 So, now let us write out the space of strategies for both players, so actions for player 1, the actions are L1, R1, player 2, actions are L2, R2, now what are the strategies, this we have to be careful about, so remember this is a dynamic game, so a player strategy is a function now of what we know, so let us first do the simple one, what are the strategies for player 1, it is L1 or R1, so let us I will write this commonly, let us write this as gamma 1 is the set of strategies for player 1, it is all those functions gamma 1 such that they are constant functions, these are just constant functions are their strategies, so they therefore reduce to actions, because he has he starts the game, he has only one information which is just which is that he is at node x, so that was how we computed the optimal strategy, the space of strategies is simply to play either L1 or R1, but which action to play is to be computed through this logic, these are two different things, let me just complete this, so here is a what we have computed is a particular strategy for player 1 which is to play L1, but the space of strategies is anyway to play either L1 or R1, yeah it is a function of the information and the function of the information is that, so the information that player has, okay let me get to this also, again we are a little bit jumping ahead, but there are two types of information, when we are talking of information there are two things, first is what is the information you have at the start of the game, at the start of the game both players know the tree, that is common knowledge, okay they would they also know what each would know during gameplay and that is another piece of information, okay the information players would have during gameplay, okay the information they acquire during gameplay is another thing altogether, in this case player 1 does not have any additional information than the one he started with during gameplay, during gameplay he just starts with x and that is it, during gameplay player 2 does have additional information, right and that is why this is a dynamic game, so player 2 does come to know during gameplay whether he player 1 has played L1 or R1, okay both players know that this is what players would know during the gameplay, is this clear, okay so strategies are functions from what from the information they would have during gameplay to the actions that they have to take during during at each node, okay so the actions the functions that so therefore player 1 would only have the you know just another information to start the game with and so he has his strategies are to play either L1 or R1, player 2 has during gameplay the information that he is either at node z or at node y and then his strategies are functions from that to actions, yeah, yeah, yeah so that is why this is therefore these both of these things that I wrote out are heuristic, so once we write out the Nash equilibrium then everything will fall you know that will give you the proper theory, this is still heuristic, right this is as I said this is we are just sort of trying to kind of motivate a mode of play, okay, so for that let us write out the strategies then it will become clear, so yeah someone else had a question, yeah, yeah, yeah, yeah so players know the entire tree okay to begin with at the start of the game, so the tree is common knowledge as we start the game, okay, so these strategies for player 2 are these, so let me write it like this, I am keeping the do nothing strategy, okay, so gamma 2 at z is always do nothing and gamma 2 at y can be either L2 or R2, so how many strategies do each do these players have, how many strategies for player 1, for player 1 it is 2 strategies either the constant strategy L1 or constant strategy R1, what about for player 2, okay, so now let us do the following, let us make the matrix then, so the first one, player 1 has 2 strategies, player 1, gamma 1 1 and gamma 1 2, player 2 has 2 strategies, gamma 2 1 and gamma 2 2, so gamma 1 1 is the L1 strategy, gamma 1 2 is the R1, gamma 2 1 is this do nothing, do nothing comma L2 and gamma 2 2 is do nothing comma R2, let us write out, write this, write this table out, if player 1 plays gamma 1 1 and player 2 plays gamma 2 1, what would we get, what would the players get, it means player 1 plays L1 and player 2 is playing gamma 2 1 which is do nothing comma L2, 0 2, right, okay, now gamma 1 1 and gamma 2 2, if it is, once player 1 is playing L1 it is 0 2 2, 0 2 for both, what about this, gamma 1 2 is R1 followed by L2 which is minus 1 comma minus 1 and this here is 1 comma 1, okay, now we can, this is your, so what we have done effectively is we have written out the payoffs of the players, we have written J1 of gamma 1 gamma 2 and J2 of gamma 1 gamma 2, these are the payoffs that are there that the players have in the space of their strategies, right, so assuming each player plays their strategy, so you consuming a profile of strategies for each player, you get these payoffs that is given in the table and as before now, we can, since the strategies remember are being picked before the game starts, we can talk of a Nash equilibrium for this game, in this space of strategy, right, so we have to then essentially look for a gamma 1 star, so a Nash equilibrium is a gamma 1 star, gamma 2 star, this is your Nash equilibrium, as before, oh sorry, since you are maximizing, let us put a greater than equal to, yeah, okay, so effectively all I have to do is then look for the Nash equilibrium in this table, this table is now just a, is a non-zero sum game with two strategies for each player, I have basically forgotten the dynamic aspect of the game altogether and I have just listed out the strategies explicitly and now I am looking for the Nash equilibrium in this space, okay, okay, so let us look at this then, what are the Nash equilibrium of this game, so this here is one Nash equilibrium, 1 comma 1 is a Nash equilibrium, okay, how does that come about, so player 1, so if player 2 is playing gamma 2, 2, if player 2 is playing gamma 2, 2 then he is basically promising to play R2 at node y and do nothing at node z, so assuming player 2 is playing gamma 2, 2, it is optimal for