 Welcome back everyone. So what we will do now is look at another example in which we will change the information structure that is that we have in the Wittson-hausen problem. So where we what we will do is suppose we will assume that we did not have a non-classical but rather a classical information structure. So suppose here this is I will suppose now that the information at the first controller is just the observation y0 but the information and the information at the second controller in Wittson-hausen problem it was only y1 but I am now going to assume that he also has access to y0. Now this now is a classical information pattern. This here is clearly a classical information pattern. Now let us see what happens in this particular problem. So in this problem let us what is the what is the second let us see what we what is going on. So remember the second controller what is the second controller attempting to do second controller which is gamma 2 what is what is he attempting to do? Well he is attempting to estimate x1 from the information that he has and earlier the problem was that he was trying to estimate x1 from y1 only and y1 was x1 plus v but now we our information pattern has changed. So he is estimating x1 from I2 but I2 has changed to now y1 comma y0. So he is estimating x1 so estimate x1 from y1 and what is y1 y1 is x1 plus v and also y0 which is simply x0 right. So essentially therefore gamma 2 gamma 2 star of I2 of the of I2 is conditional expectation of x1 given x1 plus v comma x0. Now here now what is let us see how this is this is therefore the optimal control action as far as the second controller is concerned it always remains this way that he has to just estimate these the state at time this the state given the information. Now let us think how is it that the first controller should play this. So the first controller now has to decide a gamma 1 knowing that he has to minimize that first stage cost. So gamma 1 has to minimize here this expectation of k square u1 square plus the plus this term right the expectation of plus the expectation of x1 minus gamma 2 star of I2 the whole square. Now what is gamma 2 star of I2 that is that is this conditional expectation here right and I2 remember now has access to the information that is present in that is that is present in gamma in I1 as well right. So here so let us see how this can this can be done. So my claim is that you can actually you can make this cost 0 as well ok how do we make this cost 0. So here is one possibility now one possibility is that suppose then that gamma 1 simply just is gamma 1 of so suppose you choose u1 to be gamma 1 star of x0 to be 0. So this is always so in other words regardless of the value of x0 gamma 1 star is now 0 notice that this is different from the earlier approach in the earlier in the earlier assumption we did not have any cost on u1. So what gamma 1 gamma 1 did there was he said he was trying to cancel out the effect of x0 right. So he was trying he and therefore he made x1 independent of x0 right whereas here in here what is happening is that here he is choosing u1 to be 0. So then x1 is still going to be dependent on x0 but then that is not a problem because x1 even though x1 is independent on x0 x1 now is the x0 is now known to the second controller. So what would happen in this case so u1 is equal to gamma 1 star of x0 equal to 0 this would give you x1 equal to x0 and therefore x1 plus v would also be equal to x0. So therefore this problem here this is about then becomes the problem of estimating x0 itself from y0 comma from i2 and what does i2 contain well it contains x0 and x0 plus v remember this was y0 and this is this here is y1. So the second controllers problem then becomes that of estimating x0 from x0 comma x0 plus v and this problem is obviously trivial because if you want to estimate x0 given this information since this information itself tells you what the value of x0 is we already know what the optimal error what the optimal thing is going to be it is actually going to be 0 the opt you will actually get an optimum you will get an error equal to 0. So as a result of this u1 star u1 itself this becomes equal to 0 and this second term also becomes equal to 0. So therefore this therefore gives you that gamma 1 star equal to 0 and gamma 2 star of x0 comma gamma 2 gamma 1 star of x0 equal to 0 and gamma 2 star of x0 comma x0 plus v equal to x0 is optimal. So in other words so what is this what is this taught us what we did was we looked at now the Wittsenhausen problem itself but we looked at it without the non-classical information structure. So we assume that the information structure is classical and in that what we found is that the problem actually is rather simple it is it is extremely easy to decide what to do in fact you can get the global optimal the numerically optimal cost by using a bunch of very simple controllers. So this now again let us go back to what I was saying earlier see notice that because of this because of this particular depend because x0 was present here out here in this when we were estimating x0 from x0 plus v things became very very simple for us because it was possible for the first controller to choose an action in such a way that the second controllers job becomes extremely easy. So he chooses his gamma 1 in such a way that the second controller can estimate the next state which is x1. So he can he leaves the second controller with a state x1 to estimate such that it can be estimated from x0 and this is something he can do because no matter what the first no matter you know no matter what we have second controller always has access to x0. So the first controller therefore has to does not have to bother about what information is going to be there with the second controller he just has to give the second controller the right sort of target to estimate. And this basically is what happens in a problem when you have classical information structure and the information structure is classical the first controller does not have to worry about this the about what is it that the second controller ought to know you know through his does not have to choose its actions