 So, now we will start with our study of dynamic games and the key issue we will now begin to tackle is the issue of information. So, first let me ask you this, we I kept mentioning that all in all our games the players were playing simultaneously, so when when do you say that players are playing simultaneously. So, let us take the the example that we had that we had looked at in last class right it was this is game of matching pennies, I had looked at the asymmetric version but the here is the symmetric version. So, the game of matching pennies with a symmetric version is something like this, these are the payoffs. So, the players have to put down the penny coin with either heads up or tails up and if both if both players put up the same face then player 1 gets 1 and player 2 gets player 2 gets minus 1. If both players if one player puts up heads the other player puts up tails then player 2 gets plus 1 and player 1 gets minus 1. This was the situation. Now, what this this game obviously is meaningful if so this was a simultaneous move game. So, what does it mean for players to play simultaneously? So, simultaneous if you think about it in the in the in the in the in the in its English meaning you would mean that they are actually instead putting the coin down at the exact same instance okay. So, physically at the exact same you know cosmic time that is when they are in fact putting down the coin but that that is not so the game remains strategically the same even if they put the coins down one after the other but without knowledge of what the other one has done right. So, what we mean by simultaneous really is that the player does not know what the other player has done. So, simultaneous in a in a game theoretic or strategic context has not not so much to do with time as it is to do with information right if players do not have information of what the other player has done then it is as good as simultaneous is this clear okay. So, so now now consider the following let us take the let us take another game okay. So, so this let me just write out whatever I just said that this is simultaneous and we mean that it is simultaneous really means players do not know what the other player has played this is what it means for a game to be simultaneous okay. Now, here is another game let us this is this is actually a game in with infinitely many strategies for each player. So, player say suppose there are two players that are producing some homogeneous good two firms let us say they produce a homogeneous good the homogeneous means a good they are producing a commodity or a good that that genius a good is homogeneous means that the the the the one that is produced by the the thing that is produced by one player is is indistinguishable from that produced by the other okay. So, what matters is how much they produce not what they produce so long as they are producing the same thing okay. So, so there is that means there so examples are for example electricity electricity produced by one plant is the same as the electricity produced by another plant you know you do not really distinguish between the electrons coming from one plant and the electrons coming from the other. So, there that so this this this sort of a good is a homogeneous good you know many of the other things that we you know that we use on a day to day basis are homogeneous you know cereals rise wheat etc there you know it does not matter whether this come from this firm or that firm so long as the the breed of the of the of the rice is the same it is it is the same rice right okay. So, that is not the same in other goods like for example, shampoos will be different for different shampoo made by one company will be different from that made by the other company toothpaste made by one will be different from the other cars etc etc those are not homogeneous. So, one thing that we know about when goods are homogeneous is that what matters is the total supply of the quantity in in the market that determines the price okay and the the price is can be characterized usually by some sort of a formula you would kind of know what the how the price would behave as a function the market price would behave as a function of the amount of quantity produced alright. So, in this case so we have two firms producing homogeneous good suppose Q i is the quantity produced by produced by firm i and let us say that the price okay the price the price in the market that is the is a function of the quantity is produced by both. So, it is a function of Q 1 and Q 2 and so I am going to and I will assume it has it takes a very simple form I will assume that it is takes the form 1 minus Q 1 plus Q 2 okay. So, this has been normalized to eventually give you this sort of form if the price is 1 minus Q 1 plus Q 2. So, it is a function of the total production okay that is Q 1 plus Q 2. So, if this is what the firms are faced with then the firms have what they would do is they well they they each firm has a wants to maximize its profit as a function of its profit is is a function of Q 1 and Q 2. So, that would be equal to 1 minus Q 1 plus Q 2 which is the price times the quantity it has produced which is in this case Q i minus the cost that it that it incurs in producing it okay. So, then let us suppose that is C times Q i okay. So, this is the so this is a cost okay. So, this is per unit cost per unit quantity okay. So, this is the this is what these firms produce this is what the objective of each firm is. So, the each firm wants would like to now decide the quantity it should produce in order to maximize this profit okay. Now, when the firms are trying to decide their quantity is simultaneously okay then essentially what they are doing is they are they are trying to decide what quantity they should produce without knowing what the other would would be doing right. So, then what would be the then what the way we would solve for this look for a an equilibrium a solution for this sort of a situation through the Nash equilibrium. So, can you tell me what would be the Nash equilibrium of this game? So, this U i is actually you can see is actually quadratic and concave in Q i okay. So, you can solve for this can you tell me what it would