 So, we will continue our discussion on game theory. Last class, if you remember, we discussed about the different assumptions of the game theory. What is the need of game theory generally, in which case what is the usefulness of the game theory is there in the economic analysis. Then we discussed about the structure of a game. Then we took a small example to understand what is the dominant strategy, what is the maximum strategy, what is the min-max strategy and how generally equilibrium is achieved when there are two dominant strategies. Then we introduced the concept of Nash equilibrium and Nash equilibrium, if you remember, this is the best action given by the player irrespective of whatever the opponent does, that is the best strategy, that is generally the Nash equilibrium. So, taking the example of both the firms to advertise or not to advertise, what we discussed in the previous session, the same example we are going to take to understand this Nash equilibrium. So, just to a quick recap, Nash equilibrium is a strategy for each player such that no player has the incentive to change its action unilaterally given that the other player follows the proposed action. So, generally this is a state of equilibrium, this is a state of balance beyond which whatever the effort the player they are going to put it, it is not going to change their payoff and that is why this typical combination is called as the Nash equilibrium. It is generally the other way to put it is that it is the optimal collective strategy in the game involving two or more players where no player has anything to gain by changing his strategy. So, this is the point, this is the optimal collective strategy, this is the optimal strategy for both the firms beyond which whatever may be the change, no player has to gain anything by changing the strategy. So, we will take the same example to understand this Nash equilibrium and here there are two firms firm 1 and firm 2 and it is a choice between them that whether they should advertise or whether they should not advertise. So, we will take the case of firm 2 here, here we will take the case of firm 1, here it is whether they should go for the advertisement, whether they should not advertise, here again we have taken the advertise and do not advertise. So, when both the firms they are advertising, firm 1 get a share of 50, firm 2 get a share of 20. When firm 1 advertise and firm 2 is not advertising, firm 1 is getting 60 and firm 2 is getting 10. When firm 1 is, firm 2 is advertising, firm 1 is not advertising, he gets 40, firm 1 get 40 and firm 2 get 30 and when both of them they are not advertising, they get a payoff of 65 and 25. Now, how we can decide their Nash equilibrium? Now, what firm 1 will try to do? Firm 1 will try to speculate firms to action and in return what firm 2 will do? Firm 2 will also try to anticipate what is firm 1 action. Now, to start with let us see what the firm 1 will first do. So, to start with firm 1 will presume that firm 2 is going to advertise. If firm 2 is going to advertise, it is better for the firm 1 to advertise because in that case they are getting a payoff of 50. If he is not going for advertising, even if firm 2 is going for advertising, he is getting a payoff of 40. What is best for him? Best for him to go for the advertisement because he will presume that anyway since it is a case of the market share, the optimal output at the end of the day is that there should be increase in the market share. How it will increase if firm they are going for the advertisement? So, in this case if firm 1 always presume that the firm 2 is going to advertise and if firm 2 is going to advertise, then in that case if firm 1 also advertises, he gets a market share of 50 whereas, if he is not advertising, even if firm 2 is advertising, he gets a market share of 40. So, since 50 is greater than 40, it is always best outcome or the best payoff to go for advertisement. Now, we will understand from the perspective of the firm 2. Now, how firm 2 will react to this or how firm 2 will behave in this case? So, firm 2 knows that same thought process again or the same speculation again that since it is about the market share, market share increases whenever there is a increase in the advertisement. The more it reaches to the consumer, more is the market share. So, in this case firm 2 will also think that anyway firm 1 is going to advertise because he has to increase the market share and if firm 1 is advertising, if firm 2 is advertising, it gets a share of payoff because their payoff is 20. If firm 1 is advertising and firm 2 is not advertising, it gets a share of 10. So, since 20 is greater than 10, this should be the preferred, this should be the best outcome, best strategy. So, for firm 2 also, the best strategy or may be the decision of the firm 2 has to advertise. Now, what is the Nash equilibrium? Nash equilibrium is since whatever the strategy taken by firm 1 irrespective of whatever the other firm is doing, if it is matching with the strategy, whatever is taken by firm 2 irrespective of what firm 1 is doing. So, this is what this is a similarity that both the firm, they are going to advertise whatever the other firms they are doing in a share. So, in this case, the Nash equilibrium is Nash equilibrium for both the firm is to advertise and advertise. So, this combination of firm 2 and firm 1 is generally leads to Nash equilibrium. Because the whatever the best for B irrespective of whatever the best for B irrespective of what A does or whatever best for A irrespective of whatever B does, if that equal then we get the Nash equilibrium. So, Nash equilibrium is what in