in order to influence the information of the second controller. So that is so we this is so this I if you recall I had called this the dual effect that particular phenomenon you see that here there is no dual effect this is the there is this is only a for the x0 is already available here. So the so what this second controller for the first controller does does not affect the information of the second controller and he just simply he you know so as a result of that the first controller can plan in such a way that the second controllers error becomes 0 and as a and simultaneously his stage wise cost also becomes 0. So what we are seeing therefore is that the information structure is essentially the culprit in in in this problem and Wittsenhausen is showed exactly this he in fact he so if you take a problem with classical information structure the problem is trivial the minute you take away this this the the the classicality of the information structure the problem collapses to an extremely hard problem. So let us try to understand in this this problem from a slightly different perspective and so that we you know we get some more different kind of intuition for this problem. See the the one way in which we can think of what is happening in this in this problem is that we can think of as I as I keep saying there are these two controllers one way in which we can think of what is happening here is that these are actually two different agents the the the first here the first agent here is a field agent this agent is you can say is a is a field agent he is he is trying he is the one who is in the field trying to make he is the one who can observe the the the ground state the state in the in the actual in the in the field right the the so that is your that is your agent gamma 1 gamma 2 is the one who has who wants to who wants to know a certain thing a know a certain state so he wants to know in this case he wants to know x 1 right. So gamma 2 is is this agent who is who wants to know x 1 but x 1 is not some kind is not some fixed state x 1 itself is this is is disturbed is is shaped by the by the actions of the of the first time the so the field agent has to send a signal or send some information to the to the to his supervisor which is gamma 2 the supervisor then uses that information to estimate x 1 but x 1 is also changing based on the signal or based on the information that the that the that the field agent sends to the second control to the second to to his supervisor right. Moreover there is the the so that is that term here is that this influence of where the where the field agents action ends up influencing the second controllers second controllers cost or the supervisors cost that appears based on the based on the the because of the dependence of the cost on the field agents action. So this this x 2 here remember x 2 depends on x 1 and x 1 is chosen x 1 is gets influenced by the choice u 1 which is which is just a function of x 0 which is gamma which is gamma 0 of x 0 right. So so the the the what the field agent here does influences influences u 1 and therefore in turn influences x 1 right and x 1 is the thing that this that the supervisor was trying to estimate right. Now the supervisor also has to he has to he has to estimate based on some information and that information itself is a noisy version of x 1 right. So what the information with the supervisor is y 1 and the the the the this this and this itself depends on the on the policy on the policy gamma 1 chosen by chosen by the field agent right. So the field agent has to decide what to tell the supervisor knowing that what the supervisor wants to estimate and based on which he is going to estimate and the information based on which he is going to estimate is both influenced by his action by his choice right. Now the supervisor does not hear what the field agent is saying in a perfect way. So whatever the so the field agent sends you you know chooses an action u 1 that gives you an x 1 but x 1 gets corrupted by noise on the way to the supervisor. So this field agent is talking to the supervisor in a in a noisy fashion. So there is a noisy medium that is present in between which so the supervisor does not quite get exactly what the field agent is saying that is that is what is going on here ok. Moreover the field agent also has this dilemma that he has he you know if he exerts himself too much you know to try and beat the noise where he suppose he tries to you know use a very high amplitude signal to send to the supervisor in order to you know sort of cancel out the effect of noise then he incurs a cost for that as well. So there is a cost term here notice that there is a cost term here which depends on u 1 square as well. So that field agent is therefore caught in this dilemma he cannot choose a very high u 1 he cannot he has to also signal some right amount of information to the supervisor what the supervisor estimate would estimate depends on what the field agent is signaling but the whatever the information based on which the supervisor makes this estimate is also corrupted by noise and is a and is influenced by what the field agent is saying. So it is this thing that that happens in this it is this you know popory of various dilemmas that happens in this problem and it is often been known by the dilemma of what is called signaling through action. So it means that you have to concern yourself with the information that you are providing to the to the to the future acting controllers alongside also concerning yourself with minimizing the cost. So in once the information structure is class is classical the signaling effect is not present and therefore you know you know you can focus only on choosing actions that minimize the cost and not worry about how you know what information is being sent to the to the future time steps. So this is this this here is is one more angle to the Whitson-Hausen problem. Whitson-Hausen problem can be interpreted in a in a third entirely different way which is neither control nor economics and that problem that setup is is what is is the setup from communication. So let us come let us look at this setup again. So