 So, let us continue with the study of dynamic games, last time we started with looking at dynamic games by first considering this example of two producers producing a homogeneous good and the second producer could observe the quantity that the first producer had produced. Now there is I will talk about a generalization now or maybe let us say a different model let us look at this particular example. So, the game starts with at let us say this node X X is some juncture in time this is we just denote that instant with X and at that instant it is the turn of player 1 to play. Player 1 can play either L1 or R1 if he plays L1 the game goes to node Z and then the game ends with payoff 0 comma 2 so 0 for player 1 and 2 for player 2. If he plays node Y if he plays R sorry if he plays R1 the game goes to node Y and then at Y it is the turn of player 2 to play. Player 2 has two options L2 and R2 if he plays L2 then they would get the players get minus 1 comma minus 1 both get minus 1 and if he plays R2 they each get plus 1. Players are maximizing players. Now can you tell me what should be the mode of play in this case? So player 2 firstly let us make clear our assumptions player 2 knows when it is his turn to play. So he knows when the game reaches Y he knows that it is his turn to play and he has to choose action either L2 or R2. When the game reaches node Z player 2 also knows that well it is a game over there is nothing more to do and the game ends that he has no nothing to choose from at node Z that is also something that is not player 2. Player 1 knows at the start of the game that it is his turn to play and to start the game. Every player knows this whole tree they know that these are how these are how the payoffs are going to play out this is what is going to these are the options available to each player and so on. So all of this is common knowledge and during gameplay player 2 knows that when it is his turn to play. So all of this is known so what this means is in short when it is player 2's turn to play at node Y he knows also that player 1 has in fact played R1 you can conclude that from the graph itself. So he has this information that player 1 has played R1 in the previous step. So now tell me how should the players play? Why? So the answer suggested here is one mode of play is to is player 1 plays R1 and player 2 follows it up with R2. So one way of reasoning about this is the following. So player 2's turn to play will come at when it comes when the game reaches Y and at once the game reaches Y at that stage the player 1 has already played R1 there is no going back on that R1 has already evolved the state is that the game is at state Y. Now player 2 has to decide what to play given the options that are in front of it the options there are L2 and R2 the options are L2 and R2 and in this case naturally given these two options R2 is a better option so player 2 plays R2. So following that now if you think about this fact that if the game reaches Y player 2 is going to play R2 then player 1 has the choice of whether to play L1 or R1. So if he plays L1 he takes the game to Z and then gets 0 for himself. If he plays R1 he knows that player 2 is going to then respond with R2 and then he is going to get 1 and player 1 is going to get 1. So the choice for player 1 is effectively a choice between playing L1 and then getting 0 or playing R1 and then getting 1. So between these two naturally he would pick the choice R1. So one mode of play therefore possible mode of play player 1 plays R1 and then player 2 plays R2 and why R1 because better than playing L1 better than N1 which would have given him 0. And player 2 plays R2 why because this is better than playing yeah you had a question he knows the entire tree the entire this whole whole tree is known to do all to both players. So here in this so whatever I said is specific to this structure okay here being at Y itself lets you conclude that player 1 has played R1 yeah yeah so so player 2 knows that he is at node Y okay from node Y he can conclude that the only way Y could have happened is that player 1 has played R1 all right. So that is how he concludes that player 1 has played R1 it could be that he does not know which node he is at and so on those are more complicated problems we will come to that in a moment but right this is this is what he can deduce from this from this structure okay if there were now to answer your other question which is suppose there were you know branches coming out from Z also then if player knows that he is at Z then he would know also from there that player 1 has played L1 okay. So this we have to build into the assumption of the game about what player knows then we will come to all of that in a moment okay okay so this is one possible mode of play that player 1 plays R1 player 2 responds with R2 the way you have reasoned about this is is essentially similar is a kind of backward reasoning we said we start from the end at the end player the last player to play is player 2 he has two options L2 and R2 in the in the event that the game reaches node Y he has two options L2 and R2 and at well at that node then he would play R2 and then the question for player 1 is whether you end the game right there by taking the game to node Z or you actually take it further to node Y and let player 2 play okay all right now question then is so this is what let us also write out the payoffs that the players get what are the payoffs that players are going to get through this in this mode of play player is both players get 1 comma 1 okay both players get 1 all right so question then from my question to you is is there another mode of play so there is no such there is no formal definition we have to think of modes of ways of playing and because we have not yet defined what a rational outcome of the game is yeah okay yes okay any other okay so that could be another way but we have already seen that you know minimax type of thinking is not applicable when there is a zero even there is a non-zero some game