this case specific case, whatever is firm 2 is doing irrespective of firm 1 that strategy is advertised and whatever is firm 1 is doing irrespective of firm 2, the strategy is again to advertise that is why to advertise and to advertise this combination leads to the Nash equilibrium. So, taking the same example, we can understand that how generally when the both the company they are into a competition or they are into taking a decision making that whether they have to go for it not for it. In that case, generally they look at what may be the opponent choice, but here they have to when they have to reach a decision they have to see that irrespective of whatever the other company is doing or whatever the other firm is doing what they are getting out of it. Now, in this case both the firms they have the dominant strategy. The dominant strategy for firm 1 is to advertise, the dominant strategy for firm 2 is also advertised, but may be we will get a case where firm 1 and firm 2 they do not have the dominant strategy. Now, what would happen to the equilibrium output in this case? Because it may it may happen that the payoff changes or the payoff is in such a way that there is no no dominant strategy for one of them. Suppose one is having a dominant strategy the other is not having a dominant strategy in that case what is the Nash equilibrium. Because Nash equilibrium is ideally the dominant strategy of one when it is matches the dominant strategy of the others we get the Nash equilibrium, but if any case if the there is absence of dominant strategy of one of the firm or may be the other firm in this case what should be the equilibrium output. So, we will just change the payoff slightly and we will see in this case when there is no dominant strategy how the equilibrium output is achieved. So, we will just change the payoff matrix for firm 1 and firm 2. So, this is advertised, this is not advertising, this is advertising, this is not advertising. So, this is the payoff in the first case, this is the payoff in the second case, this is the payoff in the third case, this is the payoff in the fourth case. Now, we will start it again how to whether we can reach the equilibrium looking at the strategy taken by firm and firm 2. Now, suppose if firm 1 advertises now what firm 2 will do firm 2 will also advertise because it gets a better payoff in case of advertising like 50 and 40. Now, it is 20 and 10 because 20 he gets by advertising by not advertising it gets 10. Suppose if firm 1 is not advertising now here you need to look at the options that whether it is still profitable for the second firm to go for the advertising. So, if firm 1 is not advertising now what the firm 2 will do firm 2 will also not advertise because the payoff what they are getting from not advertising is higher than the whatever the payment they are getting from the advertising. So, in the previous case since this was 25 the payoff it was profitable for the firm 2 to advertise even if firm 1 is not advertising. But in this case if firm 1 is not advertising and firm 2 is advertising they are getting a payoff of 30 if they are not advertising when firm 1 is not advertising then they are getting a 35. So, not advertising payoff is greater than advertising and that is why they will prefer not to advertise when the firm 1 is not advertising. So, if you look at here if now if you look at if firm 2 is advertising. And if this is for when firm 1 is advertising not advertising what firm 2 should do. So, in this case when firm 1 is not advertising firm 2 is also going to a strategy where it is where the second firm is also not advertising. Similarly, if you look at now if the firm 2 is advertising and if firm 1 is also advertising and if firm 2 is not advertising when firm 1 is not advertising. So, in this case specifically if you look at firm 2 it depends upon what firm does. So, if firm 1 advertise firm 2 also advertise and if firm 1 is not advertising firm 2 is also not advertising. This leads to the conclusion that firm 1 2 it does not have a dominant strategy because the strategy are different whenever the opponent changes the whenever the opponent changes the strategy even the other firm also they have to change the strategy. So, we reach to one conclusion here that firm 2 does not have a dominant strategy. Now, we will analyze the case for firm 1. So, we will just take the payoff matrix for the reference. So, this is firm 2 advertising not advertising this is firm 1 again this is advertising this is not advertising and the payoff are 50, 20, 40, 30, 60, 10 and 65, 35. So, we have already reached to a conclusion that firm 2 does not have a dominant strategy because it is dependent on firm 1. If firm 1 advertise it advertise firm 1 is not advertising it is not advertising. Now, we will analyze for firm 1 if firm 2 advertise then it is better for firm 1 to also advertise why it is better for firm 1 to also advertise because 50 the payoff of advertising is greater than 40 which is the payoff for not advertising this is one situation. Second when firm 2 is not advertising in this case if firm 2 is not advertising if firm 1 is advertising they get a payoff of 60. When firm 2 is not advertising and firm 1 is also not advertising they get a payoff of 65. So, not advertise is the payoff of not advertise is greater than the payoff for advertise. So, now what is the best for firm 1 here the or maybe if you can conclude here that if firm 1 should advertise if firm 2 is advertising then and firm 1 is not advertising when firm 2 is not advertising. So, we can say again here that firm 2 firm 1 is also having no dominant strategy because it is dependent on