 Once we define a new game, it's very natural to ask what are the equilibrium concepts in that game? And we are going to do the same exercise with this newly defined game called the Bayesian games. Now, there are two stages as we have discussed. There are ex ante stage where certain decisions are being taken. So we have so far discussed only the utilities in this ex ante and ex inter in stage of this game and and therefore when the players are taking the decision, we are also going to call the strategies which has been taken as ex ante strategies and ex ante ex inter in strategies. So the first thing is the usual Nash equilibrium. So remember that Nash equilibrium in this context sigma star is a mapping. So sigma i's were mappings from this set capital theta i to the simplex of ai. So it is defined on the type of each player. So before the player is observing its own type, we can define this equilibrium concept of Nash equilibrium which says, so this definition of utility is nothing but the ex ante utility of those players of player i. It is saying that if you look at this profile sigma i star sigma minus i star, this is going to be a Nash equilibrium with respect to that common prior p. If deviating to any other strategy sigma i star is not going to be unilaterally, it is going to be better for player i and this should hold for all sigma i primes and for all players i in n. So this is very similar to the same Nash equilibrium definition. The only difference being that these utilities are now an expectation over all these pthetas. So when we are looking at the ex ante in stage when the players have observed its own type but it does not know what the types of the other players but is the type of equilibrium that we can define is known as the Bayesian equilibrium. So what does that mean? So it means that the utility when it is actually choosing the strategy which is sigma i star. So we are still going to call the same sigma star. So notice that in these two stages ex ante and ex ante ring the sigma i's are not going to change. They are the strategies which is a mapping from their typeset to this simplex. This is going to be same even in ex ante or ex ante ring stage. The only difference that happens is whether they can observe their types or not. And in the first case we are taking when calculating the utilities we are taking the expectation with respect to p theta i as well. In the second case we are not taking that expectation because we know player i knows deterministically what that theta i is. So this left hand side is essentially that utility when it is choosing the same sigma i star but it is also evaluating it at the same theta i because it now knows the theta i and it can take the expectation with respect to theta minus i given theta i for all the other players and it is not going to be better than the utility when it is deviating to the sigma i prime and again calculating the same utility at theta i. So if this inequality holds we are going to call this sigma star to be sigma star comma p to be the Bayesian equilibrium. Now what we can say here we are going to make a small remark as we did in the case of the Nash equilibrium the mixed strategy Nash equilibrium if you remember. We are going to replace this right hand side with just an utility just an action let's say a i prime and we can say that this inequality holds for all a i prime in capital EI. And one can show that this is necessary and sufficient for this Bayesian equilibrium. So therefore we can use this also as an the definition of the Bayesian equilibrium. I am not going to go over the proof it is exactly similar as the as the proof of the equivalent definition of a mixed strategy Nash equilibrium in the in the normal form game. So you can just redo it can be left as an exercise. Now what we can see here is this and the strategies as we have already said it's just a mapping from this capital theta i to the probability distribution over EI. This two equilibrium concepts even though they are looked at in two different levels of information for the same player. These two equilibrium concepts are actually equivalent when we are looking at the finite Bayesian games. So here is the theorem that says it formally. In finite Bayesian games a strategy profile is a Bayesian equilibrium if and only if it is a Nash equilibrium. So that actually demystifies this distinction. In fact we can without loss of generality work with any one of these kind of games in the incomplete information setup. So let us prove it formally. So the first direction is the direction where we are given a strategy profile to be Bayesian equilibrium. We will have to show that it is a Nash equilibrium. This is rather an easy direction because Bayesian equilibrium is when you have more information. You also observe your own type and you are taking the best response. So you can just write it down. So we are writing the right hand side of the Nash equilibrium definition. So this part here we are writing it down. And we are expanding it with respect to this the definition. So if we are looking at theta i the probability that this theta i is picked is p of theta i because we are taking the expectation with respect. So this is the x anti-utility of player i when it is choosing theta i prime. And inside you have the x-interim utility. Now because this is a Bayesian equilibrium we know that this x-interim utility is going to be at most the x-interim utility under theta i star. Therefore we can write this inequality here. This is by the definition of Bayesian equilibrium. And then you merge this together and you get the x anti-utility at sigma i star, sigma minus i star. So this is the definition of the Nash equilibrium in the context of Bayesian game. So if we have a Bayesian equilibrium which is sigma star we know that that is going to be a Nash equilibrium as well. Now the other direction is when we are starting with a Nash equilibrium we will have to show