Game Theory

The Basic Idea

The Milton Bradley board game, The Game of Life, was not far off the mark. In many ways, life does resemble a game, in which we are all players. There are rules to follow, opponents to compete with, and teammates to cooperate with. Game theory provides models for conceptualizing interactions among competing individuals. Despite its name, game theory is not exclusive to the study of games. It has a wealth of applications, from economics to psychology. This theory is applicable in any situation in which the actions of one party have influence over the actions of the other relevant parties. A key aspect of this theory is that it is based on certain assumptions, including that all involved parties understand the rules of the “game” and that people are rational decision-makers. Of course, humans have the capacity to err and we often make irrational decisions, which means that game theory is not infallible. As a general rule, however, it is a useful tool for predicting the outcomes of certain interactions between decision-makers.

 

[Game theory is] essentially a structural theory. It uncovers the logical structure of a great variety of conflict situations and describes this structure in mathematical terms. Sometimes the logical structure of a conflict situation admits rational decisions; sometimes it does not

– Anatol Rapoport

Key Terms

Games, players, and strategies

Game theory uses certain terms to conceptualize strategic interactions between two or more people. In game theory, the word “game” refers to any interaction where the outcome depends on the actions of two or more people. The decision-makers involved in the “game” are referred to as the “players”, while their “strategy” is the possible actions they may take within the context of the “game”. Another important term is “payout”, which refers to the outcome obtained by each “player”. Lastly, the term “equilibrium” is used to describe the point where all “players” have made their final decision.

The prisoner’s dilemma

One of the most well-known concepts of this theory is the prisoner’s dilemma. There have been several different versions of this scenario but they all rest on the same basic premise. Two parties are separated and unable to communicate. They are each given the option to cooperate or not. If both parties choose to cooperate, they will obtain equally favorable outcomes. Conversely, if both parties defect, they will both end up with equally unfavorable outcomes. Finally, if one party chooses to defect, while the other cooperates, the former will achieve the most favorable outcome possible while the latter will achieve the most unfavorable outcome possible.1

The prisoner’s dilemma has applications outside of the hypothetical. It has been observed in international arms races, in competition between shopkeepers, and in agricultural sales and production.2

History

The development of game theory has been attributed to John von Neumann and Oskar Morgenstern, two mathematicians at Princeton University in the mid-twentieth century. They developed the theory to be applied in economics, a field that they felt could not accurately be captured by the existing mathematical models that were designed to be applied to the physical sciences. Von Neumann and Morgenstern observed that the interactions that occurred between competing individuals mirrored the strategic decision-making seen in games, where one player anticipates the other’s next moves, thus leading them to coin the term “game theory”.3

Since then, this theory has been expanded to include a number of specifications. First, games can be classified by the number of players involved. These types of games are denoted as n-person games, where n refers to the number of players. Importantly, one player does not have to be a single individual. Entire groups of people, even entire nations, can qualify as a single player. The next specification is that of conflict versus cooperation. For example, constant-sum games are games where there can only be one winner, whereas in variable-sum games everyone will either win or lose.4 Games can also be divided into finite versus infinite, where the former refers to games where the rules and players are fixed and the latter refers to games where they are subject to change.5 Finally, a zero-sum game refers to a game where, any time one party gains something, the competing party must make an equivalent loss, leading to a net sum of zero.

John Nash was another key contributor to the origins of this theory. Von Neumann had previously stated that in every two-player, finite, zero-sum game there is a well-defined, optimal strategy. Nash expanded on von Neumann’s research and developed the theory behind what is now known as Nash equilibrium, which states that there is a clearly defined optimal strategy for every finite, non-zero-sum, n-player game. This expands the theory beyond the scope of von Neumann’s theory by incorporating all n-player games, instead of only two-player games.6

Game theory developed further throughout the 1950s. It was at this point that the prisoner’s dilemma paradox was conceived. The classic example of this paradox is a scenario in which two people are arrested for robbery. Each of the accused would be given the option to confess or to remain silent. There are three possible outcomes. If they both confess, they each get five years in prison. If one person confesses and the other remains silent, the charges against the confessor are dropped and the other party is sentenced to twenty years in prison. Finally, if both remain silent, they each get one year in jail. It is up to each person to choose whether they want to cooperate and remain silent, or defect and confess.7 The prisoners must then make a decision, without knowing what their accomplice will decide. The best case scenario is for both parties to cooperate and remain silent, however, there is always the risk that one’s accomplice will be uncooperative and confess, thereby causing one’s silence to earn them the twenty-year-long sentence.