whatever the strategy taken by whatever the strategy taken by firm 2. So, if firm 2 is advertising firm 1 is also advertising and if firm 1 firm 2 is not advertising firm 1 is also not advertising. So, in this case also if you look at there is no dominant strategy for firm 1 because it is not the best whatever the best that depends on dependent on the rival action and in this case there is no specific action it is all the dependent on the whatever the best that is dependent on the rival action. So, in this case if you look at neither firm 1 nor firm 2 they have a dominant strategy. Now, how we should reach or how we should find out the equilibrium output over here or whether it is possible to get the Nash equilibrium. So, we have we have checked a situation we have analyzed a situation where firm 1 and firm 2 they do not have the dominant strategy. What would happen to the equilibrium output in this case? Here we will get 2 Nash equilibrium we will not get a individual Nash equilibrium for the entire situation. Here we will get 2 Nash equilibrium 1 Nash equilibrium it occurs when both companies they advertise when firm 1 and firm 2 they are advertising both of them they are advertising we get 1 Nash equilibrium and when both of them they do not advertise we get another Nash equilibrium. So, when the dominant strategy of 1 is not matching with the dominant strategy of others or maybe there is a absence of the dominant strategy for both the firms over here we will not get a individual Nash equilibrium for the entire game rather we will get 2 Nash equilibrium 1 Nash equilibrium occurs when both company advertise and second Nash equilibrium occurs when both the firm they are not advertising. And here each firm is better off if it play the same strategy as the other firm. So, if one firm is advertising the other firm should also advertise and if the other firm is not advertising this firm also should not advertise. And simultaneously both the Nash equilibrium occurs when both the firm simultaneously play the same strategy. So, it is not that when one firm is advertising and other firm is not advertising the Nash equilibrium will come rather Nash equilibrium will come when both the firms simultaneously play the same strategy. If one is advertising the other one is also advertising and if one firm is not advertising the other firm is also not advertising. And Nash equilibrium comes when both the firms they play the same strategy at a particular point of time. So, any maximum strategy profile confer to Nash equilibrium because maximum strategy generally this is the maximization of the worst payoff that confers to Nash equilibrium. And also the minimax strategy by both the player also lead to Nash equilibrium. So, in the first case maximizing the worst payoff by both the players that will lead to Nash equilibrium because it is basically in that case the player chooses a strategy which maximizes the payoff among the worst payoff. So, that in that case also we can get a Nash equilibrium and also in case of minimax strategy where the strategy is not to maximize something of the own payoff rather to minimize the payoff for the others. And that is why the minimax strategy for both the players if you look at because one firm generally try to minimize the payoff for the others. And at the same time the other firm is also trying to minimize the payoff for the other firm. And in this case generally that also leads to kind of a Nash equilibrium. Then we will discuss interesting case generally which is more common it is a kind of game generally followed to understand the human behavior rather than a typical profit maximizing firm. And how this specific game is also a part of if you look at also a part of the economic theory because it is basically individual, but when we generalize this to the individual firms or the economic agents generally they behave in that situation. So, we will start a like interesting case study or interesting kind of game where the basically the moral of this case study is that even if cooperation is profitable still they are not cooperate with each other because they feel that this is not going to the best for them. So, business dilemma generally provides an insight into the difficulty in maintaining the cooperation. And even if the cooperation is profitable still they find difficulty in maintaining the cooperation. And that is the reason they never reach to the optimal solution they always get into a suboptimal solution. So, often people or when it comes to oligopolist firm they fail to cooperate with one another even when they when cooperation would make them better up. So, they fail to cooperate whether it is a case of individual economic agent or whether it is a case of a firm they fail to cooperate with each other when the cooperation would make them better up. So, this prisoner's dilemma it is a particular game between two capture prisoners that illustrate why cooperation is difficult to maintain even if when it is mutual beneficial. So, this is a if you look at this is the case study of two capture prisoners and when they when they were captured by the authority they knew that the cooperation is going to help them out the cooperation is profitable. But it always it difficult to maintain the cooperation finally, they land into a situation which is suboptimal. But that is not the optimal solution or that is not bad not the best solution for them. So, here two suspects the story goes like this or the case study goes like this two suspects are arrested for a armed robbery they are immediately separated they are taken into prison. So, the suspects they are arrested for some robbery.