that that is also a Bayesian equilibrium. As we said the this is only talking about the strategies and the strategies are just the mapping. So we can actually talk about the equivalence of the strategies. What is different in this setup in Nash equilibrium and Bayesian equilibrium is the information about those theta i's. So we are going to prove this reverse direction by contradiction. So what we are going to assume, suppose this is not true, what we are trying to prove if a strategy profile is a Nash equilibrium it must be a Bayesian equilibrium rather we are going to show that if it is not a Bayesian equilibrium then it will not be a Nash equilibrium either. So let us do in that direction. So because it is not a Bayesian equilibrium that means we are going to negate the statement of the Bayesian equilibrium. So that means that there exists some theta i and some a i like this where this inequality is not true. So it is going in the reverse direction. So let us say that formally so that means that there exists some type of player i theta i and some action of that player a i such that this inequality is in the reverse direction. So this is a violation so we cannot say this sigma i star sigma minus i star to be a Bayesian equilibrium. Now because it is so because we know that it is not a Bayesian equilibrium now we can actually show that this sigma star b is not a Nash equilibrium as well and we show that in a constructive manner. So we construct a strategy sigma i hat for player i in the following way. So we know that we have a very distinguished type theta i. So for all the types which is not theta i from the same set capital theta i we are going to keep the probability masses probability distribution to be the same. Now for this particular type theta i we are going to put all the mass on this specific action for which this inequality holds. You can already start seeing what we are trying to do. We are actually trying to show that if we put all the masses there then that must be violating the condition of Nash equilibrium. And for all the other actions which is not equal to a i that distinguished action the probability mass under this strategy of player i mixed strategy of player i is going to be equal to 0. So now we can start with the x and a utility of player i under this strategy sigma i hat and sigma minus i star. So we know that this can be written in the following way. Now what we know is we can actually decouple it into two parts. So from this set pick out this theta i and write that separately. So this is just that specific term separately written for all the other things. For all the other things we know that theta i hat there is no difference between theta i hat and theta i star. Because that is the way we have defined it for all the other types, for all the types that is living in this set this is going to be the same as theta i star. So we can as well replace this. For this particular type we actually know that this is going to be strictly larger than the the utility at sigma sigma i star theta i theta minus i because that is the way we have constructed it. In particular if you look at this how can you write it it is only one at a i that specific a i where you have this inequality for every other place it is zero. So therefore because of this fact that at that a i it is strictly greater than the utility under that theta i star you can have this strict inequality here. So therefore putting everything together you have this thing strictly greater than this quantity. So actually you can replace this with just sigma i star and that is nothing but the x anti-unitility when player i's have this strategy of theta i star theta minus i star and therefore this strategy profile cannot be a Nash equilibrium. We already constructively shown another strategy for player i which is unilaterally better for player i. So we have actually proved both this direction. So this so we have first shown that if it is a Nash equilibrium if it is a Bayesian equilibrium then it must be a Nash equilibrium. If it is a if it is a Nash equilibrium then it is also a Bayesian equilibrium. Let us now look at the existence of Bayesian equilibrium. The Bayesian equilibrium is guaranteed to exist for every finite Bayesian game and the argument is very similar to the to the arguments that we have done in the case of existence of Nash equilibrium. So what so we are not going to prove it formally the the proof rather post an addendum which does a transformation the idea of the proof is that we are going to transform because it is finite we can transform this Bayesian game into a finite normal form game. The difference being that you have the players as the types themselves. So even though there are let us say for a specific player 1 there were three types let us say theta 1 1 theta 1 2 and theta 1 3 we are going to treat in this transformed game each of these types as one individual player and those players will have the same utility which is equal to the utility at that type for that player. So we can actually expand this the space we are going to blow up the the number of players in this way but the point is that in that game we can show that that game is a finite game action space and also the number of players are all finite and in particular once you look at the strategies the utility is there that utility can be actually shown to be equal to the utility of this Bayesian game when the type is actually theta 1 1 for that player. So please take a look at that addendum and you can see that there is a transformation which is possible and because we know in finite finite normal form games by Nash theorem and the mixed strategy Nash equilibrium exists that mixed strategy Nash equilibrium will be the the Bayesian equilibrium when we look at this game where these types are theta 1 1. So that essentially proves the fact that Bayesian game we can actually have a Bayesian equilibrium for every finite Bayesian game.