 So, now let us see, in fact now that you have asked this, let us see if there is any conflict in the two, ok. So, then this here Q1 star star, we got this from a certain set of reasoning which is gamma 2 star star and Q1 star star. Question is that an ash equilibrium of this game? So, let us check that. This is Q1 star star comma gamma 2 star star an ash equilibrium, so is this an ash equilibrium? What do we need to check in the, for it to be an ash equilibrium, we need to check these conditions here, right, these conditions. Which means that assuming player 2 plays gamma 2 star star, player 1 would want to stick to the quantity Q1 star star, ok. Assuming player 2 plays the function gamma 2 star star, player 1 would want to stick to a quantity Q1 star star and assuming player 1 plays the quantity Q1 star star, player 2 would want to respond with the function, with the function gamma 2 star star, is this clear? That is what we need to check, ok. So, let us check that now. So, let us look player 1 first, ok. So, player 1, you are, so this is player 1 plays a quantity suppose Q1, player 2 is going to stick to gamma 2 star star, ok. So, he gets a utility U1 of Q1 comma gamma 2 star star of Q1, ok. This is the payoff of player 1 when he plays a quantity, some quantity Q1. So, what he wants to do is then maximize this over Q1, right. So, he is maximizing this over Q1. Now, what is that, what is the quantity that maximizes this? It is Q1 star star itself, because Q1 star star was derived from this particular maximization itself. So, Q1 star star maximizes is in fact the maximization of maximizing quantity here. That is by construction, because of the way we constructed Q1 star star, it actually is the maximizing thing here, ok, alright, so good. So, one part is done. Now, let us look at player 2, so this was for player 1, now look at player 2, player 2, for player 2 we have to fix player 1's quantity at Q1 star star and we are asking what is the function that player 2 respond to, ok. So, suppose player 2 plays a function gamma 2, you are maximizing this now as a function of gamma 2 and so you are maximizing this, so what is what is J2 of Q1 star star comma gamma 2, I need to put Q1 star star here and Q1 star star here. So, what is this maximization and what function maximizes this, ok, but remember so is there any other function that maximizes this. So, let us take this very carefully, so remember gamma 2 star star was derived by maximizing a different expression, it was derived by maximizing this expression. And here you have Q1 not Q1 star star, so which means gamma 2 star star was the function that maximized this U2 of that maximized this here for every value of Q1, because we got you know for a as a function of Q1 we wanted to find the maximizer, whereas now we are asking not for every value of Q1, we are asking for a specific value of Q1 which is Q1 star star, see compare that with this expression here there is only Q1 star star that has been held fixed. So, fixing Q1 star star, what is the function that maximizes U2 of gamma 2 of this, so the point is see remember when we are looking for a function of an arbitrary Q1, it makes sense to ask for a function, now you are asking for the value of the function only at Q1 star star, right, so really what the what gamma 2 does at other values of Q1 is irrelevant. So, any function gamma 2 add or such that or any function gamma 2 let us say any function gamma 2 such that gamma 2 of Q1 star star is equal to gamma 2 star star of Q1 star star maximizes, I can choose any function so long as at Q1 star star it agrees with the maximizing value that came from gamma 2 star star, outside for the other values of Q1 it does not matter, it can do whatever it wants, it will still maximize this objective, right. So, this is become like a what it what has happened is become a degenerate function optimization because you have fixed the argument of the function, really you are not you know what you do not care about what the value of gamma 2 is at other values of Q1, right. So, for example, if Q1 star star was 100 what matters here is what gamma 2 of 100 is not what gamma 2 of 90 is or 105 is that is that does not affect this objective. So, all of these are maximizes all these functions are maximizes in particular gamma 2 star star is also a maximizer. So, it means that this is greater than equal to this since gamma 2 star star is also maximizer so I may as well take that and that does satisfy this. So, you see there is a bit of degeneracy that has come up here that you know after having played Q1 star star player 2 could have actually responded with any function after player 1 has played Q1 star star star player 2 could have responded with any function. So, keep this in mind we will come back to this we will come back to this again. So, this is so this what have we established then we have established that this is a Nash Equilibrium that Q1 star star comma gamma 2 star star is actually a Nash Equilibrium. So, the strategies that we got by doing some you know sort of mental calculation kind of heuristic calculations was in is in fact a Nash Equilibrium of the game. But you can see there is some the subtleties have already started emerging that there is something else also