Nash Equilibrium Game Theory

What is the Nash Equilibrium?

Nash equilibrium is a game theory concept that helps in determining the optimum solution in a social situation (also referred to the as non-cooperative game), wherein the participants don’t have any incentive in changing their initial strategy. In other words, in this strategy, a participant doesn’t gain anything by diverging from their initial strategy, which is subject to the assumption that the other participants also don’t change their strategies.

Nash Equilibrium Game Theory

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History

This game theory concept of Nash equilibrium is named after American mathematician, John Nash, who was awarded the Nobel Prize in Economics in the year 1994 for his invaluable contribution to the field of game theory.

The underlying principle is similar to what was used by Antoine Augustin Cournot in his theory of oligopoly (1838). As per Cournot’s theory, all the firms in a competitive market would choose to produce only that much output which would maximize his profit. However, the best output of one firm is dependent on the output of the others in the market. Consequently, Cournot equilibrium is achieved only when the output of each firm maximizes their profits, taking into account the output of the other firms, which is again the strategy for Nash equilibrium.

The modern concept of Nash equilibrium game theory has changed a bit as now it also includes mixed strategies, wherein the participants avert possible actions and prefer to choose probability distributionProbability DistributionProbability distribution is the calculation that shows the possible outcome of an event with the relative possibility of occurrence or non-occurrence as required. It is a mathematical function that gives results as per the possible events.read more. This mixed-strategy concept under Nash equilibrium was pioneered by Oskar Morgenstern and John von Neumann, in their book The Theory of Games and Economic Behavior (1944).

Examples of Nash Equilibrium

Example #1

Let us take the example of two rival companies – Company X and Company Y, to illustrate the concept of Nash equilibrium in game theory. Both companies intend to determine whether it is the right time to expand their production capacity. If both companies expand their capacities now, each can increase their market shareMarket ShareMarket share determines the company's contribution in percentage to the total revenue generated within an industry or market in a certain period. It depicts the company's market position when compared to that of its competitors.read more by 10%. However, if only one of them decides to expand, then it can increase its market share by 20%, while the other one will not gain any market share. On the other hand, if both the companies give up the idea of expansion, then neither of them will gain any market share. The below table indicates the payoff in this case.

Nash Equilibrium Game Theory Example 1

So, in this case, the Nash equilibrium is achieved when both the companies expand their production capacities as it offers better payoff overall.

Example #2

Let us look at another example to illustrate the concept of multiple Nash Equilibria in game theory. Imagine that two friends, David and Neil, are registering for a new semester and they both have the option to choose between Finance and Marketing. If David and Neil register for the same class, then they will able to study together for the exams. On the other hand, if they pick different classes, then neither they will lose out on the mutual benefit of group study. The below table indicates the payoff in this case.

Nash Equilibrium Game Theory Example 1-1

So, in this case, there are multiple Nash equilibria that are achieved when both David and Neil register for the same class. Thus, the outcomes are David picks Finance – Neil picks Finance, and David picks Marketing – Neil picks Marketing.

Applications

  • Analysis of hostile situations like arms races, and wars (Prisoner’s dilemma).
  • Analysis to mitigate conflict through repeated interactions.
  • Study of human behaviour to determine at what point people with different preferences can cooperate.
  • Determination of probability of currency crises and bank runs (Coordination game).
  • Design algorithm for traffic control (Wardrop’s principle).

Advantages

  • It is a well-defined quantitative approach for decision making in a competitive situation.
  • It helps in the assessment of the competitors’ reactions.
  • It is a management tool that helps in policymaking.

Disadvantages

  • The determination of the optimal solution becomes difficult with the increase in the number of participants.
  • It is more of a logical strategy and not a winning strategy.
  • The concept fails to account for uncertainties that are encountered in real-life business situations.
  • The theory expects the participants to act rationally, which is not always the case.

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