In the world of mathematics, there is a wide variety of research topics that are related to other fields of study, such as economics, sociology, health, etc. Therefore, mastering mathematical tools and finding their application in real life is essential for the future academic education and career. In this paper, mathematical probabilities and he knowledge acquired through the course are used to research the topic of Prisoner’s Dilemma.

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This topic was selected due to its popular use in the fields of Economics, Statistics, and Mathematics. It is of particular interest to the author since it not only allows to apply the principles of probabilities to explain the paradox but also relates to the behavioral science as it refrains from rationality in the behavior of human actors.

Prisoner’s dilemma is one of the best-known concepts in the game theory. The idea was first developed by Merrill Flood and Melvin Dresher in 1950s and later formalized by Albert W. Tucker (Dixit & Nalebuff, 1991). In the original setting, two criminals A and B are arrested and placed in separate rooms so that they cannot communicate. If both criminals keep silent and do not testify against each other, they will each get a sentence of 1 year. If A testifies against B, A will be set free, but B will be sentenced to 3 years, and vice versa. If both of the suspects testify against each other, they will get 2 years each.

Additional conditions include the impossibility of punishing or rewarding each other for their decisions and null influence of the decision on their reputation. In the end, both suspects chose to betray each other, even though they would be better off if they both kept silent. In this paper, mathematical probabilities related to the problem are examined, as well as the real life situations in which the Prisoner’s Dilemma was tested.

In the Prisoner’s Dilemma, the question of preferences is illustrated, which is widely used in Microeconomics and the related sciences. Both prisoners being rational would prefer to get away with no sentence to any sentence. Therefore, their decision would be to betray their partner. At the same time, both of them would prefer 1 year sentence to 2 years and would prefer 2 years to 3 years respectively. From the table, it is clear that 0 > -1 > -2 > -3. For the further calculations, the outcomes and preferences will be codified as α = 0, β = -1, γ = -2, δ = -3 so that α > β > γ > δ. The optimal decision for them would be to cooperate and keep silent. However, taking into account the previous preference, both suspects chose to betray and potentially get away with their most preferred option. As a result, both suspects chose to betray and end up with the 2-year sentence. This outcome is worse than the optimal decision; however, this is where the equilibrium in this paradox is found. The dilemma in this question is that individually rational decisions are not jointly rational (Wilson, 1986). The equilibrium, in this case, is the famous Nash Equilibrium that was developed by John Nash in 1951 (Shepsle & Bonchek, 1997). The conditional of the Nash Equilibrium stand even though in this case the players have to make their decisions simultaneously and are not aware of the opponent’s strategy.

Nash Equilibrium is one of the key achievements of the game theory. It identifies the optimal outcome of a game when the opponents are aware of each other actions and do not change their strategy after considering opponent’s choice (Shepsle & Bonchek, 1997). Nash Equilibrium is used in economics, and a range of social sciences, and therefore is important for understanding.

From the mathematical point of view, Prisoner’s Dilemma can be expressed as a set of equations with the expected outcomes for both of the players and probabilities of their respective actions (Dixit & Nalebuff, 1991). In this case, A shall be the probability that prisoner A betrays, and B will be the corresponding probability for prisoner B. We will also introduce a variable E(B) corresponding to the expected value of the sentence for prisoner B. The expected value function will be formed using the defined probabilities and outcomes specified earlier in the paper:
Now, this function is considered in different scenarios of the game. Firstly, it is assumed that player B chooses to betray. Therefore, the value of variable B in the equation is 1. After plugging in B=1, the function p is simplified to the following:

From this expected value, it can be observed that the prisoner B has two possible scenarios for the final outcome of the game: γ and α. The final outcome, however, depends on the choice of prisoner A. If he chooses to betray then A=1, and therefore the expected value is:

The outcome γ corresponds to 2 years of imprisonment. If prisoner A chooses to keep silent, A=0 and the expected value for prisoner B is the following:

This outcome corresponds to 0 years imprisonment for suspect B and is individually optimal in this case.
In order to prove the existence of Nash Equilibrium in this dilemma, it is necessary to compare the possible outcomes for prisoner B if he decides to betray or stay silent. The expected value for suspect B if he decides to be silent is the following:

The possible outcomes for him are now δ and β, depending on the decision of the prisoner A. For the prisoner B to consider cooperating and keeping silent, the expected value of keeping silent has to be higher than the one of betraying. That is:

As we can see, the solution of the inequality depends on A since all other variables are constants corresponding to the sentence term. Solving for A we obtain the following inequality:

It should be noted that A as a probability has to take only positive values between 0 and 1. For this to be true, in the following condition has to be met:

