 So, this is in this basically what it has done is, so apart from modeling the responsiveness of the player, it is also modeling his skill or precision at picking the best strategy and at playing the best strategy. So, if you are again as I said because the sort of the main utility or the main value of this sort of theory the way I see it is that it is not merely as a reasoning tool, but rather as a tool for modeling when you have real data. So, if you have real data of say players that are playing you know PUBG or something like that what is going to be the skill with which they are going to play a particular pure strategy. You want to model that based on past data this is the sort of model that will be very useful. So, this lambda of models the responsiveness of the player and also models the skill or precision of the player, the precision with which he plays pure strategy. So, let us, so I will just write out now the more the general theoretical concept here. See essentially we can build up a quantum response equilibrium using any what anything what is what is called a quantum response function. You can take a quantum response function and using that you just build you can build up a quantum theory of quantum response. So, what is a quantum response function? So, quantum response function is one that satisfies the following. So, it is firstly a function that takes your vector of payoffs and produces a mixed strategy. So, r i will map a vector of payoffs for player i and produce a mixed strategy for player i. Now, r ij there is the jth component of that mixed strategy mixed strategy is going to be a vector of probabilities for each pure strategy r ij is the probability that it gives to the jth strategy. So, r ij has to firstly be greater than 0 for every j. So, for every pure strategy gets some positive probability r ij of vi if you look at this as a function of vi which means the is a function of whatever you put is his vector of payoffs. It is as a function of that it is continuously differentiable. So, it is a continuously differentiable as a function of this which means that if your payoffs change slightly you should not have a big jump. So, remember the best response does not have this property if you have two pure strategies that are equally good and one of them becomes slightly worse then that slightly worse guy is gone from the best response. You know it is a sudden jump. So, this continuous differentiability is the way is this smoothing that it does from allow where it makes some allowances for suboptimal strategies. That is the essential nature of a quantal response as opposed to a best response. So, this has to be continuously differentiable. The other two properties are important for it to be a valid response to begin with. Firstly, if you change vij slightly means if you increase vij, if you increase vij the probability with which you play the jth strategy would also have to increase. So, all things kept all other things kept constant if you change vij slightly, if you increase vij slightly then the probability with which you play the jth pure strategy would have to increase. This is another demand naturally because you want to play strategies that would give you the ones that give higher utility are being have to be played with higher probability. In fact, and the fourth one is actually saying exactly that that if vij is more than vik then it has to be that the probability you assign to vi to the jth strategy is more than the probability that you assign to the kth strategy. So, which means that if the if pure strategy j is more profitable than k then it must get a higher probability. So, a quantal response in that sense is not just is not anything arbitrary it has it is not just some model of response with you know where player can do whatever he wants. We have there is some logic to sort of some method to the madness in the sense that it is responsive means that you increase the utility the probability actually increases. If you compare two different utilities then the one with the higher utility gets a higher probability. All of these nice properties are there in the quantal in the in a quantal response now what you do is you come up with these response functions which satisfy these properties. And now from here in one step we will be able to show that there is always a quantal response equilibrium ok. So, what you do is you take now you take R as R1 of V1 dot dot dot Rn of Vn R of V of R of V is defined this way ok. So, you put together these responses now there is no question of it being a set or anything like that because the responses actually tells you the exact value that the probability should R tells you the probability with which you are going to be playing various pure strategies ok. So, this is this just defines for you the mixed strategies directly and remember the Vs themselves are functions of the probabilities. So, what you have is you have what you get is that these guys are themselves functions of what you of all these right. So, what your so a definition of quantal response equilibrium is a mixed strategy profile sigma star such that sigma star is a response to so you put in sigma star to evaluate the expected utilities and compute sigma star from the response function you should get back sigma star. So, compute the response from the response function that should be equal to sigma star. So, it is precisely the fixed points type of property that we had for an IH equilibrium but now under instead of the best response you have the quantal response ok yeah R of V sorry what every term should yeah so V1 actually I should have been careful here and what V i actually let me write it here V i is V i 1 till V i I wrote V ij right so it is V i this so this is the vector of expected payoffs for from his various pure strategies. So, a quantal response equilibrium is then just is a mixed strategy that is preserved under this quantal response operation. So, you give a give a mixed strategy compute from there a quantal response to that mixed strategy if you should get back the pure mixed strategy that you started with and that is your quantal response equilibrium ok. So, so sigma star is a quantal response equilibrium if it is a quantal response to itself rather than being a best response to itself ok. So, now if you see R was defined to be continuous right then in that case consequent because R was continuous you get we have there and because we have assuming finitely many pure strategies and so on then R composed with V is actually a continuous function. So, R composed with V just maps sigma to sigma the set of mixed strategies to set of mixed strategies yeah is a mixed strategy profile it is R i is already taken to be having image here it is a mixed it is already a mixed strategy R i is belongs to sigma i so this is a map from here to here it is it is continuous and sigma is close convex and compact close convex and bounded. So, then again from Brower just like we did today in the start of the class Brower fixed point theorem implies that there exists a quantal response equilibrium for each quantal response function. So, you can see there is a little too much flexibility if you see because you can define whatever quantal response function you want and you will get a quantal response equilibrium ok. Now, this too much flexibility is good again when you are