that is going on it is not as simple as just you know this fellow responses and let me do this and so on it is not just not only that there seems to be more at play. So, let us ask is there another Nash Equilibrium or can you think of another Nash Equilibrium yes which is no. So, this was the I mean so the this was the bit and you fell for it. So, the so I mean you might think that for example I could just replace in place since there was so much of flexibility here to take any gamma 2 right. Can you take any gamma 2 really and will that be still a Nash Equilibrium. So, suppose I took gamma 2 some other gamma 2 which was not optimal for so suppose Q1 star star is 100. I took a gamma 2 which is which is not optimal for 90 it is optimal for 100 but not optimal for 90. The question is then is will that gamma 2 and Q1 star star which is 100 will that still be a Nash Equilibrium no exactly see it is still a best response to Q1 star star. So, that that other bogus gamma 2 that I just mentioned is still a best response to Q1 star star. But Q1 star star is not a best response to that there is no guarantee Q1 star star is obtained by assuming Q1 star star is layer 1 plays Q1 star star because player 2 is playing gamma 2 star star. It is optimal for rather let me put it this way the optimal quantity for player 1 to play is Q1 star star when player 2 is playing gamma 2 star star. If player 2 decides to play something else then the quantity for player 1 would also change. So, although there are infinitely many best responses here for player 2 you cannot does not mean that player 2 can really pick whatever he wants. This happens again we have seen this also in when the finitely many strategies for the player for each player. That is not through the Nash Equilibrium. I mean of course that is so whatever he wants in Nash Equilibrium it is obviously. So, this is something we saw even when there were finitely many for when we are looking at mixed strategies right. A player will have in general infinitely many best responses but not all of them will actually constitute a Nash Equilibrium. So, then now tell me what is tell me another Nash Equilibrium. So, let us go back to the definition this is what we are looking for. So, let us go back and see this. My contention is this is also a Nash Equilibrium. But I will we have to interpret it appropriately. This here this remember this what was this? This was the Nash Equilibrium of the simultaneous move game when players were deciding the quantity simultaneously. Now player 2 is observing what player 1 is producing. Yet I am saying that this one which was this equilibrium of the of the simultaneous move game is still a Nash Equilibrium of the dynamic game. Let us see how that happens. Firstly that equilibrium was mentioned in the space of quantities right. So, I have to first produce for you an equilibrium in the space of strategies here. So, it is so let us first do that. So, my claim is that for I am going to take for player 1 I am going to still take the quantity that he was producing in the simultaneous game. For player 2 I am going to take a function gamma 2 star function gamma 2 star and it is a this is going to be a function of q 1. But I will take it to be a trivial function I will take it to be a constant function. So, gamma 2 star is defined like this gamma 2 star is a constant function equal to q 2 star. What was q 2 star? It is the quantity that player 2 would have produced in the equilibrium of the simultaneous game. It was 1 minus C by 3 that quantity. So, this is now a this is identically equal to this. So, gamma 2 star is a constant function identically equal to q 2 star. Is this clear? So, this I am going to claim this here is a Nash equilibrium of the dynamic game. In the it is a it is a equilibrium of the dynamic game which means that it satisfies this. Yeah, it does not actually do not even matter, but let us but this is what is being going to be shown. So, that this is a Nash equilibrium of the dynamic game. So, again I need to check best responses. So, let us start with player 1. So, player 1 when he plays q 1 star and player 2 is responding with the function gamma 2 star. So, now what is the utility that he gets here? He is going to get the utility q 1 star comma gamma 2 star of q 1 star. But what is gamma 2 star of q 1 star? Gamma 2 star is a constant function. So, this is equal to q 1 of q 1 star comma q 2 star. Now, suppose player 1 let us suppose player 1 played some q 1. What happens to this? What is gamma 2 gamma 1 star of q 1? It is still q 2 star because gamma 2 star is a constant function. So, irrespective of the value of q 1 it is still going to produce the value is q 2 star. So, this is still equal to q 2 star. So, now you maximize this over q 1 what would you get? You are effectively just doing the best response to the Nash equilibrium simultaneous Nash equilibrium strategy of player 2. So, what is this? So, if you maximize this now over q 1 this maximizer here is q 1 star. So, which means that if player 2 plays the constant function gamma 2 star it is optimal for player 1 to play the quantity q 1 star. So, which means player 2 plays the constant function equal in which the value is equal to the Nash equilibrium from the simultaneous game