 Apart from the fact that correlated equilibrium models certain situations, certain real world situations much better than mixed strategy Nash equilibrium as we have discussed in the case of the friends, two friends watching a footballer soccer game or in the case of traffic signal game, correlated equilibrium also had another advantage which is in its computational tractability. We saw that mixed strategy Nash equilibrium is difficult to solve and in particular that is a hard problem, computationally hard problem but that does not apply to correlated equilibrium. So let us look at what are the set of inequalities and equalities we need to solve, we need to satisfy in order to make a correlated strategy to be a correlated equilibrium. So this is this condition one that we have written here is directly coming from the first principles. So we have just written down the definition of correlated equilibrium and this inequality is satisfied for all every si and si prime and for all players i in n. And the second set of inequalities is just the feasibility that these are valid probability distributions. So each of these numbers pi of s should be non-negative and they should sum to 1. So now notice that instead of the variables being individual probability masses of over all the strategies of a specific player, this is now the correlated strategy is a probability distribution over strategy profile. So therefore the optimization variables here is just pi of s. So which is which is appearing on this side here and also here. So contrast this with the similar expression which we had for mixed strategy Nash equilibrium there we had sigma 1 s1 multiplied by sigma 2 s2 and so on. So that was actually giving rise to a product form of these variables which is a non-linear optimization problem and that was not very easy to solve. While here all the inequalities are actually linear inequalities and that is giving so that that makes the problem of finding correlated equilibrium a linear program. So now let us look at how many inequalities that we are trying to solve. So just comparing the number of inequalities we can get one idea that how easy it is to solve. Of course we are not considering this non-linear optimization problem in the case of Nash equilibrium. We are going to compare this just from the counting the number of inequalities that we need to solve. So here we have because this inequality here has to be satisfied for all si si prime. So there so there will be m square number of inequalities if we assume the cardinality of each of these players cardinality of the strategies sets of each of these players to be equal to m then there should be m square such inequalities for every player. So therefore together there will be order of nm square number of inequalities. The second inequality in the second condition where we have the non-negativity constraint this number becomes a little larger because there are m to the power n number of possible strategy profiles in this case. So therefore this actually increases accordingly and the final one is just one. Now these inequalities together represent the feasibility LP. So even in the case of mixed strategy Nash equilibrium we solve them feasibility LP where we have written down the conditions for becoming a mixed strategy Nash equilibrium and feasibility LP just means that the objective function is a constant. So you do not really care about the objective function you just care about solving this inequalities. So solving this feasibility LP will give you one correlated equilibrium in this case. So let us now contrast to that how many number of cases inequalities that it is solving in the case of CE and vis-a-vis with the MSNE. So in MSNE the total number of support profile itself was 2 to the power m times n. So if you remember we defined a number called K which was exhaustively listing all possible supports and there were 2 to the power cardinality of Si minus 1 number of supports for every player and we have taken the product over all players. So since this cardinality is equal to m and you are multiplying all this n over n players this becomes 2 to the power m times n. So this many number of support profiles you have and for each of those support profiles we are going to solve one feasibility program which is a non-linear feasibility program. So not considering that fact still you have a lot of search a lot of support profiles to search over. So in the case of correlated equilibrium we have only the order of m to the power n that is the only bottleneck because if you look at this this is much smaller this number of how many number of inequalities we are handling for the case of for satisfying this condition one the only bottleneck is in the second bar. So even if we look at these two numbers order of 2 to the power mn and order of m to the power n we can see that this is actually exponentially smaller because if you look at the log of these two numbers the log of this will give you mn while log of this will give you n times the log of m and even though the first term is same here the second term is exponentially smaller in the case of correlated equilibrium. So correlated equilibrium not only gives you less number of inequalities it is also giving each of this inequalities are actually linear inequality so it is much easier to solve. On top of that you can also look at some of the optimization objectives because we have seen that there exists multiple correlated equilibrium for instance in the case of the traffic signal game we have seen that there are multiple correlated equilibrium for football or cricket game also there are multiple correlated equilibrium. Now the question comes which correlated equilibrium should we choose and that can be resolved by solving the same feasibility program with an objective function. So now it is a full-fledged linear program with a properly defined objective function on the in the constraint set we have all the constraints for becoming this correlated equilibrium so all these constraints that are written here but I we might be interested in for instance finding that correlated equilibrium which maximizes the sum of the utilities of all these players. So that could be one feasible objective and that gives rise to different kind of different kind of objectives gives rise to different correlated equilibrium. Now we have defined a new equilibrium concept in correlated equilibrium it is a little different from the previous concepts that we have discussed because all of those were strictly looking at the agent level view and not cooperating with each other but the correlated equilibrium is in some sense not cooperating directly but via mediator. So it is a correlated equilibrium that is why the name also says that it is coordinating with the multiple players and taking that decision. So how does it compare with the how does it connect this equilibrium with the mixed strategy Nash equilibrium concept? We can actually show that for every mixed strategy Nash equilibrium sigma star there exists a correlated equilibrium pi star and the construction of that pi star so what does this mean that we can find a correlated equilibrium for every game where there exists a mixed strategy Nash equilibrium and the construction is really simple you just take the product of all these sigma sigma i stars so we had the so let us say we have a mixed strategy Nash equilibrium profile sigma 1 star sigma 2 star and so on to sigma n star all that we are doing is to define the pi let us say pi star correlated equilibrium of s we are just taking the corresponding product sigma 1 star of s 1 that is the probability mass that you are associating with that strategy of player 1 times sigma 2 star of s 2 similarly sigma n star s n. So this particular structure of the correlated strategy is claimed to be a correlated equilibrium as well and I leave that as an exercise it is not very difficult all that you can use you need to use here is a hint you will have to use a MSN characterization result the characterization result which says that for all the strategies on the support of a mixed strategy Nash equilibrium your utility should be same expected utility should be same and that should be greater than equal to all the strategies the expected utilities at all the strategies which is outside that support in fact you can write the inequality for all the strategies and that and that is in s si so this using this fact that you are using this MSN characterization and using this substitution for pi i star you can show that pi star pi star is essentially a correlated equilibrium. So that theorem essentially gives us this bigger picture so we have already seen that strongly dominant strategy equilibrium is also an weekly dominant strategy equilibrium weekly dominant strategy equilibrium is also PSNE PSNE is definitely a special case of an MSNE. Now we have also shown that MSNE for every game that has an MSNE you can always construct correlated equilibrium. So therefore this structure so this is the space of games which admits an SDSE this is the space of games which admit an SDWDSE and so on and this set keeps on increasing in this way. So the largest set is the correlated equilibrium it is the most relaxed equilibrium concept and therefore the the set of games that has a correlated equilibrium is the largest. So let us make our short summary so far we have so far discussed only normal form games and we have looked at the definitions like rationality, intelligence, common knowledge and we have discussed with examples. We have distinguished between what is a strategy and what is an action looking at certain kinds of games we have looked at dominance which were strict and weak and the corresponding equilibria concepts were defined as a strongly dominant strategy equilibrium and weekly dominant strategy equilibrium. Then we went to the definition of unilateral deviation so if one player unilaterally deviates and does not get better off then we call those those strategy profiles as pure strategy Nash equilibrium its generalized version is a mixed strategy Nash equilibrium and its existence is guaranteed. Then we also looked at the characterization result of MSNE but the computing side we have seen that it is hard to compute then we went to this scenario where we use the trusted mediator and go to correlated strategies and there we also defined a correlated equilibrium.