Prisoner’s Dilemma

The Basic Idea

Prisoner's Dilemma

Adam and Bianca are arrested for robbing a bank and placed in separate holding cells. The prosecutor individually presents both of them the following proposition: “You may choose to confess or remain silent. If one of you confesses and the other remains silent, the confessor will walk away free of charges, while the other faces 20 years in prison. If you both confess, both of you will face five years in prison. If neither of you confesses, then you both will face one year in prison.”

What is the best move for Adam to make, regardless of Bianca’s decision, if he is only looking out for his own well-being? What will Bianca’s decision be, given the consequences of all the scenarios? It seems like Adam and Bianca have found themselves in a prisoner’s dilemma.

People who fight fire with fire usually end up with ashes.

– Abigail Van Buren

Key Terms

Prisoner’s dilemma: A paradox seen predominantly in game theory in which two individuals acting for their own self-interest do not produce the optimal outcome.1

Game theory: Analysis of strategic decisions made by interacting players, as a method of modeling behavior using a mathematical approach.2

Nash equilibrium: A concept in game theory, in which the optimal outcome of a game occurs when there are no incentives from any player to deviate from the initial strategy.3

Dominant strategy: Asserts that the chosen strategy of a player will result in the best results out of all possible strategies that can be used by said player, regardless of what other players do.3

History

The Prisoner’s Dilemma was conceived in 1950 by mathematicians Merrill Flood and Melvin Dresher in military and strategic context for a class of psychology majors at Stanford University.4 In the original scenario (similar to Adam and Bianca’s) two individuals are charged with the same infraction of the law and are held separately by authorities. Each is told that if one confesses while the other does not, the former will face no jail time while the latter will be punished with “two units” of jail time. If both confess, then both will face “one unit” of jail time. However, if both parties remain silent then both will walk away free. This non-cooperative two-person game is recognized as an early example of game theory in economics globally.

The prisoner’s dilemma has been heavily popularized in culture and media. In the 2001 film, A Beautiful Mind, Russel Crowe, who plays the influential mathematician John F. Nash, famously explains the concept of the Nash equilibrium. He uses an analogy of men approaching women at a bar and states that instead of approaching the most beautiful woman, the men should pursue other women in the group to ensure less competition and a favorable outcome for everyone (if one makes the assumption that everyone’s goal is to end up in a pair and that the women are not able to pursue the men).5

The film misses a key component of the prisoner’s dilemma (not to mention, applies it to a sexist and dehumanizing scenario). It implies that the unfavorable outcome, or equilibrium, can be avoided; however, a fundamental assumption of the prisoner’s dilemma is that everyone acts with self-interest. In the film scenario, all the gentlemen would compete over the “most beautiful woman”, as each man would act in their own self-interest – the dominant strategy. So perhaps we should all stick to learning about behavioral economics outside of popular films.

People

 

Melvin Dresher

A Polish-born American mathematician, Dresher’s notable work includes the development of the game theory model known as the Prisoner’s Dilemma with Merrill Flood. They developed this model at RAND Corporation, which is an American non-profit global policy think tank, in 1950. Dresher received his Ph.D. from Yale University, worked as a mathematics instructor at Michigan State College, a statistician, a mathematical physicist, a mathematics professor, and a research mathematician. He was also the author of many RAND research papers on game theory, as well as his widely acclaimed The Mathematics of Games of Strategy, Theory and Applications.6, 7

Merrill Flood

An American mathematician, Flood is recognized as a pioneer in the development of operations and management sciences. He is most known for his collaborative efforts with Melvin Dresher in the development of the Prisoner’s Dilemma in the 1950s After completing his studies in algebra from Princeton University, Flood was hired by the institution as a mathematics instructor. Flood also worked as Chief Civilian Scientist in the United States War Department during World War II and for the Office of Naval Research. With fellow American mathematician John Tukey, Flood is also credited for the naming of the discipline, “linear programming.”8

Albert W. Tucker

Albert W. Tucker was a Canadian-American mathematician, most recognized for his contributions to topology, game theory, and nonlinear programming. Tucker is recognized for naming and popularizing the Prisoner’s Dilemma, by devising the commonly used and rephrased situation similar to the one seen at the beginning of this reference guide. He received his Ph.D. from Princeton University and went on to become a mathematics professor at the institution in 1946. During World War II he worked with Merrill Flood’s Fire Control Project and managed the pre-radar research and optical range finders for anti-aircraft systems.4

Consequences

Paradoxically, the best-case scenario for Adam and Bianca’s overall well-being would be to both remain silent. However, if they act solely considering personal well-being, they will choose to confess. This scenario represents the non-cooperative Nash equilibrium. The irony, however, is that when each acts selfishly, they experience worse outcomes than if they cooperated.9

The prisoner’s dilemma conceptual framework can be applied across various disciplines which hold similar characteristics to those observed in the model. It is seen primarily in business situations. A commonly-used example of two shopkeepers competing for business on the same block with the decision to raise or lower their respective price of service. If one shopkeeper lowers their price below that of their competitor, they will attract the rival’s customers and increase profit. However, if both lower their prices, then neither will attract new customers and will end up with smaller profits.

The prisoner’s dilemma can also be a useful instrument for predicting the behavioral outcome of certain situations, such as the implementation of new marketing strategies or the decision to discontinue old products.10 For businesses, this concept allows for the prediction of competitor behavior and its consequences.

