Game Theory and the Prisoner's Dilemma

Edited by Keith Poon.

Game theory is a branch of deeply rooted applied mathematics that provides a framework for analysing situations and decision-making. It is a study on how individuals and parties make decisions about their situations to conceive social scenarios. Parties that are involved make decisions that are interdependent, and this interdependence causes each party to consider the other party’s possible decision to form a strategy. (Brams & Davis, n.d.) From some aspects, game theory is a science of strategy, or at least it's an optimal decision made in a strategic setting. 

John von Neumann, a Hungarian-born American mathematician, and his Princeton University colleague Oskar Morgenstern, a German-born American economist, first invented game theory to address economic issues. (Brams & Davis, n.d.) They discovered that economics is quite similar to a game in which participants anticipate each other's movements, prompting the birth of an entirely novel branch of mathematics — game theory. This was then further developed in the 1950s by American mathematician John Nash, who came up with the mathematical concepts behind this theory, turning it into a field of mathematics that studies rivalry between rivals with conflicting objectives. 

Figure 1. Dilemmas represented in game theory

Since then, game theory has found practical applications in a variety of real-world situations to lay out various problems and predict the most possible outcome, providing insights into strategic decision-making. For instance, in the field of economics, game theory is applied to better understand market behaviour and pricing strategies. In political science on the other hand, it is used for negotiation tactics, conflict resolutions, and internal relations. (Jain, 2024) With regard to everyday decision-making, this theory maximises one’s outcome by helping them choose the most optimal strategy. 

The Prisoner's Dilemma is a popular game theory scenario in which two people must decide whether to cooperate, for mutual benefit, or to betray one another to gain personal advantage. The dilemma occurs when an individual's ideal approach results in an undesirable end result for both participants, resulting in a "lose-lose" situation. 

This concept is simplified to give two parties two decisions each — resulting in 2*2=4 possible outcomes. The following scenario and outcomes are:

1. If prisoner 1 testifies and prisoner 2 remains silent, then prisoner 1 will walk free while prisoner 2 will get 10 years in prison.

2. If prisoner 2 testifies and prisoner 1 remains silent, then prisoner 2 will walk free while prisoner 1 will get 10 years in prison.

3. If both prisoners betray each other and testify, they will each get 6 years in prison.

4. If both prisoners remain silent, they will each get 1 year in prison for the lesser charge.

These four outcomes are displayed in the Prisoner’s dilemma payoff matrix. (The Prisoner's Dilemma: Using Game Theory to Achieve the Optimal Solution, 2023)

Figure 2. Payoff matrix of possible outcomes

Here, understanding the strategic dynamics of the Prisoner's Dilemma requires a strong foundation in mathematics. The payoff matrix is a simple way to represent the potential outcomes and associated consequences based on each player’s choices. (Payoff Matrix | Definition, Theory & Uses - Lesson, n.d.) By giving numerical values to each result, game theorists can strategize the benefits and costs of various options, or identify the Nash equilibrium.

Nash equilibrium is a state in which no player can gain anything by changing their own strategy. In almost every game, the Nash equilibrium is attained gradually. (Viswanathan, 2014) However, once it is established, it cannot be changed — which does not make any sense. Hence, it is also referred to as a “no regrets” outcome regarding the consequences. It is a mathematical insight that demonstrates the underlying tension and contradiction between individual rationality and group benefit.

Figure 3. Possible strategy leading to Nash equilibrium

To recapitulate, game theory is the study of how competing strategies and parties’ actions affect the outcome of a scenario. It illustrates the complicated thought process of decision-making. Thus, mathematics is a vital tool to navigate the complexity of these situations. Understanding the fundamental mathematical basis of this allows us to gain many new insights into cooperation and betrayal, which may serve as a guide for us to make wiser decisions. 

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References:

1. Brams, S. J., & Davis, M. D. (n.d.). Game theory | Definition, Facts, & Examples. Britannica. https://www.britannica.com/science/game-theory

2. Jain, S. (2024, April 23). Real-Life Applications of Game Theory. GeeksforGeeks. https://www.geeksforgeeks.org/real-life-applications-of-game-theory/

3. Kelly, R. (n.d.). Game Theory: A Comprehensive Guide - Economics. Investopedia. https://www.investopedia.com/terms/g/gametheory.asp

4. Payoff Matrix | Definition, Theory & Uses - Lesson. (n.d.). Study.com. https://study.com/learn/lesson/payoff-matrix-economics-theory-calculation-template.html

5. The Prisoner's Dilemma: Using Game Theory to Achieve the Optimal Solution. (2023, December 10). Owlcation. https://owlcation.com/stem/The-Prisoners-Dilemma-Using-Game-Theory-to-Achieve-the-Optimal-Solution

6. Viswanathan, B. (2014, July 16). Nash equilibrium: How did the mathematical concept impact game theory and the world? https://slate.com/human-interest/2014/07/nash-equilibrium-how-did-the-mathematical-concept-impact-game-theory-and-the-world.html