be? So, let us let us just differentiate this or let us write this for U 1 and U 2. So, I let us differentiate this and put this equal to 0 with respect to Q 1 and let us also put this equal to 0. What do we get here? So, I will get yeah can it will be what this would be? So, this will be 1 minus 2 Q 1 plus minus Q 2 yeah this should be Q 1 sorry this is Q 1 minus Q 2 minus C equal to 0 and then I a similar equation for the from the second one would give me would give me 1 minus 2 Q 2 minus Q 1 minus C equal to 0. So, can let us solve this simultaneously and tell me what you get 1 minus C by 3 right. So, Q 1 you get Q 1 star equal to Q 2 star equal to 1 minus C by 3 okay. So, this is the Nash equilibrium. So, the Nash equilibrium of this if these of the game where if these players were playing simultaneously would be that they would each choose these quantities 1 minus C 1 minus C by 3 okay. Now, let us take this one step forward and let us ask what if player 2 what if this game was not simultaneous okay. So, what this means is suppose player 2 okay suppose player 2 can observe suppose player 2 can observe what player 1 plays if player 2 can observe what player 1 plays okay then what is the strategy of player 2. So, let me go back here. So, first tell me here in this game what were the strategies of the players they had to decide the amount of quantity to be produced what is the space of strategies in this case it will be said well I have not put said it explicitly but it is it is all non-negative numbers right any any any any basically the space of strategies is 0 to infinity is a space of strategies is for each player okay. Now, what is the now here player 2 can observe what player 1 plays okay player 2 still has the same objective it is still profit profit is still given by the same formula okay. So, he is still the the the price that will come in the market is still going to be 1 minus Q 1 plus Q 2 alright the cost is still C times Q C times Q 2. So, on the face of it nothing seems to have changed in the problem in terms of its specification right it seems like well the utility the utility function is the profit expression is still the same the the price is the same the the cost is the same but somewhere something has changed when P 2 can observe what P 1 can has played right because when P 2 can observe what P 1 has played he will obviously play differently. So, where is that manifested now. So, what are this let me ask again this question what are the strategies now of the players in the other earlier case the strategies of the players was 0 to infinity now what are the strategies of the players okay very good why right right. So, what are the strategies for the players see there are so this this is your first introduction to a simple dynamic game this is a dynamic game is one where players have some there is some player who has some information which is different from another player in a simultaneous game every player starts with the all players start with basically null information 0 information with right whereas here there is player 2 who has the information of what player 1 has played and when he has this information it makes sense that he picks his quantity then as a function of that information right if he had say see here in this case the information is the amount of quantity that player 1 has produced if the information was something else like for example it was the information is about the weather or something like that which player 1 did not have then he would pick his quantity as a function of that right. So, the point is that now that player a player has additional some additional information his strategy is not merely to decide how much quantity to produce but it is to decide how much quantity to produce as a function of what he knows. So, the strategy is not really the quantity the numerical quantity itself but the function that maps his information to a quantity okay. So, the players player 2 strategy now strategy is a function from 0 infinity to 0 infinity. So, here we distinguish between two different terms. So, this is what we call a strategy in a dynamic game the quantity that he actually produces and that is what in fact eventually influences his cost and his eventually influences his profit is that is what is called action okay. The quantity that he produces is the action. So, Q2 here this is his action and Q2 is chosen as a function of his information information being Q1. So, let us call this function gamma 2 this here is his strategy and it is chosen as a function of Q1 this here is his information is clear. So, the so strategies now are whenever you are in a dynamic game strategies now necessarily become functions okay because there is someone has some information which is not trivial right. So, the the strategy for that player is a function then from that goes from his from his information to his action all right. So, the the immediate shift that you see when you go from a static or simultaneous move game to a dynamic game is that the problem necessarily goes into function space because now everything has to be dealt with in the space of functions all right. So, those of you have taken my stochastic control course you would have seen this over there as well when you are when we write out we cannot write a stochastic control problem in terms of only actions you have to write it in the space of policies right. And that the space of policies there is basically the again the space of functions okay. So, then the reason for that is again because stochastic control is about at the end of the day about information it is about what information you have at at instant in time. So, because of and because you have to map that information into into an eventually a control action this issue is usually moved in the case of deterministic control because there you have to simply see take a sequence of actions and the information and the information is available to you as a look ahead because of the lack of randomness in the problem all right okay all right. So, I will not go into that those digressions but the point is the so now that you are now that p2 knows q1 it makes sense that he chooses his he tunes the quantity that he