player 1 to play gamma 1, 2, okay, so and likewise assuming player 1 is playing gamma 1, 2, okay, it is optimal for player 2 to play gamma 2, 2, okay, so this is, so you can compare 1 with minus 1 and 1 with 0, okay, so this is therefore a Nash equilibrium, okay, what about, are there any other Nash equilibrium, the first one here, this is also a Nash equilibrium, so this here is a Nash equilibrium and why is this a Nash equilibrium, this is because assuming player 2 plays do nothing and L2, it is optimal for player 1 to play L1, okay and because he is comparing 0 with minus 1 and assuming player 1 plays L1, it is assuming one player 1 plays L1, player 2 can play anything because they both give him, give him 2, okay, so this is also, this is a Nash equilibrium, so you can see here that this is basically once we, once we wrote this game out as formally in term in the space of strategies and look for the Nash equilibrium, the Nash equilibrium as pointed as exactly to these two outcomes, you have one outcome here which is the first mode of play that we thought of that player 2 plays R2, maybe I will just write this here, this is your R2, this is L2, this is L1, this is R1, okay, so the first mode of play here was where player 1 played R1, player 2 responded with R2 and the second one which is the threat one where player 1 played, promised to play L2 and player 1 played L1, okay, so what this means, so this is actually one of the, this is again I think really a feather in the cap for the Nash equilibrium that it is able to produce all of these, that it is able to point you to these kind of outcomes which would have, which you know would have escaped are a very sort of a naive reasoning, you know most people would not even have guessed that there is such a this thing, but a fantastic prediction comes out from the Nash equilibrium when you follow through it, when you follow through the logic formula, okay, there is another thing to point out, is there any dominance here, anything dominates anything, R2 weakly dominates L2, right. Now, go back to your reasoning, some of you were very wanted to say that, you know player 2 should play R2, right and player 1 should know that and therefore player 1 should play R1, right. Effectively what you are saying is, you are saying that I should eliminate L2, I should eliminate gamma 2 1, so this is not just 2 for this particular game, this is 2 in general for any dynamic game, what you will find is that the threat equilibrium, you know there is going to be this feature here that the threat you know has this, will have this feature that the fellow who is issuing the threat, right, it does not matter to him what he was doing elsewhere, so he would get the same payoff regardless of what happened, he would have multiple strategies would which would give him the same payoff and that would mean that there would be some weak dominance, so if you remember in our last class we looked at this producer's game, in that if you remember when we looked at the Nash equilibrium from the static game, and we said that the best response for the second producer, if the first producer is playing at the static Nash equilibrium level, the best response for the second producer was to produce, was to come up with any function, so long as at this level, at the first producer's level it matched with the static best response, at this for the first producer's level it should be giving you the static best response, elsewhere it could be anything and in fact elsewhere it was taken as constant, it was taken as a constant thing, he played the same thing regardless of what, so this particular feature comes up again and again and what that ensures is that what that basically does is it gives you weak dominance, so ignoring the threat equilibrium, so in my view ignoring the threat equilibrium then amounts to eliminating a weakly dominated strategy. Now we have seen before that whenever we would eliminate weakly dominated strategies you land the, you stand the risk of eliminating equilibria as well and that is effectively what is happening, when you stick to a particular line of reasoning which is coming from this backward reasoning or the backward induction or dynamic programming it is essentially keeping you blind to the other possible equilibria that are there and they are getting eliminated. This is also one of the reasons why I have conviction in this as a possible, as a completely valid mode of play, that the same reasons why we could not we do not justify the elimination of weakly dominated strategies should also justify why this particular equilibrium should also should be treated at par with the other equilibrium. Yeah, what do you mean? No, but we never said well he is trying to maximize his payoff and the payoff is given in this matrix which is the point, the point is that whether that are you, is this an obligation that you should be rational at every stage if you want to be considered as rational or can you be, as I said is there a next level of rationality which involves some selective rationality that so all of this is exactly this as I said this is a matter of, this is a matter of debate I think people will, there are some people believe one way, one will believe the other way. The point is that really we are actually at some level the you can say the issue is that beyond this you cannot really say from the standpoint that game theory has taken. The game theory has taken this standpoint that we are going to be observers of the game and we want to strictly speaking from the assumptions we want to come up with come up with what we think is the outcome and with no additional assumptions this is as far as we can go. If you want something more you need additional assumptions. I mean the debate that is going on is whether these assumptions are enough to somehow rule out the L1, L2 type of equilibrium. No, then that is a different thing altogether. I understand so if the game occurs multiple times say for example you know this kind of altercation with on a geopolitical thing you know you keep having negotiations peace talks and every time so if you know that there is a multiple there is an iteration involved then the