we can think of this whatever I was calling you calling here as the field agent. This field agent is is can be thought of also as a as a transmitter right and whatever I was calling as a supervisor can be thought of as a receiver. Alternatively this can be thought of as a encoder and this can be thought of as a decoder and so on. There are there are many the you too can be thought of as a decoder these are these these are different ways in which you can think of what is happening here. So in a typical communication setting you again have a transmitter who knows some information present in a remote location. This information is what is to be known or is desired to be known by by the receiver here. The receiver wants to know this this piece of information and the the and the two the two the transmitter and receiver are their their dilemma is that okay how much energy should we be spending on communicating and how should we be defeating the noise that is present in the media. So this this dilemma is again the one that we have seen already in the in the Witson-Hausen problem. There is however one subtle difference which is that although there is it is essentially that kind of a setup where where you have where you know there is a transmitter and there is a receiver here. But the the the essential difference is that the receiver in a in a in a usual communication setting the receiver is concerned with knowing this here which is x0. He is he is concerned with knowing precisely what information is there at the other end. Whereas if you see the Witson-Hausen problem in the Witson-Hausen problem the second controller is trying to estimate x1 from x1 plus v. So the second controller's goal is not to know x0 from x1 plus v but rather x1 from x1 plus v. So this is where the Witson-Hausen problem is is is a sort of a beast of its own. It is not it has a lot of similarity to the communication to the communication setup. But it is not exactly the vanilla communication problem either. So it is essentially a different type of problem which is not quite been studied in communication theory. But if you think of the communication problem and these this problem in one light you can see that well these are all problems with non-classical information structure. These all of these problems involve involve a two different agents separated by a noisy medium where what is known to one agent is not known to the other. But maybe with different goals Witson-Hausen's goal is is is where the second controller wants to know x1 from x1 plus v. In the communication setup the second controller wants to the second agent which is the receiver wants to know x0 which is what is known to the first agent. So as a result of this what we can see is that there are actually tremendous parallels between problems once we allow for various information structures tremendous parallels emerge between problems in in organizational structure which is the kind of problem I just mentioned with the field agent and the supervisor and so on that the the theory of of of of teams and firms and also the and the theories of communication and finally the theory of decentralized control. All of these start look are actually closely related to each other and are variations and there are there are essentially variations that arise when we vary when we vary the underlying information structure or the underlying intent or the cost function in the in the problem. So this is this is this is another thing that we that we sort of take back from the Witson-Hausen problem which is which is that it tells us how if we if we view control the in under various information structures it automatically lends itself to its to similarities in in problems with a number of that are present in a number of other fields. The final other point about the Witson-Hausen problem here is because I mentioned the angle of communication essentially notice that what is really happening in the Witson-Hausen problem is implicit communication. You know although that has not exactly been I have not precisely used that word but really it is some kind of communication is implicit here because the first controller is trying to give some information to the second controller in order to make a better decision in order to eventually minimize you know that they can collectively minimize this cost. The second controller has to just do conditional expectation but conditional expectation of what and given what that is being decided by what the first controller is sending. So there is so the so implicit in the in a problem with non classical information structure is the is an element of communication no matter you know how how you look at it that is always this element that communication play where communication is somehow happening. This is again known through in a different by a different word in the in the economic community they call it signaling the in in in other it is it is known it is known by signaling sometimes it is known by persuasion and so on but this is this is essentially what is happening in this you know in the Witson-Hausen problem as well. So what the main legacy that we take back from this problem is also another legacy that we take back from this problem is is that we cannot think of control as purely in terms of action but also have to think of control through communication as if there is an element of communication present in you know in a in a sort of subliminal way in with in control problems. So once we pose once we look at a control problem with non classical information structure it automatically has brings in it an element of communication. So this is this is another another key aspect of the Witson-Hausen problem. So what we will do in the next class now that we will discuss the Witson-Hausen problem at depth what we will do in the next class is we will we will look at we will we will we will go over certain some more variations of the Witson-Hausen problem and to see you know the dual effect in in in more in a much more pronounced manner we will also quickly review the proof that Witson-Hausen had for the optimality of non-linear strategies so that is coming up in the next class.