then it is optimal for player 1 to also respond with his own Nash equilibrium from the simultaneous Nash equilibrium component from the simultaneous game. Is this clear? So, this is this is done we are done with. So, for player 1 this is done. What about player 2? So, for player 2 we have now q 1 star fixed and player 2 is going to pick any function gamma 2 and the utility he gets is u 2 of q 1 star comma gamma 2 of q 1 star. And now you are maximizing this overall functions and like we had in the earlier case the same thing something similar is coming up here also you can see. So, like we had in the earlier case so, what is the what is the what is the optimal gamma 2 for player 2 for player 2 to play it does not be essentially the he can play any strategies. So, long as any gamma 2 so, long as at q 1 star it is the it corresponds to the best response simultaneous best response to q 1 star. So, any gamma 2 such that gamma 2 of q 1 star is equal to q 2 star is optimal is the best response. So, in particular you can take the constant of strategy the constant strategy where which regardless of what the value of q 1 is always takes value q 2 star is also is also a best response. So, in particular gamma 2 star which is identically equal to q 2 star is also optimal. So, this is also optimal. So, in short what do you conclude we have got that now for player 1 and player 2 we have concluded that this here this is also a Nash equilibrium. Now, this is actually a very interesting and very general fact you take the Nash equilibrium of the simultaneous move game it will be inherited upon suitable interpretation as a Nash equilibrium of a dynamic game. So, let us understand what what is happening in this what is happening is in this in this sort of a game in this situation is player 2 is basically saying so, let us first interpret the earlier Nash equilibrium in the earlier Nash equilibrium what was player 2 doing in the earlier Nash equilibrium player 2 was responding optimally to every quantity that player 1 could have played that was his strategy gamma 2 star star was a strategy which gave him which was optimal which was an optimal response to every which produced an optimal quantity in response to anything that player 1 produced and and then player 1 then chose his quantity keeping this you know sort of in response to this in particular is this clear but in retrospect with player 1 having played q 1 star star player 2 could have played anything else but then anything else would not have first player 1 to play q 1 star star. But look at what is happening in this in this equilibrium in this equilibrium q 1 star q 1 star and q 2 star the single starred ones are a very special quantity a special quantities because they are Nash equilibrium of the simultaneous game. So, what what is happening here is player 2 is is playing this constant strategy saying always which is always equal to q 2 star. So, player 2 is basically saying does not matter what quantity you produce I am going to produce q 2 star and he is picking a quantity equal to that which would which would have which he would have played if he had if in the simultaneous movie. So, player 2 what he is doing is although he has the information of q 1 is playing as if he has no information right he is ignoring the information that he had he has the information of q 1 but he is playing a constant strategy which makes no use of that information but he is playing pegging it his quantity at the level that he would have played in the event that he had no information. Now, in response to that then player 1 is also sort of has to is is compelled to play as if there was no information available available to player 2. So, effectively this has become a this has become a this is also become a Nash he is also a Nash equilibrium of this game. Now, there is something strange going on here that player in the earlier Nash equilibrium in this in this Nash equilibrium player 2 was guaranteed to respond optimally to every quantity that player 1 was producing right the action that player 2 was producing was optimal in in gamma 2 star star the was the optimal action regardless of what player 1 played. Here in this one the action is q 2 star which is optimal only when player 1 plays q 1 star it is not optimal generically it is not optimal in general. So, there will be okay there will be there are values of q 1 for which this is not the optimal strategy optimal response from player 2. So, player 2 is actually by ignoring information or whatever is actually playing you know irrationally in some sense in the sense that he is being irrational in in what sense he is saying is that he is you know for for quantities other than q 1 star that is he he is there is no guarantee that he is actually producing the optimal quantity for himself. But then in as a result of that what happens is the quantity that player 1 produces is q 1 star itself and in that case he has the optimal response available. So, it is like this that there are in a dynamic game it will always happen that many possible paths can emerge from a game okay player 1 could have played any potential quantity from 0 to infinity okay the system could have evolved in any many different direction. But what actually gets evolved