to several criteria. Today, “game theory is a sort of umbrella or ‘unified field’ theory for the rational side of social science, where ‘social’ is interpreted broadly, to include human as well as non-human players (computers, animals, plants)” (Aumann 1987).
Traditional applications of game theory attempt to find equilibrium in these games. In an equilibrium, each player of the game has adopted a strategy that they are unlikely to change. Many equilibrium concepts have been developed (most famously the Nash equilibrium) in an attempt to capture this idea. These equilibrium concepts are motivated differently depending on the field of application, although they often overlap or coincide. This methodology is not without criticism, and debates continue over the appropriateness of particular equilibrium concepts, the appropriateness of equilibria altogether, and the usefulness of mathematical models more generally.
Nash equilibrium
In game theory, Nash equilibrium (named after John Forbes Nash, who proposed it) is a solution concept of a 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 his or her own strategy unilaterally. If each player has chosen a strategy and no player can benefit by changing his or her strategy while the other players keep theirs unchanged, then the current set of strategy choices and the corresponding payoffs constitute a Nash equilibrium.
Prisoner’s dilemma: a study in conflict and cooperation.
The Prisoner’s Dilemma is the best-known game of strategy in social science (Dixit & Nalebuff, n.d.). This dilemma represents a common problem in achieving cooperation in any number of social settings. The dilemma “illustrates the tendency toward noncooperative behavior, despite general advantage from cooperation” (Lee & McKenzie, 2006, p.233). The Prisoner’s Dilemma classic game scenario as well as several related real world situations is presented below.
The prisoners’ dilemma is a well-known problem in game theory.
In a classical game, two people are apprehended as suspects for a major crime. They are separated from each other and interrogated. There are two options available to each of the two suspects. Each can either confess, thereby implicating the other, or keep silent. No matter what the other suspect does, each can improve his own position by confessing. If the other confesses, then one had better do the same to avoid the especially harsh sentence that awaits a recalcitrant holdout. If the other keeps silent, then one can obtain the favorable treatment accorded a state’s witness by confessing. Thus, confession is the dominant strategy for each. But when both confess, the outcome is worse for both than when both keep silent (Dixit & Nalebuff). However, each prisoner chooses to defect even though both would be better off by cooperating, hence the dilemma. In the classic form of this game, no matter what the other player does, one player will always gain a greater payoff by playing defect. Since in any situation playing defect is more beneficial than cooperating, all players will play defect, all things being equal .However, it should be noted that multiple repetition of the game will lead to different results (“Prisoner’s Dilemma”, n.d.).
The Prisoner’s Dilemma has applications in business and economics.
Example 1
Suppose there are two firms, A and B, selling similar products. Each has to decide on a pricing strategy. Both firms are better off when they both charge a high price; each makes a profit of $10 million per month. However, if one firm cheats and sells its product for lower price, it wins a lot of customers from the competitor. Assume its profit rises to $12 million, and the competitor’s profit fall to $7 million. If both set low prices, the profit of each is 9 million. In this situation, the low price strategy is like the prisoner’s confession, and the high price strategy equals to keeping silent. If we call the low price strategy cheating, and the latter cooperation, then cheating is each firm’s dominant strategy. However, the result when both cheat is worse for each than if both firms were to cooperate (Dixit & Nalebuff, n.d.).
Example 2
Lee and McKenzie (2006) gave an example of a Prisoner’s Dilemma game with respect to the healthcare decisions we make. An employer typically buys insurance policies with low deductibles. This feature of insurance policy has encouraged excessive use of healthcare services. This, in turn, drives employee’s insurance premium up. As a result, some workers can not afford to have the insurance anymore. We are in a Prisoner’s Dilemma with respect to our healthcare decisions. Collectively, we would be better off if we all moderated the amount of health care services. But because of insurance and government subsidies, it is in the interest of each of us to ignore most of the cost when we choose how much healthcare to demand (Lee & McKenzie, 2006).
Example 3
An example of a real world situation we have been observing for a number of years is the use of performance-drugs in professional sports, particularly those that are forbidden by the organizations that regulate competitions. For example, the 2007 Tour de France was rocked by a series of doping scandals.
- Pre-race favorite Alexander Vinokourov (Kazakhstan) tested positive for blood doping after winning the Stage 13 .The incident led his Astana Team to quit the Tour after Stage 15.
- Cristian Moreni (Italy) tested positive for testosterone after Stage 11. When his positive test was announced after Stage 16, his entire Cofidis (cycling team) team pulled out of the Tour. Moreni acknowledged his offense, choosing not to have his B sample tested. He was detained by French police, who searched the hotel rooms where the Cofidis team was to spend the evening after Stage 16.
- After the end of the Tour, it was revealed that Spanish rider Iban Mayo tested positive for EPO late in the race. (“Doping in Sport”)
Doping is considered to be unethical by most international sports organizations and especially the International Olympic Committee “…because of the health threat of performance-enhancing drugs, the equality of opportunity of the athletes and the exemplary effect of “clean” (doping-free) sports in the public” (“Doping in Sport”). Moreover, there are disciplinary actions employed against athletes tested positively on the doping drugs usage. However, using drugs in professional sports continues because of the strong incentive. It is a classic Prisoner’s Dilemma (Scheiree n.d.). To illustrate, Schneier (2006) gives the following example:
Suppose there are two competing athletes: Alice and Bob. Both Alice and Bob have to individually decide if they are going to take drugs or not. Imagine Alice evaluating her two options: “If Bob doesn’t take any drugs,” she thinks, “then it will be in my best interest to take them. They will give me a performance edge against Bob. I have a better chance of winning. Similarly, if Bob takes drugs, it’s also in my interest to agree to take them. At least that way Bob won’t have an advantage over me. So even though I have no control over what Bob chooses to do, taking drugs gives me the better outcome, regardless of his action.” Unfortunately, Bob goes through exactly the same analysis.
As a result, both athletes cheat, taking performance-enhancing drugs and neither has the advantage over the other. If they could trust each other, they could abstain from taking the drugs and maintain the same non-advantage status. They both would be better off since they would escape any legal or physical danger. But competing athletes can’t trust each other, and everyone feels he has to dope in order to compete (Schneier, 2006).
As Lee and McKenzie (2006) have pointed out, “Overcoming Prisoner’s Dilemmas is a pervasive problem in the development of social and management policies” (p.41). Studying principles of the game theory and its application to business will assist managers in choosing the most effective business solutions.
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References
Dixit, A. & Nalebuff, B. Prisoners’ dilemma. The Library of Economics and Liberty. Retrieved November 8, 2007 from http://www.econlib.org/Library/Enc/PrisonersDilemma.html
Doping in sport. Wikipedia. Retrieved November 7, 2007 from https://en.wikipedia.org/wiki/Doping_(sport)
Lee, D.R. & McKenzie, R. B. (2006). Microeconomics for MBAs. New York.
Cambridge University Press.
Prisoner’s dilemma. Wikipedia. Retrieved November 8, 2007 from https://en.wikipedia.org/wiki/Prisoner’s_dilemma
Schneier, B. (2006). Drugs: Sports’ Prisoner’s Dilemma. Retrieved November 8, 2007 from
http://www.wired.com/politics/security/commentary/securitymatters/2006/08/71566
Related article:
Mike Shor’s lecture notes for a course in Game Theory taught at the Owen Graduate School of Management at Vanderbilt University. Page contains links to lecture notes and supporting materialshttp://www2.owen.vanderbilt.edu/mike.shor/courses/game-theory/
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