takes based on what q1 is and you have to allow for him to have that that flexibility and so therefore his strategy then is a is a function that maps q1 to is a function that maps q1 to q2 all right. So, so the space of strategies then for player 2 is then the space of all such functions space of strategies let us call it capital gamma 2 is a is these functions gamma 2 and gamma 2 maps now what is the space of strategies for player 1 yeah the space of strategies for player 1 is still to pick a quantity right he has no information to start off with. So, there are there are two different questions here one is what is the space of strategies okay means what is the allowable space through which he is picking his picking his strategy the other is what is the optimal strategy that he should be playing okay now I am not asking you what the optimal strategy for player 1 is I am asking you what are this what is the space through from which his strategy has to be chosen and the space from which his strategy has to be chosen is simply a quantity player 1 has to pick a quantity player 2 then observes that quantity and then decides. So, in terms of so the strategy for player 2 is a plan effectively right it is a plan which says that okay you know if if player 1 may produces 100 quintals I will produce so much if player 2 produces 200 player 1 produces 200 quintals then I will produce so much player 2 produces 300 quintals then I will produce so much essentially for every value of q 1 he has a he has a response planned right he has a he has a he has a plan that gives him a quantity to produce as a function of what player 1 has would hypothetically produce player 1 does not have any need to make such plans because he has no you know there are no contingent informations for there is no contingent information for him he starts he is the one who starts the game he plays player 2 observes and player 2 responds okay now that does not mean player 1 does not have to strategize that is a different matter altogether but he does not need to have a contingent plan which says that you know this is my information then this is what I will do that is my information then that is what I will do etc. So, player 1 space of strategies for player 1 is this is still 0 infinity okay all right now that we know that these are the space of strategies for the players. So, now tell me how should these players play okay so now P2 so P2 does so he will pick fine P2 looks for a quantity such that Q2 star such that if he looks at his utility from Q2 star that is better than playing any other Q2 now this looks oddly like the Nash equilibrium but is there something that I have that is different yeah so if this is a potential mode of play is what I am writing here so what P2 could do is that he could say well I will play a Q2 star that maximizes this expression and the what in this expression what do you observe about Q1 Q1 is not fixed at a star value so this is just some value of Q1 right so he is going to just take up a hypothetical value of Q1 find whatever is the maximizing Q2 right maximizing Q2 from here and take that as the Q2 star so his so then as a result Q2 star will necessarily be a function of Q1 is this clear so this is this is then the function of the dependence the functional dependence that we just said so the so potential mode of play for P2 then is to say well he could play this strategy gamma 2 star let me call this gamma 2 star where what is gamma 2 star gamma 2 star is is the is that gamma 2 star of as a function of Q1 produces a quantity Q2 star where Q2 star satisfies this right so so gamma 2 star for every Q1 will specify is the quantity Q2 star which which optimizes this is clear so okay so now with this in mind so this is this is the mode of play for for player 2 and what is the mode of play for player 1 if player 2 is going to respond is going to play Q gamma 2 star which is basically specifying the quantity that maximizes his utility as a function of Q1 okay now what is the mode for how should player 1 play okay so a mode of play for player 1 would now be would now we see like like she asked does does player 1 know that player 2 is going to respond okay or player 2 does player 1 know that player 2 can observe player 1's quantity well so this is that that that is known to player 1 so the players know that this is the this is what will be known to them okay during the game so now that player 1 knows that player 2 is going to respond with Q gamma 2 star what player 1 should then play is well he can say well let he can play he can essentially anticipate that that is going to be the response of the player okay so a potential mode of play for player 1 now is to play Q1 star so let me use star star here so that there is no confusion okay so what player 1 would then say is he plays a quantity Q1 star star such that it maximizes the utility that he would get assuming player 2 would respond with gamma 2 star star okay so assuming taking Q2 to be Q2 as to be equal to gamma 2 star star of Q1 he maximizes this over Q1. So you see one thing that has happened here so the there is now a dependence here the the quantity of the other player is not just the quantity that the other player is producing is not fixed in this optimization the quantity that the other player is producing now is a function of what player 1 would produce so player 1 is anticipating that that player 2 is going to respond in an adaptive way so he is going to adapt to whatever quantity I produce and so keeping that model in mind he optimizes his is he maximizes his profit is this clear so his so he is maximizing this knowing that Q2 star star is a function of so knowing that Q2 star star is a function of Q1 okay so now in a game we always assume that the utilities are common knowledge we assume that who knows what at various times who would know what at during the game during gameplay is also common knowledge so gamma 2 star star is something that player that player 1 can in fact compute on his own right he can compute that that is that is what player 1 would play and then respond and then come as a result he can then compute this whole expression is this clear so he can compute this and get his gamma Q1 star star