Furthermore, it was in the 1950s that game theory began to be applied to disciplines outside of economics. It was applied to fields like philosophy and political science, which allowed the theory to achieve the broad scope it is known for today.

People

 

John von Neumann

Born in Hungary in 1903, John von Neumann had a gift for applied mathematics. In 1928, he published a paper titled “Theory of Parlor Games”, which began his work in the development of game theory. In 1944, he published one of the most influential books on game theory, Theory of Games and Economic Behavior, which he wrote in collaboration with Oskar Morgenstern, a fellow mathematician.8 While the concepts behind game theory have a lengthy history, he is largely credited with the development of this theory.

Oskar Morgenstern

German-American mathematician Oskar Morgenstern is best known for co-authoring Theory of Games and Economic Behavior alongside John von Neumann. In this book, they applied the foundations of game theory von Neumann had developed to business settings. His contributions to Theory of Games and Economic Behavior have earned him the reputation as one of the trailblazers of game theory.

John Nash

Winner of the 1994 Nobel Prize for Economics, Nash is known for the work he conducted on game theory in the 1950s. He expanded upon von Neumann’s theory and established the mathematical principle known as the Nash equilibrium to explain the interactions between threat and action between competitors.9 The 2001 film A Beautiful Mind is a fictionalized biography of his life, research, and struggles with schizophrenia, in which he is portrayed by Russell Crowe.

Consequences

Game theory provides a conceptual framework on the basis of which players can make rational decisions. It can be applied across disciplines to help people choose the best response to a given situation. For example, this theory has been hugely impactful in business. It has become a useful tool for predicting the outcomes of certain behaviors, such as adopting new marketing strategies or choosing to discontinue old products. This allows businesses to identify the strategy with the greatest probability of leading to the most favorable outcome possible. This is particularly useful for businesses, as they are often faced with many choices that can have significant consequences for their growth.10

Controversies

Controversy arises from game theory’s assumption that all players are rational decision-makers. As evidenced by the extensive research conducted on different biases and heuristics that can impair decision-making, this is not always the case. This theory attempts to account for this through bounded rationality, which suggests that there are limits to the extent to which we can be rational, based on the mental energy, information, and time available to us.11 It has been suggested that, by factoring bounded-rationality into the models for predicting behavior, we can continue to act on the assumption that humans are rational agents.12 Despite these efforts to account for irrational decision-making, game theory still does not fully capture the extent to which we are influenced by cognitive biases.

Furthermore, this theory assumes that human beings act exclusively out of self-interest which, of course, is not always the case. We have the capacity for altruism and may certainly prioritize the welfare of others at our own expense, which is not something game theory accounts for when making predictions about behavior.13

Case Study

Tobacco industry

In the early 1970s, cigarette companies were prohibited from running advertisements on television in the United States. This interdiction was part of the government’s efforts to discourage smoking and thereby reduce the prevalence of the diseases associated with it. Interestingly enough, the attempt to cut cigarette sales backfired completely; each of the four major tobacco companies in America at the time reported a boost in sales compared to the previous year.14

Companies that invest in advertising have an edge over those that do not, because advertisements promote customer loyalty and shape peoples’ perception of the company. As a result, if no tobacco companies are allowed to advertise, they will all benefit. Game theory can be used to understand this phenomenon, as it is similar to the prisoner’s dilemma paradox. Like in the prisoner’s dilemma, all tobacco companies are better off if they cooperate. In this case, that is if no one advertises. However, if one company advertises and another does not, the one that invested in advertising their brand will reap a greater reward, in the form of more sales. Finally, if both companies invest in advertising, they will likely perform equally well but turn lower profits than they would have if neither had advertised. Because there is no guarantee that their competitors will choose not to advertise, most companies choose to play it safe and invest in advertisements.15

Using game theory, we can understand why certain public health campaigns, like the ban on advertising cigarettes, do not always have the desired effect on people’s behavior.

