Nash equilibrium and The Prisoner’s Dilemma

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Nash equilibrium was a concept developed by John Forbes Nash who was awarded novel prize in economics in 1994 for his contribution in developing game theory. It is the concept in game theory that determines the optimal strategy in a non-cooperative game in which each player lacks any incentive to change his/her strategy. It conceptualizes the behavior and interactions between game participants to determine the best outcomes and also helps in predicting the decision of players if they are making decisions at the same time and the decision of player takes into account the decision of other players. Under Nash equilibrium, a player does not gain anything from deviating from the initial chosen strategy assuming that the other player keeps their decision intact. There may be Single or Multiple Nash equilibrium


Single Nash equilibrium


Imagine two competing companies: A and B. Both the companies want to determine whether they should launch a new advertising campaign for their products. If both companies start advertising, each company will attract 100 customers. If one company decides to advertise, it will attract 200 customers while the other company will attract none. Similarly, if none of the company advertises, none of the companies will attract new customers. This situation is presented in the following payoff table.


Nash Equilibrium  



 


According to the above payoff matrix, company A should advertise its products because the strategy provides a better payoff option than not advertising. The same situation exists for company. Thus, the optimal strategy when both the companies advertise their products this situation is called as Nash equilibrium.


Multiple Nash Equilibrium 


Under certain circumstances, a game may feature multiple Nash equilibrium (equilibriums). For example, Let Ram and Shyam are registering for a new semester. They both have the option to choose either economics or Psychology course. They have only 30 seconds before deadline so there is no time for consultation. If Ram and Shyam register for the same course, they have benefit of sitting in exam together. However, if they choose different courses, none of them will benefit.



In this example, there are multiple Nash equilibrium. If Ram and Shyam both register for the same course, they will benefit to sit together for exams. Thus, the outcomes are Economics/Economics and Psychology/Psychology which are the two Nash equilibrium in this scenario.


Thus, Nash equilibrium is the set of strategies or action such that each Player is doing the best it can given the strategies of actions of the opponents. Once the Nash equilibrium is attained, no player has incentive to change their decision because each player has made the best decision possible.


Prisoner’s Dilemma


In game theory, the Prisoner’s Dilemma shows that two people acting “rationally” can produce irrational results. The setting for Prisoner’s dilemma is police interrogation in which the prisoners are taken and interrogated separately. The police offer each the same deal in terms of prison time to get them confess and each knows that the same offer has been made to the other. Both prisoners are assumed to be individually rational in the sense that they want the most for themselves without thought about social norms and obligations. It means individual is rational and selects the payoff that is best suited for him or her or to minimize the jail sentence. We also assume that an individual has two choices to confess or not to confess. Both the individuals have choice to confess or not to confess.



The outcomes of the decision represent what happens to the payoff when the player confesses. When both the player confesses then, both players will get 7 year prison sentence. Similarly, when both the player gets 4 years prison sentence. When Player A confesses but player B do not confess then, player A will get 2 years prison whereas B gets 10 years prison. Similarly, when player B confesses and player A do not confess then, player B will get 2 years sentence and player A gets 10 years. Thus, both the players will maximize their payoff by confessing rather than not confessing.


 For example, player B confesses, then A should confess as well because by confessing he can minimize the prison time by 3 years. Again, let us suppose that player B does not confess, then, the best case of A is to confess because by confessing, he can only spend 2 years in prison whereas the other player spends 10 years. Thus, the best case for both the players is to confess from where s/he can minimize the prison time for himself. Thus, such game is set so as to get confession from either of the players.     


 


 


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