dealing with data sometimes you get erratic data and you know use no matter how much you try you will not be able to fit the you know the demanding very demanding theory of games to it but so this kind of a more a softer model like this can you can can fit in your data ok. So, another let us I will give end with one more example of a of a response function. So, you can do the following you can consider any CDF like this ok consider a CDF this is a cumulative distribution function this is a CDF ok and I am going to assume that if this has some sort of a symmetry property. So, f of x should be equal to 1 minus f of minus x which means that if you look at something like this. So, because this is a CDF it means that it eventually you know as x tends to plus infinity you are going to get this is going to tend to 1 and x tends to minus infinity is going to tend to 0 ok this is that is that is the that is your f. Now by when I write this symmetry property what this basically is saying is that you know I should be able to if I take a if I take an x like this here this height here is fx if I take the mirror image of this here this x which is minus x and I look at this height this height is 1 minus f of minus x right. So, then in that what it is basically saying is that this shaded height is equal to this shaded ok. So, you there is a sort of a flip and a and a rotation around the that is being that sort of symmetry is being ok. Now, if you have if you take any such function then what you can do is I will write this for our Majinthani's game you can write out a quantal response in this kind of form you can take sigma i 1 to be f of lambda ui 1 minus ui 2 ok. So, player i plays his strategy one way with this probability f of this ok. Now, in particular you can take fx to be 1 by 1 plus e raised to minus x and this is something that all of you would have seen some time or the other if you have done some machine learning this is what would give you logistic regression right. So, you the what you get from here is you get a quantal you get a logit response quantal response given generated by the logit function ok which is and it looks like something like this you get e raised to minus lambda x yeah so that so the exponential whenever it comes up many interesting things come up it is the bold it is it is it is it is part of the partition function. So, you can now it turns out unfortunately because of the nature of this we cannot solve for these sigmas in closed form but you can we can try to plot them and I will just show you what is interesting is how they how they tend to look. So, I have just I can just draw this quickly and show you. So, on the on the on the left on the x axis I have the probability that the column player plays h and on the right y axis I have the probability that rho plays h. So, if you if this is one and this is being plotted for x equal to 1 by 10. So, if x is equal to 1 by 10 the this column guy plays x h remember with probability 1 by x plus 1. So, it is so this becomes it is almost equal very close to 1. So, this so and this is and the row player plays h with probability half. So, this this dot here is is your Nash equilibrium. So, this is 1 by x plus 1. Now, you can also do plot the the logit response or the quantum response to of each player as a function of the other guy's strategy. So, I can plot for example, the logit response of the row player as a function of the player as a function of what the what the column player is playing. So, I take the column player is this thing. So, this has the independent variable and plot the row player strategy. It turns out it is something that it actually passes through the Nash equilibrium and it has this sort of form. So, this is the logit response of row player to column player strategy column players and I can similarly plot the logit response of the column player to the row player strategy and the place where they intersect is your quantal or logit response equally. So, here now I am taking the domain of this one is going to be this is the domain. The vertical axis is the domain for the green line, the horizontal axis is the domain for the red line and wherever they intersect this is now your logit QRE and it is distinct for it turns out to be distinct from the Nash equilibrium. People have tried to fit in data and all that into this and they found that the data seems to be somewhere here. Now, it is anybody's guess or whether it is closer to Nash or closer to logit. But the interesting thing is what happens as X changes. So, as X changes this figure changes quite a bit. I will just quickly draw that as well as when X if I if you change X to from 1 by 10 to say 9. So, then in that case then the firstly the Nash equilibrium shifts here. So, the Nash equilibrium will shifts because the one coordinate remains at 0.5 but the other coordinate has shifted. So, this is becomes your new this becomes the new Nash equilibrium and the these red and the red and green curves actually shift end up looking end up behaving something like this. This becomes the red curve and the green curve becomes something like this and this point here is your new logit QRE. So, the good thing is because there is this so one of the dilemmas that we have when we think about games from the point of view of data sciences is that what how do we interpret past data? Is the past data something that has is that only an exploration by the players just some trial and error trying to know what these strategies are, what the opponent's capabilities are you know what might the opponent be interested in etc. Is it just what we call exploration in reinforcement learning and so on is it just that or is the past data already having a strategic intent built into it. So, has the past data been arrived at with players playing with strategic knowledge and with strategic intent and strategic you know emphasis. So, the all these strategies that they have all that they have played in the past is not exploratory only it is actually intended as a strategic move by the players. So, if so if you interpret the past data in one way if you think of it as pure exploration then the opponent actually does not come into the picture the opponent is just there as noise effectively whereas if you think of the past data as being strategic then the past data itself represents a game and what you are seeing over there is the outcome of the game and so therefore you do not know what the players could have otherwise done and you do not know the space of the game outside of what has actually come out as the outcome. So, what the quantum response actually in my view what its main sort of benefit that I have found in my own work and so on is this is that it gives you some this kind of a soft in between model that is partly exploratory sort of models a little bit of imprecision lack of skill or lack of experience on part of the players because by bringing in that factor lambda and so on and at the same time it has some intrinsic merit as a solution idea. So, it is not completely non-strategic either. So, because if it is in between like this it lets you it gives you an opportunity to model this the game from its previous data from whatever you have seen players have done previously. So, that is one of its utilities whenever you want to do any modeling exercise form data this is one direction to go.