Controversies

Though the prisoner’s dilemma is held in high regard and can be applied to certain real-world situations, there are certain assumptions that threaten this model’s ecological validity and applicability outside of the 2×2 table it is usually presented in. First, it is incorrect to believe all players are completely rational. Humans experience bounded rationality, which means they have limited thinking capacity dependant on available information and time.11 Although efforts have been made towards accounting for irrational decision-making in economic models, the game theory still has a long way to go before it can capture the extensive impact of cognitive biases on human behavior.10

The prisoner’s dilemma also assumes that players will only act in order to optimize their own well-being, however, humans are social beings who care about the well-being of others. Altruistic behavior, sometimes at the expense of personal well-being, is not unbeknownst to society, and is something game theory models have yet to account for.12

Case Studies

The Tobacco Industry

In 1971, tobacco companies were banned by the American Government from running cigarette advertisements on television. One would assume that tobacco companies suffered from this restriction as they received less exposure, but in fact, the tobacco industry saw higher profits than before. Why did this happen?

First, let’s backtrack to before the ban on advertisements. Tobacco companies that advertise will have a slight advantage over those that do not since advertisements shape brand perception and can influence customer preference for one brand over another. Tobacco smokers will smoke regardless of whether companies run advertisements, it is just a matter of which company they chose to buy from. It turns out that all tobacco companies would fare better without advertising, under the condition that all other tobacco companies do not advertise as well.13

Comparing directly with the prisoner’s dilemma framing, the three possible scenarios are as follows (assume there are only two tobacco companies in the industry): if one company advertises while the other does not, then the one who advertises will reap greater profits. If both companies advertise then both will nonetheless profit but to a lesser degree than if neither had advertised. The last and most optimal outcome would be if neither company advertises, leaving both with greater profit compared to the scenario in which both companies advertise. Here, we can see that the dominant strategy would be to advertise. However, with the ban from the government, the optimal cooperative outcome is forced and all tobacco companies end up profiting more.10

Arms Race

In the early days of the Cold War, there was a nuclear arms race between the United States and the Soviet Union. Through the fight for supremacy, both funneled trillions of dollars and resources into manufacturing nuclear weapons. This period in history can be iterated down to the common structure of the prisoner’s dilemma.14

Both the US and the Soviet Union can choose to either arm or disarm. As a result, there are three possible scenarios: If one arm while the other disarms, the one that arms will reap the greater benefits. If both chose to arm themselves, they both will face more unfavorable consequences compared to if both chose to disarm, which is the third and most optimal outcome for both parties involved, saving resources and keeping the threat of nuclear warfare at bay. However, since the assumption under the prisoner’s dilemma is that both parties are only acting for their personal benefit, the dominant strategy for both is to arm themselves.15

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Sources

  1. Chappelow, J. (2021, May 30). Prisoner’s Dilemma Definition. Investopedia. https://www.investopedia.com/terms/p/prisoners-dilemma.asp.
  2. (Behavioral) Game theory. BehavioralEconomics.com | The BE Hub. (2019, March 29). https://www.behavioraleconomics.com/resources/mini-encyclopedia-of-be/behavioral-game-theoy/.
  3. Chen, J. (2021, May 19). Nash Equilibrium. Investopedia. https://www.investopedia.com/terms/n/nash-equilibrium.asp.
  4. Informs. (n.d.). Tucker, Albert W. INFORMS. https://www.informs.org/Explore/History-of-O.R.-Excellence/Biographical-Profiles/Tucker-Albert-W.
  5. Hartley, J. (2015, May 26). John Nash’s Indelible Contribution To Economic Analysis. Forbes. https://www.forbes.com/sites/jonhartley/2015/05/25/john-nashs-indelible-contribution-to-economic-analysis/?sh=3d 0c361d4d8a.
  6. Obituary, July 2, 1992 issue of the Palisadian-Post newspaper (Pacific Palisades, California).
  7. “In Remembrance”, July 9, 1992 issue of RAND Items (a biweekly publication for employees of RAND).
  8. Informs. (n.d.). Flood, Merrill M. INFORMS. https://www.informs.org/Explore/History-of-O.R.-Excellence/Biographical-Profiles/Flood-Merrill-M.
  9. Encyclopædia Britannica, inc. (n.d.). The prisoner’s dilemma. Encyclopædia Britannica. https://www.britannica.com/science/game-theory/The-prisoners-dilemma.
  10. Game Theory. The Decision Lab. (2021, February 5). https://thedecisionlab.com/reference-guide/economics/game-theory/.
  11. Bounded Rationality. Behavioral Economics. https://www.behavioraleconomics.com/resources/mini-encyclopedia-of-be/bounded-rationality/.
  12. Hayes, A. (2020). Game Theory Definition. Investopedia. https://www.investopedia.com/terms/g/gametheory.asp.
  13. YouTube. (2017). Game theory lessons – Historical example: Tobacco companies. YouTube. https://www.youtube.com/watch?v=27scYCyhd5o&ab_channel=365Careers.
  14. United States vs. Soviet Union: Prisoner’s Dilemma. Cornell University. (2015, September 11). https://blogs.cornell.edu/info2040/2015/09/11/united-states-vs-soviet-union-prisoners-dilemma/
  15. Plous, S. (1993). The Nuclear Arms Race: Prisoner’s Dilemma or Perceptual Dilemma? Journal of Peace Research, 30(2), 163–179. https://doi.org/10.1177/0022343393030002004

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