Nash equilibrium is an end result of a video game such the no player can gain by unilaterally changing its strategy. It is completed when each player by the optimal strategy given the strategy the the various other player.

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Nash equilibrium is called after john Nash, a renowned game theorist played through Russel crow in ‘A Beautiful Mind’.

The most essential property of Nash equilibrium is that it is self-enforcing. It is the outcome which 2 rational players A and B should at some point reach in a non-cooperative game. Player A reaches Nash equilibrium by employing a strategy i m sorry is his best response to the strategy preferred by that opponent, Player B. But since the enemy i.e. Player B additionally choses a strategy which offers him the maximum payoff given the strategy the Player A, the video game gravitates towards an unavoidable outcome. This result is referred to as a Nash equilibrium.

Even despite it is in the ideal interest of each player to adopt a strategy dictated through the Nash equilibrium, the is not important that the Nash equilibrium would certainly maximize the combined payoff. Prisoners’ dilemma is a standard example the this phenomena.

A Nash equilibrium is stable because no player have the right to improve the payoff by transforming its strategy.

### Nash Equilibrium vs leading Strategy

A leading strategy is a form of Nash equilibrium.

A leading strategy is a strategy which outcomes in the best payoff for a player no issue what the other firm does however a Nash equilibrium to represent a strategy i beg your pardon maximizes payoff provided what the various other player would certainly do. A Nash equilibrium is conditional top top the other player’s finest strategy, however a leading strategy is unconditional. Us reach a Nash equilibrium through assuming that the other player is rational, however we can follow a dominant strategy there is no forecasting supposed strategy that the opponent.

A video game has a Nash equilibrium even if over there is no dominant strategy (see example below). That is also feasible for a video game to have actually multiple Nash equilibria.

## Finding Nash Equilibrium

Following rules are useful in identifying a game’s Nash equilibrium.

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If one player has a leading strategy, the cabinet in the dominant strategy row or pillar in i m sorry the other player has the preferably payoff is the Nash equilibrium.

## Example

Let’s think about two firms A and B who should decide about their advertising budget. The complying with payoff matrix reflects net boost in profit of each firm under different scenarios:

There is a dominant strategy in this game for firm A i.e. To advertise. The is since the maximum payoff for heat player in all columns occurs in the last row.

Firm B has actually no dominant strategy since its preferably payoff doesn’t take place in the same column. Once Firm A cuts its declaring budget, firm B’s maximum payoff occurs once it likewise follows. Similarly, as soon as Firm A doesn’t readjust its declaring budget, for sure B does finest by cutting its advertising. Yet if firm A increases its advertising spending, certain B’s finest strategy is come not change its current advertising budget.

There is no dominated strategy either for Firm B because there is no shaft in which its payoff is constantly worst.

Using the rules disputed above, we recognize that the Nash equilibrium should exist in the last heat i.e. The leading strategy of certain A. In other words, Nash equilibrium in this game would take place when for sure A advertises. Offer this fact, we need to uncover which cell offers us the preferably payoff for Firm B in the last row. It is the 2nd column which represents Firm B not transforming its declaring budget. Heat 3 and Column 2 hence show a Nash equilibrium because:

Firm A has no incentive to adjust its strategy because increasing proclaiming is its dominant strategy.Since certain A will certainly advertise in any case, for sure B’s best solution is come not readjust its budget because this gets it the maximum payoff. If firm B cut its advertising budget plan or rises it, its payoff will certainly drop to \$20 million in each situation which is worse 보다 \$50 million.

by Obaidullah Jan, ACA, CFA and also last amendment on Mar 27, 2019