nature of the game differs because you would play differently knowing that this is going to be an interaction that is going to happen multiple times. So that is what is called a repeated game. If we have the time we can look at repeated games. Repeated games lead to a different you will have to model it appropriately. It is a particular form of a dynamic game but with the same structure repeating again and again. There you will have depending on the actually there are many strange results that come up. For example if you allow enough repetitions pretty much anything can become a Nash Equilibrium and which is which is also once in some sense a critic of game here criticism of game theory that you know if you can give if anything can become a Nash Equilibrium with enough interactions and what is the meaning of having such a theory as well. So that is that kind of thing can also happen. But the reason for that is because basically when the game becomes large enough if you have enough number of repetitions then this number of strategic trajectories you can take become very very large and player by issuing threats for saying that form if you do not agree to this from here onwards I am going to punish you that sort of strategy can be applied for pretty much any type anything. So if there is a large enough repetition then you know such threats can be issued and pretty much anything can be done. So the repeated games is a different matter altogether we are talking of the game that we have at hand which is just this that is we are playing once this is the tree and these are the pair. I mean that is a good thought I mean I have not thought about this whether you can and so I have thought about something else which is this and I mean but I have not written a paper on this I mean if anyone sorry because I have not had time to work it out. But there is this possibility of thinking about what you can say are ulterior motives in games. So ulterior motive would be that you have one objective which you project but your actual objective is something else. So the game is then played with one objective but you play but so in this game you look irrational but you have there is a bigger master strategy going on in a larger game in which you are rational. So effectively that becomes like a you know an optimization and with over another game when you are playing for you are looking to pick the best equilibrium out of this out of a particular game for the purpose of optimizing your actual objective or ulterior. So I think that could be a way to sort of build in selective irrationality as a choice it would give a more formal framework for doing that. But yeah I mean we can talk offline if you want but I mean since this is like like now getting to very cutting edge stuff because it is not even I do not there is no one who has written a paper on this also so I cannot even we cannot discuss this. But yeah I mean this is this could be one approach I mean that we you could try to where you switch on and off your rationality you know as part of a you know strategic choice. No no why not it feels it is a game of it is a game of complete information actually no no no issues with no no it is it is this equilibrium. No there is no see this is this is in fact my point is that this equilibrium is in fact coming through very clinic if you just follow clinically the logic of game theory right this equilibrium comes out as an ash equilibrium. So it is it is a it is a valid prediction I mean there is no in fact you do not need anything else to make this occur it is a it is an ash equilibrium of the game. What you want to do is maybe you want to put in additional assumptions and refine it further to then get you to get to this if you know if you so desire if you want to say that okay under what assumptions is this more logical you might have to put in some additional assumptions that is okay that is a different thing altogether. But but this is the just to produce this outcome which is your L1 L2 outcome we did not need anything else L1 L1 no so that is just that is no no no he does not have that information it is an outcome the way we we would have we would have the way we would have otherwise reasoned about this game. So suppose if I had just given you this game this table and asked you to find the ash equilibrium what would you find you would find these what you can do is exposed having found the ash equilibrium mathematically you can try to say what is actually happening strategically in the game and that is this I do not need to have you know a threat letter with me to actually to in fact play this play this particular strategy. Okay see the the as I said the issue is essentially I think my my own feeling is that given given how little we have assumed right this is as far as we can go. If you want to refine further you need some additional this thing some additional criterion has to be there okay now again we that is another topic again which is called refinement of ash equilibrium refinement of ash equilibrium is that you put in some additional requirements you as we said ash equilibrium is a is a basic requirement is a necessary condition now you can ask okay is there can we put in some more demands and refine that set okay. So one of the demands is for example you can say well these payoffs are not perfect okay this is this is actually 0 comma 2 plus epsilon and then you look for an equilibrium which is as a function of epsilon and then let epsilon go to 0 okay now it turns out that some some Nash equilibria get vanish in the limit they do not they do not remain equilibria anymore. So you will get equilibria for so or some Nash equilibria cannot be approached through such a limit which means that they are equilibria for this specific numbers but if the numbers had were slightly off then they would not remain equilibria okay which means that they are you know in the in the language of this thing there is not stable in in some sense it is not stable to to small perturbations or they are not robust or whatever you want to call it okay. So then those can be eliminated okay now that is a one type of reasoning that is basically saying that you know imperfections in