The Prisoner's Dilemma: Understanding Cooperation and Conflict in Game Theory, Guide, Progetti e Ricerche di Inglese Giuridico

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INDEX

1) WHAT IS THE “PRISONER’S DILEMMA”

2) THE NASH EQUILIBRIUM AND PARETO OPTIMAL

3) SOME EXAMPLE: THE PRISONERS

4) ECONOMIC EXAMPLE: THE BRANDS

6) CONCLUSIONS

BIBLIOGRAPHY

The Prisoner’s Dilemma

1) WHAT IS THE “PRISONER’S DILEMMA”

The Prisoner’s Dilemma is a hypothetical no-cooperative strategy game set up showing a situation where players will push until work alone to save themselves in a very difficult situation.

As J. Nash wrote: “One may define a concept of an n -person game in which each player has a finite set of pure strategies and in which a definite set of payments to the n players corresponds to each n -tuple of pure strategies, one strategy being taken for each player. For mixed strategies, which are probability distributions over the pure strategies, the pay-off functions are the expectations of the players, thus becoming polylinear forms in the probabilities with which the various players play their various pure strategies.”^1

Watching players bluffing in a poker game, inspired John von Neumann--father of the modern computer and one of the sharpest minds of the century--to construct “Game Theory”, a mathematical study of conflict and deception. Game Theory was readily embraced at the RAND Corporation and in 1950 they developed with Melvin Dresher “The Prisoner’s Dilemma” that was formalized by the mathematician Albert W. Tucker.^2

(^1) John F. Nash 1950. Equilibrium Points in N-Person Games. Proceedings of the National Academy of Sciences 36 (1): 48–49. 2 W. Poundstone, Prisoner’s Dilemma: John von Neumann, Game Theory, and the Puzzle of the Bomb. New York: Doubleday, 1992.

We can apply this game in all domains, for example in the marketing domain or in the business one. For that Kashi Abhyankar wrote: “This discipline concerns itself with the behavior of decision makers(players) whose decisions affect each other. The analysis is from a rational, rather than a psychological or sociological viewpoint.“ 3

2) THE NASH EQUILIBRIUM AND PARETO OPTIMAL

We can resume that the The Prisoner’s Dilemma is one of the best strategy to know in which way the human social behavior can change.

Basically to realize the optimal scenario all the players have just to both deny or simply not confess. But as we saw they haven’t loyalty or more simply they don’t trust in the other part, so actually the confession or the not built is the second optimal scenario that we can call “Nash Equilibrium”.

The Nash Equilibrium was given by John Nash (Nobel prize for Economics 1994). It’s a game theoretical concept very fancy where the player will take the best decision for himself based on what he thinks the others will do.

In the Oxford dictionary we can find the definition of Nash Equilibrium:”... a stable state of a system involving the interaction of different participants, in which no participant can gain by a unilateral change of strategy if the strategies of the others remain unchanged.”^4

For all payers is definitely always better to confess or not built because in any case:

• if the Player 1 confess and the Player 2 too, they will have the best option;
• if the Player 1 accept and the Player 2 refuse, the first will have the maximum of the payoff

(^3) K. Abhyankar, Smale Strategies for Prisoner’s Dilemma Type Games, 2004; (chap 1, pag. 56 (^4) The Oxford Dictionary

So if we look at the different states of the system we can always prefer the stable optimal scenario where everyone accept (3/3) to the unstable optimal scenario where everyone refuse (4/4), because the first is more stable without strategy change and it will not be affectable and it will never ship into the worst situation the player who accept.

The Prisoner’s Dilemma explain how not every time the good decision for the individual can be good for the group, so not every time the Nash Equilibrium coincides with best decision for the group,.

But it is too important the concept of the “Pareto Optimal”. For the Cambridge Dictionary the Pareto Optimal is “using or dividing resources (= time, money, employees, etc.) in a way that results in a situation where nobody is doing worse than before, and at least one person is doing better.” 5 (in the Oxford Dictionary we can find the definition of Pareto Optimal as a “relating to or denoting a distribution of wealth such that any redistribution or other change beneficial to one individual is detrimental to one or more others.”^6

We can realize that not all Nash Equilibria are Pareto Optimal and the Prisoner’s Dilemma is the most important example. In this case, when the results in a situation where nobody is doing worse than before but nobody can doing better is called “Pareto Efficiency”.

3) SOME EXAMPLE : THE PRISONERS

(^5) The Cambridge Dictionary (^6) The Oxford Dictionary

In this case,as we can see, the Nash Equilibrium is just if the two prisoners confess and this is also the Pareto Efficiency.

Otherwise if all the prisoners deny it, it is another Nash Equilibrium but it’s a Pareto Optimal and this is unstable and very difficult to have in the reality.

4) ECONOMIC EXAMPLE: THE BRANDS

The prisoner’s dilemma is not always presented as we have seen in this case. Payoffs Matrix can vary,

For example if there are 2 companies competing for the same market, Pepsi and Fanta, and one of the companies (for example Pepsi) decides to flood the public with advertisements. Pepsi will have to pay more money for all those advertisements but will earn a lot more money than the other company Fanta if he don’t be the same.

If the two companies fight an advertisements war then they’ll both have to pay more money for the advertisements, but they won’t have much increase in their earnings since they’ll have to share them between each other.

Both companies just need to make enough advertisements for the general public to be aware of their products and to want to use them which is when they are both cooperating. But at any time either company can increase their profits by producing excessive advertisements however once both companies start doing that too they will both end up losing money due to advertising costs.

6) CONCLUSIONS

We can definitely identificate the important role of the “strategy” in these games. As Jonathan Weinstein wrote in his conclusion: “... the result must be interpreted as stemming from the factors that are outside of the model, such as irrationality, psychological anomalies, and super-game concerns.”^8

The whole points of The Prisoner’s Dilemma are: 1.) If the second player cooperates it’s better for the first person to defect, but if the second defects it’s better for the first person to cooperate; 2.) If both player’s cooperate it’s the best collectively and if both player’s defect it’s collectively the worst.

As we saw we can apply the Prisoner’s Dilemma in all domains.

(^8) J. Weinstein, Reputation without commitment in finitely separated games, 2016 ; pag.

BIBLIOGRAPHY

1. John F.Nash. Equilibrium Points in N-Person Games. Proceedings of the National Academy of Sciences, 1950.
2. W. Poundstone. Prisoner’s Dilemma: John von Neumann, Game Theory, and the Puzzle of the Bomb. New York: Doubleday, 1992.
3. A. Smith, The Theory of Moral Sentiments. 1976.
4. K. Abhyankar, Smale Strategies for Prisoner’s Dilemma Type Games, 2004.
5. The Oxford Dictionary
6. The Cambridge Dictionary
7. J. Weinstein, Reputation without commitment in finitely separated games,
8. Richard H. Mcadams, Beyond the Prisoner’s Dilemma:Coordination, Game Theory,and law, 2009.