Nash Equilibrium

Nash equilibrium is a concept from Game theory which establishes that a set of strategies followed by economic agents within a game is in equilibrium if, holding the strategies of all other economic agents constant, no economic agent can obtain a higher pay-off by choosing a different strategy. For example, when firms operate within an oligopoly, once Nash equilibrium has been reached, none of them will want to change their strategy because by doing it they cannot obtain a higher profit. In other words, a Nash equilibrium is a solution in which no player can improve his/her pay-off given the other’s strategy. In other words, each player’s strategy is a best response against the other player’s strategy, that is given player A’s strategy, player B can do no better, and given B’s strategy, A can do no better. The Nash equilibrium is also sometimes called the non cooperative equilibrium because each party chooses that strategy which is best for itself, without collusion or cooperation and without regard for the welfare of society or any other party. 

In the solution concept of Nash, each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only his or her own strategy unilaterally. If each player has chosen a strategy and no player can benefit by changing his or her strategy while the other players keep their unchanged, then the current set of strategy choices and the corresponding pay-offs constitute a Nash equilibrium.

According to the Nash theorem every game with a finite number of players and a finite number of strategies will have at least one Nash equilibrium. For this to hold, however, there has to be the possibility of some random elements to strategies. A Nash Equilibrium is a set of mixed strategies for finite, non-cooperative games between two or more players whereby no player can improve his or her pay-off by changing their strategy. Each player’s strategy is an ‘optimal’ response based on the anticipated rational strategy of the other players in the game. 

The theory of Nash equilibrium has two components: (i) the players act in accordance with the theory of rational choice, given their beliefs about the other players’ actions (i.e., the player makes rational decision-making in the absence of cooperation), and (ii) these beliefs are correct. If every player/ participant knows the game he/she is playing and faces incentives that correspond to the preferences of the player whose role he/she is taking, then difference/ deviation between the observed outcome and a Nash equilibrium can be blamed on a failure of one or both of these two components. If a Nash equilibrium is established by any means whatsoever, no firm (player) has an incentive to exit/ move from it by changing its own behavior. It is self-policing. It is self-policing in the sense that there is no need for group behavior to enforce it. Each firm has self-interest to continue (keep up) it because any move that it can make on its own will not improve its profits, given what other firms are currently doing. 

The Nash equilibrium can be illustrated by making some modifications in the pay-off-matrix given in the table. Now we assume that action and counter-action of advertising (AD) between Firms A and B is a regular phenomenon and the pay-off matrix that appears finally is given in table. The only change in the modified pay-off matrix is that if neither Firm A nor Firm B increases its ad-expenditure, then pay-offs change from (15, 5) to (25, 5).

Pay-off Matrix of the Game 
                                                     
B’s Options 
                                                      Increase AD        Don’t Increase 


A’s Strategy      Increase AD        A            B            A              B 
                                                   20           10           30             0 
                         Don’t Increase    A            B            A              B 
                                                  10           15           25             5 

From the payoffs matrix, we can see that Firm A has no more dominant strategy. Its optimum decision depends now on what Firm B does. If Firm B increases its advertising-expenditure, Firm A has no option but to increase its advertisement expenditure. And, if Firm A reinforces its advertisement, Firm B will have to follow the suit. On the other hand, if Firm B does not increase its advertising-expenditure, Firm A does the best by increasing its ad-expenditure. Under these condition, the conclusion that both the firms arrive at is to increase advertising expenditure if the other firm does so, and ‘don’t increase’, if the competitor ‘does not increase’. In the ultimate analysis, however, both the firms will decide to increase the ad-expenditure. 

The reason is that if none of the firms increases advertisement, Firm A gains more in terms of increase in its sales (Rs. 25 million only). And, if firm B increases advertisement expenditure, its sales increase by Rs. 10 million. Therefore, Firm B would do best to increase its ad-expenditure. In that case, Firm A will have no option but to increase its ad-expenditure. Thus, the final conclusion that emerges is that both the firms will go for advertisement war. In that case, each firm finds that it is doing the best given what the rival firm in doing. This is the Nash equilibrium.

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