Recalling the initial setting of the problem, we can observe that this condition contradicts the preference structure of the prisoner. Therefore, a conclusion can be made that in order for the prisoner to choose to keep silent, a different setting of the problem has to be introduced (Dixit & Nalebuff, 1991). In particular, the sentence term in case of the betrayal of both prisoners has to be greater than the sentence term if only one of the suspects chooses to betray: α > β> δ > γ. Therefore, the new values have to be assigned to the determined constants. Let α remain 0, and β remain -1. In the new setting, δ = -2 (if A betrays and B keeps silent, A gets off with 0 years and B spends 2 years in prison); and γ = -3 (if both A and B betray, they get 3 years in prison each). In this case, the graph representing the non-equilibrium will be the following:

The worst outcome for both prisoners, in this case, is achieved when they decide to betray. It should be noted that in this case the probabilities assigned to the decisions change since a choice of betrayal is now associated with the probability of getting the highest penalty. Therefore, if prisoner B is not aware of the strategy of the opponent, he should reconsider his choice. He would want to avoid the maximum penalty and therefore is more likely to cooperate than before. The probability of cooperating for prisoner B will depend on the maximum penalty he would face. In this case, the probability is found by solving the equation with the established values of α, β, δ, and γ:

Therefore, if prisoner B knows that the probability of prisoner A betraying is greater than 0.5, he should choose to cooperate to avoid the maximum penalty. A conclusion can be made that in non-equilibrium conditions the prisoners should make opposite decisions, which is proved by the mathematical calculations based on probabilities (Dixit & Nalebuff, 1991).

The topic of Prisoner’s Dilemma is of a particular interest for the author because of its mathematical and behavioral components. A further investigation was conducted to determine whether the real life experiments prove the findings of the game theory. A research conducted in University of Hamburg involved female inmates for the best illustration of the actual prisoner’s dilemma, and a group of students as a control group (Nisen, 2013). The findings of the study did not confirm the theory, since the cooperation rate among the participants was much higher than expected. Both groups of participants were put through two stages of the game: in the first one they played simultaneously, and in the second one they were aware of their opponent’s actions. It turned out that 37% of students cooperated, while the cooperation rate among the inmates was at astonishing 56% (Khadjavi, Lange, 2013). The findings of the research prove the importance of behavioral sciences in economics and social studies. Despite the expectations of rational behavior, especially when it comes to prisoners who are thought to be practical and count on themselves, it was proved that socialization has a great effect on the decision-making process. However, the limitations of the study are that it was not held in the real setting, namely the participants did not risk several years of their lives when making the choice (Nisen, 2013). Even though it is not possible to hold all the conditions of the theoretical setting of this game, the participants may reconsider their choice and act more rationally when a larger benefit or loss is at stake.

In summary, it can be concluded that Prisoner’s Dilemma is a very important part of the game theory and is one of its profound concepts. In theory, it assumes rational behavior from the participants and gives way to the Nash Equilibrium, at which the decision of the player is not altered regardless of the strategy of the other one. Under the initial condition, it was proved that both prisoners would choose to betray since they would be aiming at getting the maximum benefit and avoiding the penalty. However, when the conditions change and the maximum penalty can be reached in a case of mutual betrayal, the prisoners will choose the opposite strategies in order to avoid the maximum penalties, if the probability of betrayal is high enough. However, when researchers from Hamburg University produced the real life test of the theory, it turned out that people tend to cooperate more eagerly due to social relations between them (Shepsle & Bonchek, 1997).

This gives rise to the behavioral theories and expands the opportunities for the future research questions about the factors influencing the decision process. It can be assumed that the players would be most rational when they would have no consideration for their opponent, or a very high loss or gain will be at stake. However, such games are only hypothetical, and the real life solutions can hardly be achieved. In the future, it would be interesting to expand the research on the Prisoner’s Dilemma, study more real life examples of its application, and conduct a survey or an experiment to compare the results of a real life setting with the theoretical findings.

References
• Dixit, A. & Nalebuff, B. (1991). Thinking Strategically: The Competitive Edge in Business, Politics and Everyday Life (“Introducing the Prisoner’s Dilemma: Go Directly to Jail”), pp. 11-14. (V*2).
• Shepsle, K. & Bonchek, M. (1997). Analyzing Politics: Rationality, Behavior and Institutions, chapter 8 (“Cooperation”), pp. 198-218.
• Khadjavi, M. & Lange, A. (2013). Prisoners and their dilemma. Journal of Economic Behavior & Organization. Volume 92, August 2013, Pages 163-175.
• Nisen, M. (2013). They finally tested the ‘Prisoner’s Dilemma’ on actual prisoners — and the results were not what you would expect. Business Insider, July 21. [Online] Available at: http://www.businessinsider.com/prisoners-dilemma-in-real-life-2013-7 [Accessed 30 Aug. 2017]
• Wilson, J. (1986). Subjective probability and the Prisoner’s Dilemma. Management Science. Vol. 32, No. 1, January 1986.