 So, this here is a potential solution of the game. That means the solution being that player 1 plays Q1 star star and player 2 comes up with a quantity, comes up with a plan to play gamma 2 star star and the actual quantity that he ends up producing is gamma 2 star star which is a function of Q1 star star. The actual quantity that he produces is gamma 2 star star of Q1 star star. So, this here is the action, these two are the strategies. So, then if this seems like a very natural gameplay, player 2 is going to respond with by maximizing his profit as a function of anything that player 1 would have played, keeping that in mind, player 1 would then pick a quantity that would appropriately then maximize his anticipated profit from such action. Now, tell me, is there any other gameplay possible? So, we have gone about this kind of in a somewhat intuitive way and what we have said is that we have said this is what player 2 knows. So, let him just optimize his profit as a function of Q1 and we got a certain function and we said that should be his optimal strategy, assuming that we said what should be player 1's optimal strategy. There is a pitfall in this is that we have not really formally written out what is, what do we want from a solution, just the way we did for simultaneous move games. See, in a simultaneous move game, we said, what do we want from a solution? We want this and we concluded that what we want is this property that in the absence of communication, no player should have an, since there is no communication, no player should have an incentive to unilaterally deviate from his strategy. We have not really formally thought about this problem in that way. We just sort of solved this more like a puzzle, like what seems more like a newspaper puzzle. So, let us now formally write out again what is it that we would want as a solution of this game. So, if you want to talk about solution of this game, we need to first ask, we decided what the strategies are. The strategies space, the space of strategies are, space of strategies for player 1 is just 0 to infinity, space of strategies for player 2 is all functions from 0 to infinity to 0 infinity. This is now fixed. Now, what are players then trying to do? How are we solving for the game in this space of strategies? Yes, but okay, so players, so okay, let me ask you the following questions. When are these strategies being chosen? So, are these strategies chosen? When are these strategies being decided? So, in particular, let us look at a simultaneous move game, let us take matching pennies, both players would, each player, the players play simultaneously or without revealing the one's action to the other. Now, here the action of one is being told to the other. So, there is, there seems to be like there is some communication going on in this game. So, if there is some communication going on, then do we need to revisit the Nash equilibrium? Yeah, or I mean, so can we apply all the thinking that we had for the Nash equilibrium for this game? So, the point is that because now this, there is an observation involved here, player 2 can observe what player 1 is doing and the simultaneous nature is lost. Then, you know, all our earlier thinking of unilateral deviations, etc., etc., is that still valid? Okay, why, why is it still valid? See, that is why the earlier question that I asked matters. When are these strategies being chosen? That is the main thing. See, there are two different, let us say, time instances where things are happening. One is when the strategies are being decided. Okay, that is when players decide to play their, when player 2 decides to play gamma 2 star star and when player 1 decides to play gamma 1 star, sorry, q1 star star. Okay, so that is, that is the, that is one instance. The other instance is when the actions are decided. Okay, the action for player 1 is trivial, it is equal to his strategies. So, you can say it is chosen at the start of the game. For player 2, the action is not equal to his strategy. For player 2, the action is equal to, equal to a function of his strategy and for that function to be evaluated, player 1's action has to be put in as an argument. Right? So, player 2's action is chosen during gameplay, but the strategies are chosen before gameplay. So, the communication that is happening, if at all, you know, or any transfer of information that is happening, is happening during gameplay, not before gameplay. So, at the time when the strategies are still being chosen, that means at the time when q1 star star and gamma 2 star star are being decided, at that time the game was still a simultaneous move game, because they are being chosen without knowledge of information of the other, what the other one has done. Action is being chosen later using the information of the action of the other player, but the strategies are being chosen simultaneously in much the same way as the strategies are being chosen, as the kids are deciding whether to play heads or tails in the time in the matching pennies game. Is this clear? The chronology, so the point is the, this is actually very important that you know, you realize that the strategies are being chosen at the start of the game. Now, why is it well defined to talk of strategies at the start of the game? Does this make sense? Is it meaningful to define strategies at the start of the game? Do you have all the information needed to define strategies at the start of the game? You do. So, on the other hand, you do not have the information needed to define actions. At the start of the game, player 2 cannot say how much he is going to produce. He can only say how much he is going to produce as a function of what the other one would do. You know, he will say, oh, based on what the other one would