so if you if each player thinks of the worst the case damage that the other player could do and that is that does not that is not the right way of reasoning because it will lead you to regret it at the end of it all okay so this this is not a zero sum game so therefore that that logic is not applicable is there any other way by which you can we can reason about what is going on in this game okay so I will claim that there is in fact another plausible outcome to this game okay and the other the outcome is the following the outcome is that player 2 another plausible outcome the outcome is that player 2 responds player 2 when it when the game comes to y player 2 actually plays l 2 okay player 2 plays l 2 at y l 2 at y and player 1 and player 1 plays plays l 1 at x now this seem this may seem kind of absurd okay because player 2 plays l 2 at y is what I have said here he plays l 2 at y okay but player 1 is playing l 1 at x now if player 1 is playing l 1 at x okay player 1 plays l 1 at x the game never reaches y and the game actually goes to z and and the game and then it is game over right the game ends there so the player 1 play playing l 1 at x means that there is nothing for player 2 to do but yet player 2 is playing l 2 at y but let us just think through this and see how how this plays out suppose player 2 plays l 2 at at y okay what does that mean the player 2 has promised to play l 2 at y okay now having promised to play l 2 at y what would what would player 1 do the if player 2 has promised to play l 2 at y player 1 would not want to take the game to node y but instead want to take the game to node z by playing l 1 at node z it is of course true that player 1 player 2 has nothing to do okay the game has ended player 2 has has no action at at node z okay but but let us think about this from the point of view of player 2 player 2 has basically got got player 1 to go to node z and end the game by promising to play l 2 at node at node y right of course by promising to play l 2 at node y he he is essentially ensured that node y never actually comes and the game in fact goes to z and the game ends now let us think in terms of a Nash equilibrium here so assuming player 2 plays l 2 what is the best response of player 1 if player 2 is going to play l 2 here what would player 1 want to do naturally he would want to play l 1 and if player 1 is playing l 1 what would player 2 want to do he has nothing to do he will he just then he can just stick to his you know promise of playing l 2 at node y right so player 2 has only committed to playing l 2 at node y but it is a commit it is a commitment which you can say is not credible in some sense because the game never really goes to node y and no one can ever verify that he would have in fact played played l 2 at node y the game in fact because of that commitment because of that promise he is in fact he is he is in fact he is he is ensured that the game in fact goes to node z no no we will come to that so what we know we we just want to solve for a mode of play we will come to a formalistic what as of now all that players know is that this is the tree okay that the the tree looks like this and we want to come up with modes of play so what you you can think of see so so let us look at the other one okay now assuming player 1 is playing playing r 1 what would player 2 want to do r 2 r 2 is the best response to r 1 and assuming that player 2 is going to play r 2 what would player 1 want to do he would want to play r 1 so there is a kind of equilibrium property to r 1 r 2 but there is also an equilibrium property to l 1 l 2 in the same sense if player 2 is going to indeed play l 2 it is better for player 1 to play l 1 rather than go to node y and if it is if it is the if player 1 is playing l 1 then player 2 can promise whatever the hell he wants at node y including playing l 2 but playing l 2 is the one that makes player 1 go to l player 1 go to node z so he has so the so we will get to the information part later but but let us look at the rationality now okay now of course there is something troubling about this l 1 l 2 equilibrium and what is the troubling part the troubling part is this promise of playing l 2 right if the node if the game does go to node y it is not optimal for player 2 to play l 2 if the game does go to node y player a player 2 would want to play r 2 but by promising to play l 2 if his promise remains unverifiable by promising to play l 2 he ensures that the game never actually goes to node z node y it goes to node z is this clear so so there is a so what player 2 has done has promised to play irrationally at this node but then player 1 gets scared of that because he realizes that if the if the game goes to node y and this guy does indeed play l 2 then he is going to get minus 1 and minus and then he is better off therefore playing l 1 which gives him 0 is this clear so there is a there is clearly something troubling here which is that that the there is a there is an element of that the where player 2 you can say in some sense is thinking irrationally here is or playing irrationally but he is he in fact playing irrationally or he is he just committing to play irrationally because he is in fact what when it comes to the actual play that plays out he never actually is irrational he never has to play irrationally he never has to take an irrational action at any node this is the effect of commitment is just promising that he will play irrationally is promising to I will do this and and so on but then therefore this the he never comes the player 1 never actually comes to node y and his promise remains you know you can say unverifiable hmm yeah so let us let us let us go through now since you may since you are bringing up rationality let us bring up rationality in in this equilibrium r 1 r 2 equilibrium what did players get players got 1 and 1 and now player 2 by throwing these tantrums has been able to get 2 by saying that he is going to play irrationally at node node y by committing to play irrationally at