 So far, we have only considered strategies which are picked by the individuals themselves. So we have talked about pure strategies, we have talked about mixed strategies, but those pure or mixed strategies were picked by the players themselves. And we have seen several kinds of equilibrium concepts involving those pure and mixed strategies. Now in this model, we are going to discuss a slightly different strategy, which is not picked by one individual agent, rather it is collectively decided, but via a device which we are going to call a mediating device. So what does it mean? So let's remember one of our examples, the playing the cricket, going to watch a cricket or a football game, which was happening between two friends, and they were deciding independently. So far in our discussion, we have assumed that those decisions were made independently, and therefore we can find out something like pure strategy Nash equilibrium and mixed strategy Nash equilibrium in those kinds of games. Now if you look back and think about that, does it really the way friends decide what to watch? So sometimes they might decide together that they will go to watch a football match today and a cricket match tomorrow, but they do not take independent decisions and end up in a situation where one is going for watching cricket and the other one is going to watch football. So in some sense they can kind of coordinate, and if they are not able to decide which game they should watch, they can just toss a coin and decide based on that outcome of that coin, which decision to take, which match to watch. So this actually helps us in giving some sort of an alternative explanation of player rationality. So there are certain kind of situations like this one where it is not meaningful to ask players to take decisions independently, rather they can collectively decide even if it is via a mediating device or that randomization device, which is the coin in this case. So this is the motivation or this is the intuition through which we are actually going to define the correlated strategy and its equilibrium. So the first reason is this alternative explanation of rationality. We will also see that utility of this players can get better if they are choosing such kind of a decision, where they are coordinating with each other and giving the task to a randomization device. And finally we will see that there is some advantage on computational tractability as well. So we have seen that mixed strategy Nash equilibrium, even though it exists, it is guaranteed to exist for finite games via Nash's theorem, but it is computationally very expensive to find and there does not exist any easy method of computing mixed strategy Nash equilibrium. While we will see that correlated equilibrium is relatively easier to compute. So on both these aspects I think this is a right time to start discussing about correlated strategy and equilibrium. So one of the classic examples for discussing correlated strategy is the following. So suppose there are two cars, we are going to denote that as player one and player two respectively, they are at the crossroad, busy crossroad and they are deciding whether to wait for the other car to pass or continue moving. So there are two strategies for each of these players, wait and go. And this matrix is essentially showing that what is the utility for each of this. If both of them are waiting, then they are just wasting time and nobody is moving. So they get some utility which is let's say represented by zero. Now if one player, so let's say player one is going and the other player is waiting, then the player who is actually moving, it gets a little higher payoff. While the other player is also getting some positive payoff because it knows that once the other player has moved, other car has moved, it will be that car's turn to move. So similarly the opposite thing happens when the other car is going and this car is waiting. But if both of them go, you know what is going to happen, they will collide into each other and that will give them a very large negative payoff which they really want to avoid. Now you can calculate, so this could be a very good example, good exercise to calculate what is the Nash equilibrium. So from inspection you can find out that both go, wait and wait go are pure strategy Nash equilibrium that is almost immediately observable. But you can use the previous method to find out what is the mixed strategy Nash equilibrium and you will see that there exists a non-degenerate mixed strategy Nash equilibrium in this case which interestingly gives some positive probability on this outcome, go, go. So even though it is small but there is a positive probability that it will go and collide into each other and that is clearly not very predictable outcome of this game. So have you ever seen in any of the crossroads cars ramming into each other? That doesn't happen, right? So we cannot explain this kind of a situation using mixed strategy Nash equilibrium rather what we will do, there is a device in practice which is a traffic light or a traffic police that guides the players and players essentially agree to this plan. So the police or the traffic light might ask one of these cars in one direction to go and the crossroads to wait and then maybe in the next round it is allowing the crossroads cars to move and this road's car to wait. Now this third party which is the traffic light in this case is a trusted third party. So we are assuming in this setup of correlated equilibrium, correlated strategies that there exists a trusted third party like the traffic police or the traffic light and we are going to call that as a mediator. It is like it's not that the players are coordinating with each other directly but they are coordinating using a mediator. So the role of the mediator is to randomize over the strategy profiles. So and this is the important difference that earlier what was happening we were asking each of these players to randomize over their individual strategies but this mediator is not only randomizing over strategies rather it is randomizing over the whole strategy profile. So maybe it is randomly picking one strategy profile and whichever is the outcome it suggests those individual strategies to those players. So and we will show that in the when these strategies are enforceable then it becomes an equilibrium which we are going to call the correlated equilibrium. So let's make this definition of a correlated strategy a little more formal. So what is a correlated strategy? It is essentially a mapping from the set of strategy spaces, so strategy profiles. So S as you can remember S is nothing but the Cartesian product of S1 cross S2 up to SN. S is nothing but N players and it is assigning the mapping pi or the function pi is assigning probability masses on the entire strategy profile and therefore it is always going to be between 0 and 1 and the sum over all the strategy profiles of pi will be equal to 1. So let's look at one very simple example go back to our wait and go game. So what we can say is that let's say the probability on wait-wait and go-go is 0 and wait-go and go-wait it is half. So that means half the time you are asking one of these stretch to move, one of these cars in this straight to move and the other half of the time you are asking the cars in this straight to move and that is a correlated strategy, a valid correlated strategy. Now what is a correlated equilibrium? So when it is an equilibrium it has to be self-enforceable in some sense. So therefore the correlated strategy, a correlated strategy defined in this way. Notice that this is a strategy not of any individual player rather it is the strategy of the randomization device or the mediator whatever