do, I would do this. But he cannot say, we cannot put a number and say, I am going to produce this much quantity. So, what he, but he can say that at the start of the game, what is going to be my plan, that if he produces so much, I am going to produce so much, etc., etc. So, the entire function can be specified. So, a function gamma 2 that maps 0 infinity to 0 infinity can be specified at the start of the game for player 2. Similarly, a quantity can be specified for player 1. So, these strategies can be decided at the start of the game and therefore, it is possible to move this game to the space of strategies in which and in to a situation where the strategies are being chosen simultaneously. Is this clear? So, let us write out now the utilities of the players as a function of their strategies. So, let us write out first for player 2. So, what are the strategies of the players? Player 2 strategy is a function gamma 2, player 1 strategy is a quantity Q1. I can even write, you know, for if you want for symmetry, I can even write that the quantity that player 1 strategy is also a function. It is a function of no variable. So, it is basically a quantity. It is a constant. So, it is not a function of any variable. So, it is a trivial function. But you know, so this is also, you know, in some sense a function gamma 1 of no variable. So, in other words, it is a constant function. So, what is this gamma j2 of gamma 2 comma Q1? So, this is equal to U2, U2 of we are trying to write the payoff of player 2 as a function of his strategy, not as a function of quantity. U2 was in the space of quantity, right or was in the space of actions. So, now as a function of the strategy, I want to write this. So, as a function of this, yeah, so it will be gamma 2 of Q1 comma Q1. Now, question is this well defined, is this a number? It is a number because if I give you a quantity Q1 and a function gamma 2, this evaluates to a number. If Q1 was a function of something else, then I would need to give you that something else for this to become a number. But Q1 is already a number. So, this is now a number. Likewise, this is also well defined. So, now I can do the following. I can say I will forget about this thing that I have on the right, this thing in this that I have on the right and just think of this these as my payoffs. So, essentially now I have a game in which player 2 is choosing a value of a function or player 2 is just choosing a gamma 2 from a space capital gamma 2, player 1 is choosing a Q1 from 0 infinity, player 2 wants to maximize J2, player 1 wants to maximize J1 and these choices are being made simultaneously. So, now I can ask for a Nash equilibrium in the strategy space. The Nash equilibrium of this game now is and what is the Nash equilibrium now? The Nash equilibrium is that no player would want to deviate given the other player. What the other player is playing? It is a profile of strategies such that no player would want to unilaterally deviate. So, which means that player 1 would want to stick to playing Q1 hat assuming the other player plays gamma 2 hat and player 2 would not want to switch to a different function assuming player 1 sticks to a quantity Q1 hat. So, player 2 wants to stick to this function. So, player 1 does not have an incentive to change its quantity assuming player 2 is going to respond with this plan the gamma 2 hat plan. And player 2 would not want to switch its plan assuming player 1 does not want to change its quantity from Q1 hat. Is this clear? So, this is for all Q1 in 0 infinity and this is for all functions gamma 2 now and gamma 2. This is now, this is a Nash equilibrium. This is the notion of a Nash equilibrium. And it can be justified in just the same way as we justified a Nash equilibrium for a simultaneous move game. Although there is a exchange of information during gameplay, we can still justify this Nash equilibrium as a valid solution concept because there is no exchange of information at the time of choosing strategy. Is this clear? Yeah, player 1 does not know gamma 2. You mean the gamma 2 that player 2 will so we can justify this as a notion just the way we justified the Nash equilibrium for a simultaneous move game. It is a point from which no player would want to unilaterally you know we justify it as if any point has to be a solution then it has to be one where under the communication constraints deviation is not possible. And just that deviation now is deviation in the appropriate strategy space. For player 1, it is in the space of quantities. For player 2, it is in the space of functions or plans. We will come to that. So, little while back I said P1 can compute gamma 2 star star that which is a specific strategy. It is a device for him to compute his own quantity which is Q1 star star. So, I was just trying to sort of do a mental reasoning can this be a potential way of solving the game. And I said in that as a step in that I said we can actually do this because this is a well-defined thing for to compute a quantity is Q1 star star it is I need to solve Q player 1 has to solve this optimization for which he has to compute he needs gamma 2 star star for which he that he can solve by from the above optimization. Is this clear? So, it was just a sort of like you can say a mental calculation to just arrive at some you know candidate strategy. But after that I said what we said was that you know this is not a very formal reasoning because it is a very you know sort of a case by case thing we just sort of heuristically reasoned that this is how it should work out. But then if we think of the problem in the space of strategies formally really what all we are looking for is a Nash Equilibrium now in a different space and that is how we should be solving for the game is it okay.