node y he has taken effectively ensured that the game goes to node z and at node z he gets 2 now can you really say that player 2 is being irrational in this no not necessarily which player so play you can say well there is a really the question at hand is not about rationality but whether rationality can include commitments of irrational behavior within it ok thus promising to play irrationally include can that be included as a possible mode of play in a completely rational setting you understand because selective or posturing irrationality or posturing to be irrational pretending to be rational promising to be rational you can give it whatever way word you want but this kind of a commitment to play irrationally is that yeah it can maybe that is part of a higher level of rationality right so once there is a dynamic game and the game is non-zero sum a whole lot of new phenomena start coming up and the reason this is happening is because there is a there is the game is dynamic there is information that is that is there that is being that is that is present as an element of the game if player 1 if player if these players were playing simultaneously ok this none of these issues would arise ok so so let us let us now think about this formula and let us see how we can so these what all I have said here these are two plausible modes of play but we need to think formally and say what write this down formally as a as a game write out what the set of strategies are for both players and let us see if we can get to this through the Nash Equilibrium ok so let us let us write this formula what are the strategies for the players first first ok let us let us take this one step back what are the actions for the players what are the actions for the players at player at at node at node x the actions at node x the actions for player 1 are l 1 and r 1 at node y what are the actions l 2 and r 2 for player 2 player 1 does not have any action at node y player 2 has actions l 2 and r 2 at node z it is you can say well the game ends but we can also say that we can put in a a kind of fictitious action here we can say it is player 2's turn to play but he has only one action which is to do nothing ok let me just put it like this he has only one action here a fictitious action and that is to do nothing and it gives payoff 0 comma 2 to the player ok so it is this is completely equivalent it is like yeah yes yes no no no so but see the so I will I will come ok we will what I meant was that these phenomena do not occur ok so this in fact l 1 l 2 since you brought it up yes it is in fact an equilibrium that arises out of ignoring information if the essentially player 2 is acting as if you know it never mattered whether player 1 has played r 1 or not he is acting like he is playing the way he would have if he did not have the information of player 1's action ok so but but we will have to you know exactly how because you know if he does not have the information of player 1's action then how does he know that he has to do nothing at z and and this and why all of that has to be written out properly but loosely speaking essentially what player player 1 is player 2 is effectively doing is ignoring the fact that player 1 has that at why player would have player 1 would have he the game would have come to know why only when player 1 would have played r 1 this has been completely ignored so he is just simply come committing to play l 2 no matter what now one way of doing this if you since you have brought it up let me I sorry for jumping a little bit ahead but one way of doing this is to you know I can think what I call do nothing here no I can just relabel this as l 2 so what he is effectively doing is regardless of what has happened I will just play l 2 I just relabel it I just change the name of the action I will call it l 2 I mean doing nothing is playing l 2 we can yeah in fact we can go one step ahead we can add another branch r 2 here and give the same pay of 0 comma 2 to r 2 I can even do this so this is the all of these this is a little I mean it is not as slick as I am making out out to be but but general I mean you can in this case you can do all this we have added a couple of fictitious actions bogus actions which give which have no strategic implication it is the same game as before because okay so now effectively what so now the player 2 has the same choices l 2 and r 2 at both nodes and he is saying regardless of what you play I am going to play l 2 even though he has the information he is in fact going to play l 2 okay he could have said well if you come if it comes to know if the game goes to node z then I will play this if the game goes to know why then I will play that but he is ignoring that information and that is now reminiscent of what we said in the previous lecture which is the equilibrium of the game simultaneous move game is inherited as an equilibrium of the dynamic game so that is your l 1 l 2 is then the equilibrium of the simultaneous move okay okay yeah yeah so this is so there is so here that is another possible mode of play and that gets you to gets you to the blue mode of play that I have written here it gets you to this so what this there is a term for this in in English have you heard of this this is actually borrowed from French it is what is called faith or complete what does faith or complete means faith or complete means that okay yeah player 2 is making this threat okay but it is in some sense a bluff right because player 2 is effectively saying that that he is promising to play rationally player 1 can say I call your bluff I will take the game to know why now show me if you can actually play if you can actually play l 2 right and that gives you another solution of this game so there are many people who disagree about various whether which one is more which one is more reasonable and so on okay there is there continues to be debate or in amongst game theorists about this my own stand about this is that both are actually valid games of mode of play okay and