is a result of the strategic decisions that players can play. So, player 2 here is if is is promising to play irrationally or carelessly or whatever word you want to you know give for it for other values of q 1. But by promising to play in a in a kind of suitable way you know but that promise is not totally arbitrary it is a specific promise it is a promise to play q 1 q 2 star that particular value then forces player 1 to play q 1 star. And then in once player 1 is playing q 1 star q 2 star remains optimal anyways okay. So, what happens is this this promised irrational play at other points never really materializes in in the in the gameplay. What happened the gameplay actually sort of player 2 has effectively just navigated the game to the point where you know voila you know this is what I want anyway wanted to do you know I am optimal at this for this particular strategy for this quantity anyways okay. So, this is you will see more examples of this this is this is this is a very this is a very this is a very this something that happens generically in in dynamic games that there will be paths in a dynamic game that in retrospect might seem you know irrational irrational but those paths never actually occur and it is the irrational promises on those paths that makes certain that decides which paths will actually occur okay. So, I mean I mean it is a it is sort of you know it is an enlightened form of rationality in the sense that you are selectively being irrational to be rational at another level you know that is that is effectively what what a player is doing in a dynamic game. So, his player 2 is is deliberately but very very specifically being irrational he is not being you know arbitrary he is picking a very specific irrational response for other values of Q1 and that makes player 1 play Q1 style. Okay now next time I will show you actually a game in which this sort of irrationality actually benefits player 2 okay so that is so it is so you might argue that oh well the earlier one seems like the more natural way of reasoning okay you know we calculated the best response we use all the calculus we know etc etc but you will you will you will see that there are there are there are games where this second you know where this sort of equilibrium is actually more beneficial to the player so where ignoring information can actually benefit the player yeah finished your calculation means not necessarily because you have to assume a form right you have no no no you have to know you have to you have to see so if you want to solve these these kind of games you have to first assume a the a form of dependence. So you will have to basically assume whether Q2 depends on Q1 or not so you will get you will you know if you want to use I mean since you brought this up you want to do this with calculus right you will get things like derivative of partial derivative of this with respect to Q1 you will then have to fork from there is this equal to 0 or is it equal to something okay and that tends that tells you the form of whether something is being assumed to be a function of this or not yeah so it this is it is it is yeah it is not always the case in this case player so not even in this case actually the game that I will show you next this time this is it is not that clear but in there are games where ignoring information is beneficial there are games where ignoring information is not beneficial see when in a so the the main lesson is this they see in so now that we have we will be keep we will keep talking about information again and again the main lesson is that when you have a single player problem right more information helps you know you want to take a you want to decide which route to take if you want to go from one point to the other more information about the traffic more information about various conditions the road condition more information about you know where maintenance they breakdowns etc all of those things will actually help you okay more information only gives you a better decision in the case of a single legion you know once there are multiple players involved more information does not necessarily help it is strategic sometimes to act like you have not heard anything that is also a part of play that could have in fact benefit you also I am not suggesting you how to play in real life but but these are these are some of the consequences of game theory I mean once we follow through the logic right you get all these insights which are which are really incredible that so this is this is actually of you you will see that that you in fact the game that I will show you next time this is a fantastic conclusion that you can in fact you know act like you are like you are crazy and then get others to act to your line effectively yeah gamma 2 star star of q 1 star is equal to gamma 2 star star was the this thing right yeah is equal to q 2 star that is right because it is the response right to this to q 2 star yeah but see the thing is but player 1 does not play q 1s player 1 does not play q 1 star player 1 plays q 1 star star the double star 1 because he because of this dependence right there is a this is where the anticipation is happening is he is anticipating how the how player player 2 would respond okay