the data and so on which are leading you to this sort of a could have led you to a slightly different table then they they do not retain the certain equilibrium so you pick the equilibrium that is amongst all equilibria pick the one that has that is remaining because that is actually at least the equilibrium that that is that is remains up valid outcome even despite this perturbation. Now you can think of it in a slightly different way also you can say well if so here the perturbation is because you are not sure of the data as a this thing okay as an observer of the game you are not sure of the data and so you the other other could be that the players are themselves not clear okay so there is therefore noise in the data okay wherein that the so the player do not know which value of the game is being played and there is in fact so what you have is a is is an ensemble of games one of them is being chosen and then you let the noise go to zero that is another way of choosing. So these are all different ways of doing what we call equilibrium selection okay yeah and there are you can well there are also more rudimentary logics like you can say well take this take this thing for example the dear dear rabbit game in that it turns out that dear dear is better for both players as compared to rabbit rabbit rabbit rabbit gives both players half dear dear gives both players two okay you or one or whatever it was it gives both it is uniformly better for both players to play dear dear so then why would they play rabbit rabbit so you can say from that logic you try to eliminate but see the more the more you start putting layers like this right the class of games for which this remains applicable becomes smaller and smaller okay so the real challenge always is to say come up with a universal logic that is on par with Nash's logic and still you know gets you a smaller set okay so I will give you an example a paper I wrote in about what 14 years ago I think now what yeah 14 years ago this was from my from my PhD days so there that time we were I was looking at a certain set of games in which there were a continuum of equilibrium not not just multiple but a continuum infinitely many equilibrium okay and I argued that in that so over there so then the equilibrium could be interpreted over there as if it was being interpreted by an administrator who was charging some prices okay in some extreme you can look at a like a thought experiment take some extreme case in which the equilibrium had this interpreted that it was it it was coming out by through the imposition of some set of prices okay that some extreme example had to be taken to do that and there I argued the following I said that well now suppose this sort of a game arose in in say an internet in the internet where players are anonymous okay now if this if this equilibrium arose in the game where players are anonymous then the then the administrator cannot distinguish between the identities of the players he only knows the actions but he cannot distinguish between the players IDs because players identities are anonymous and they can be spoofed transform so basically then I what I said was that you can think of a theory in which games you can think where the equilibrium is robust to identity fudging okay the and then from there so what happened then is that this continuum of equilibria that are there it turns out that only one out of them remains robust to this it has this sort of symmetry proper some kind of a symmetry property and all these other equilibria that are there they involve they require the the administrator to know the exact IDs of the players that this is player 1 this is player 2 this is player 3 etc so his the pricing then is non-uniform in the players and it it requires you to know the ID identity of the player and so therefore on that on those grounds you can now eliminate all of these equilibria because all the other equilibria because this one is the only one that is implementable in a in a certain extreme okay so it has it carries therefore more merit or more meaning so this was a way for and and then you have to argue that such an equilibrium always exists you know the one that is implementable is always exists and so on and so that is then I propose that as a refinement okay so this that is that is that is one that is a that is one of my one of my results where I basically refined the Nash equilibrium by using this particular logic now what this does is it basically takes the full set of equilibrium where you had this continuum of equilibrium and produces for you a single one okay so the now but but that does not mean okay that does not mean that you always have to face with situations where players are anonymous I have argued that this carries more meaning because of this you know this thought experiment that I have done but that does not mean the others are devoid of meaning in fact if players are not anonymous others are just as good right and then you could you could go back to those equilibrium which is something similar here and so this is the threat this is the sort of challenge with refining the equilibrium the Nash equilibrium all the time you start with something you you take it to some kind of extreme right take some epsilon way to zero this that whatever you do and then you pull out a Nash equilibrium but that does not mean that this epsilon had to go to zero or there was in fact a perturbation to begin with right so then then you are back to square one so this this whole refinement exercise is very interesting very challenging and when you get a refinement you get a lot of attention and all that but you but it is not I mean since Nash I do not know if anyone has been able to you know move forward on this anything better than that because it is really you need that gold standard where it is applicable across the board no questions asked and yet yet carries a universal meaning because because people are not trained there there is not too many people who do who know who know about the subject you know who for that needs yeah yeah so the game theory does not have enough analysis anyway so today we are just just chatting so I think we will end the class here