the reason for that is because we have never know the whether rationality can include what I said promises of irrational behavior is not something that we have been able to rule out completely rationality you can take as common knowledge so nationality is common knowledge but issuing threats is is not part of irrationality is not being irrational so so so this is some people argue the way you do and they say that therefore this is the only way to solve the game okay that one should so the so this is what they says that rationality should also be applied recursively at every step the player is rational okay whereas whereas I am completely comfortable with this idea that rationality can include a kind of yet you can have an enlightened form of irrationality rationality where which includes a pretense of irrationality also okay so these are these are open questions okay no none of this is something that we everyone's been there is consensus on in this in the field I have this is I am teaching you my what I think is my what I feel is the more reasonable way of thinking about this okay you might find some books saying that this is the more credible equilibrium the reason because the reason is because this is as this this adheres to a stricter form of rationality that rationality that holds at every stage given the set of options the player is rational if he is picking the best option at each step but then what then question is then essentially all other equilibria which are which involve some form of threat will there be law okay and I find it I am I do not find it reasonable that these should be ignored okay anyway since I am using this this term the faith accompli here basically refers to the situation where I you know the faith accompli means that the you your options are locked because the deed is done you know you this guy just moves and then he called moves to player 1 moves to why takes R1 moves to why and then player 2 has no choice and then therefore at that stage invoking rationality given the limited choices then he he has to play art okay this second equilibrium is this here is you can say is recursively rational there essentially if you see the way the field has evolved also right in many ways the second one has become more attractive because because it has easier tools okay you are effectively what is happening is you are doing you are recursively computing from the end so it is a form of dynamic programming that is being done okay those of you know dynamic programming you are basically saying okay what would you have done at the last step then you say okay from then what would you have done at the step the previous step and previous step and so on so people from operations research control theory etc find this very is something they find this to be a solution that they can compute using the tools that they know okay so therefore this has become a very popular concept now popular does not mean right okay these are very different things and but unfortunately it is the one that that also comes up in a lot of in a certain type of literature okay whereas if you want to really explore this the second concept if you want to allow for threads then you need to compute things in a very different way and then that will that you know because the tools are in there the analysis is not there you know that also becomes a problem okay but this is but there are enough examples and it is well understood that the threat equilibrium is something that occurs generically in a dynamic game okay so this you can so the second one as I said can be thought of as this one can the first one can be thought of as recursively rational the second one is can be thought of as a as a threat as a threat equilibrium that is also in this another word that is also used is that this is in fact a non-credible threat a non-credible threat is a threat that does include a promise to be irrational so threat can still be you know could you may want to you may still be rational but but but damaging to the other player so threat just includes the damage right to the other player the irrationality part of it is is is encoded in the world non-credible non-credible threat non-credible means essentially you cannot a player the if the game does indeed come to know why then player would player 2 would not be able to carry out this threat before we go to more formalization let me also mention another another concept which comes up this comes up in in geopolitics for example have you heard of the word mutually assured destruction or assured mutual destruction right so so what that differs to is essentially this it is basically what player 2 is basically saying here is he remember player 2 is going to get minus 1 but in the process player 1 also gets minus 1 and it is that which player player 1 gets influenced by I mean that is what makes him move to l1 okay now essentially what he player 2 is promising to do is is if the game comes to know why then we will I will not only destroy myself I will also destroy you okay this is this mutual destruction that is being promised and that player 1 being player 1 being you know sort of being rational and scared of that comes to brings the game to know that okay now this this this assured mutual destruction is or mutually assured destruction this is this is a term that is used in geopolitics to refer to exactly these situations where countries or groups you know terrorist organizations etc you may want to negotiate with them and so on but at at the same time there is always this fear that they could explode a nuclear weapon or something like that and then you know not only damage themselves but also damage everyone else okay so dealing with countries that that that do not have a single point of governance or a single or a trustable rationality brings up this this particular issue okay and you can think of this as a way of for which by by which you know you can try to justify justify this the right equilibrium as a valid outcome also because it in fact does play out