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Payoff Matrix for Nash Equilibrium

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Nash equilibrium refers to the level of outcome where change of strategic would not provide extra benefits to a player if other players do not change their strategies.

Nash equilibrium can occur multiple times in a game. It is invented by John Nash and can be applied in many fields, such as ecology and economics.

Let us understand the concept of Nash equilibrium with the help of an example. Suppose two organizations, P and Q want to increase their profits by increasing their advertisement expenditure. In such a case, each of the organization can adopt two types of strategies. One is to increase the ad-expenditure and the other is to keep ad-expenditure constant or no change in the ad-expenditure.

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The payoff matrix of two organizations is shown in Table-8:

Payoff Matrix for Nash Equilibrium

In Table-8, neither of the organization has a dominant strategy; therefore, their strategies depend on each other’s strategies. For example, if organization P increases the ad-expenditure, then organization Q also needs to increase its ad-expenditure. In case, organization Q does not increase its ad-expenditure, then the outcome of P increases.

Therefore, in the present case, it can be concluded that both the organizations, P and Q should follow the same strategy increase in ad-expenditure) to get the maximum benefit. This is because if none of them increases their ad-expenditure, then organization P has the sales increase of Rs. 25 million, while the sales of organization Q would be Rs. 5 million only.

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However, if organization Q increases its ad-expenditure, its sales increase to Rs. 10 million. Therefore, it is good for organization Q to increase its ad-expenditure. Now, organizations P also need to follow organization Q. This is the case similar to an oligopolistic market condition. In an oligopolistic market, if one organization reduces its product prices, then other rival organizations also need to reduce their prices, so that they can retain customers.

Prisoners’ Dilemma:

Prisoner’s dilemma refers to the game in which even by adopting best strategies, players are not able to get the best outcome. In dilemma, every player wants to minimize his/her risk on the cost of increasing the risk for the other player(s). In the reducing the risk for oneself, the players get into a worst situation, which may be avoided if they cooperate with each us understand the prisoner’s dilemma with the help of an example.

Suppose there are two individuals, named John and are involved in drug smuggling. They are arrested by police as per the information provided by one of the informers of Now, the police are interrogating both of them separately.

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They were told three outcomes of their responses, which are as follows:

i. If both of them remain silent, then they would be set free because there is no evidence of their involvement in drug smuggling.

ii. If they confess about their evil activities, they would be imprisoned for five years.

iii. If one of them confesses and the other remains silent, then the confessor would be imprisoned for two years, while the other would be imprisoned for 10 years.

Now, the problem with John and Mac is that they are not aware whether their partner would remain silent or confess. Therefore, it is a situation of confusion.

The payoff matrix in the present case is shown in Table-9:

Payoff Matrix for Prisoner's Dilemma

As shown in Table-9, the two individuals have two options either to confess or remain silent. The decision of confessing by both the individuals depends on the period of imprisonment. The best option for both of them is to remain silent. However, due to uncertainty about each other’s decision, they are not able to remain silent. Therefore, it is a situation of dilemma for John and Mac.

In the present case, it can be anticipated that both of them would confess to reduce their time period of imprisonment. Therefore, the best option in this situation would be to confess and get imprisoned for five years. This is because if in case one partner confesses and the other denies, then the one who denies would be imprisoned for 10 years.

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