James Holzhauer

In 2019, James Holzhauer made waves after winning over 2.4 million dollars across 33 episodes of the hit game show, Jeopardy. Now one of the highest-winning Jeopardy contestants of all-time, Holzhauer set the record for most money won in a single game – an impressive $131,127.16

While game theory has a much broader scope than the name implies, it most certainly still applies within the realm of real games, like Jeopardy. Jeopardy is an example of a game where there are fixed rules and fixed players and, as is always the case with game theory, all players have the ability to influence the outcome and the behavior of the other players. Holzhauer’s successful run on the gameshow can be, in part, attributed to his expansive knowledge of trivia and his timing with the buzzer, but some credit must be given to his use of this theory.

Holzhauer brought the concept of game theory into the mainstream, leading many people to claim that he “broke” Jeopardy. With a Bachelor’s degree in mathematics and a successful career as a professional gambler, the knowledge of statistics required to apply game theory models to the game of Jeopardy were well within Holzhauer’s capacity. Holzhauer’s strategy was to select the questions that were worth more first in order to build an early lead. He would also strategically search for the Daily Doubles, questions that can only be answered by the contest who selected them and for which they get to wage any value ranging from 1.00$ to the total amount they have accrued up to that point. On the Daily Double questions, Hozhauler’s average wager was $900017, which is quite high in comparison to your typical contestant’s bet. Game theory provides the basis for Holzhauer’s reasoning behind his unconventional approach to the game. Before starting his run, he did a deep dive into the best strategies for obtaining an optimal outcome in the game of Jeopardy. It is safe to say that his efforts paid off.

Not only was Holzhauer’s run on Jeopardy a landmark moment in pop culture history, but it brought game theory into the spotlight and provided an accessible example of how it can be applied in real life.

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Sources

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  2. Davis, M.D. and Brams, S.J. (2020). The prisoner’s dilemma. Encyclopaedia Britannica. https://www.britannica.com/science/game-theory/The-prisoners-dilemma
  3. Davis, M.D., and Brams, S.J. (2020). Game Theory. Encyclopaedia Britannica. https://www.britannica.com/science/game-theory
  4. See 4
  5. The Infinite Game … Most games have finite boundaries, and measures of success … but not business. The Genius Works. https://www.thegeniusworks.com/2019/01/the-infinite-game-most-games-have-finite-boundaries-and-measures-of-success-but-not-business/
  6. Najera, J. (2019). History & Overview. What is Game Theory. Towards Data Science. https://towardsdatascience.com/game-theory-history-overview-5475e527cb82
  7. See 2
  8. Poundstone, W. (2020). John von Neumann. Encyclopaedia Britannica. https://www.britannica.com/biography/John-von-Neumann/Princeton-1930-42
  9. John Nash. Encyclopaedia Britannica. https://www.britannica.com/biography/John-Nash
  10. Hayes, A. (2020). Game Theory Definition. Investopedia. https://www.investopedia.com/terms/g/gametheory.asp
  11. Bounded Rationality. Behavioral Economics. https://www.behavioraleconomics.com/resources/mini-encyclopedia-of-be/bounded-rationality/
  12. Wolfram, E., Torsten, H., and Henning, S. (2015). The Microeconomics of Complex Economies: Evolutionary, Institutional, and Complexity Perspectives. p. 193-226 https://doi.org/10.1016/B978-0-12-411585-9.00008-7
  13. See 10
  14. 365 Careers. (2017, Jul 3). Game theory lessons – Historical example: Tobacco companies [Video]. YouTube. https://www.youtube.com/watch?v=27scYCyhd5o
  15. See 14
  16. James Holzhauer Tracker. Jeopardy.com. https://www.jeopardy.com/contestant-zone/james-holzhauer-tracker
  17. See 16

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