you want to call it. So that strategy is a correlated equilibrium when no player gains from deviating while others are following that suggested strategy. And we are also going to assume that this correlated strategy once it has been picked it is a common knowledge. So something like you know the traffic light is going to be half the time on on this direction and half the time on on the other direction. So what is a correlated equilibrium therefore? So correlated equilibrium and let me just spend a little bit more time on this. Like the probability the mixed strategies we are not looking at individual probabilities anymore rather we are looking at this correlated strategy or the probability distribution which is defined over the strategy profile. So suppose you are player i and you have been suggested this strategy si. Now you are contemplating whether you should follow this strategy or you should do something else but remember that no matter whatever strategy you pick you cannot change the outcome of the of the randomization device. So because it is a trusted third party when it has given you si that means the randomization device gave some outcome where the ith component was si. So therefore your that device has actually suggested you to follow the strategy si. So that is the reason on both on the left hand side and on the right hand side this probability mass that you are going to multiply with is si and this is one of the most frequently asked questions in correlated equilibrium why on the right hand side it is also si it is not si prime because you are you are not controlling that si in any way whether you pick si prime or something else does not change the outcome of the randomization device and that is why on the right hand side also you will be expecting with respect to the same si s minus si and the rest of the thing is very easy. So on the left hand side when you are following that strategy so the only thing that you can change is by choosing your si and you can get that specific utility and you are looking at the expected utility because you know what has been suggested to you that is you know si but you do not know what has been suggested to the other players. So all that you can do because pi is known to you you can take an expected utility. You do that expected utility when you are following the strategy si and you also calculate the expected utility when you are not following and playing something else let us say si prime and then this correlated equilibrium is saying that you will never be better off by not following that si which has been suggested to you and this inequality should hold for all si si prime and for all players i in n and if that happens then we are going to call that pi a correlated equilibrium. So let me make one small remark that this mediator suggests these actions after running its randomization device pi so this is no no subscript here just pi because there is no component wise thing here every agent's best response is to follow it if others are also following it so the whole intuition of this correlated equilibrium can be summarized into this sentence. So let us come back and look at some of our very favorite examples so this football and cricket game we have seen several times in the past and if you have done that exercise carefully then you can find out that the mixed strategy Nash equilibrium looked something like this so player one was picking f with two-third and c with one-third and player two was picking this f with one-third and c with two-third that was a mixed strategy the the non degenerate mixed strategy Nash equilibrium of course there were two pure strategy Nash equilibrium. Now let me try to ask the question that if we look at this particular thing so pi so now we are talking about correlated strategy so we will be putting probability masses directly on the strategy profile so on this profile c comma c and f comma f i am putting half and half so therefore this f f f on this we are putting probability half and this c comma c on this outcome we are putting probability half is that a correlated equilibrium so what do we need to do to check whether this is a correlated equilibrium or not we'll have to see whether this inequality of this definition holds or not so what is what is it so I'm just giving you the way how you should do it but I mean I will not do everything so let's say we are looking at player one so and s1 that has been suggested to it is f and then the only thing that we need to check is s1 prime when it is c right so this is from the point of view of player one so now we can write down so this the same expression that we have here on in the definition we can we can write it down in this case and we see that the probability so the first thing is the probability of f comma f and we are going to multiply the utility of player one when it is following that f and the other player is also playing f and then so I'm just writing down the left hand side so then the other thing will be pi of f comma c and utility when the the that player so it is still playing f and the other player is playing c now we know that u1 of fc is 0 so this part is 0 and also this this probability is also 0 under under this correlated strategy and then you have this this number which is half and this number u1 f comma f which is 2 here so together this becomes equal to 1 now if you change so if you are playing s prime s1 prime as c what changes is this quantity becomes c so this part does not change this part also does not change all that changes is these two things become this becomes also c here and also this becomes c now we know that u1 c comma f is 0 and here pi of f comma c fc is 0 so on the right hand side we will have something like 0 which is strictly less than the utility when it is following that suggestion which has been given by the by the mediator similarly you can do it for c so when the suggestion was c for player 1 whether it is beneficial for for that player to follow that or not and also do this for player 2 and you can see that in all these cases this inequality is getting satisfied the inequality that we have said in this definition of correlated equilibrium and therefore this half and half so putting half probability here and half probability here is indeed a correlated equilibrium what you can also observe let us now compare the the the utility the expected utility here so let me just erase this part a little bit so the expected utility here under MSNE you can see that that is going to be two-third for both the players while for correlated equilibrium all that you need to do is to find out the corresponding strategy the so with probability half you are getting so when you are in the correlated equilibrium for player 1 you are getting with probability half you are getting 2 and with probability half you are getting 1 so it gives you this this expected utility which is larger than the mixed strategy Nash equilibrium so certainly if you are following a correlated strategy you can do better in an expected way so similarly you can take a look at the wait and go game in the traffic lights so and I have given a different correlated strategy which is putting probability of one-third under over this three strategy profiles wait wait wait go and go wait and the question is whether this is a correlated equilibrium follow the same strategy as before you can you can check whether there are other correlated equilibrium the one that I have just suggested a while ago and which is very commonly used in the in the traffic lights that is you put half so the the example that we have given so this example put half and half on wait go and go wait and zero for the other two cases whether that is a correlated equilibrium or not