 From the discussions of the previous module, we know what a Bayesian game is. This is essentially given by this tuple where you have the players, you have the type sets for each of these players. There is a common prior with which the whole type profile is chosen and for each type profile, lowercase theta, there is a normal form game given by gamma of theta. Now with all this, now we are going to look at what are the different strategies in this game. And we will also see some examples to understand how these strategies are being used. Remember that Bayesian game is a somewhat extension of the normal form game with the thing that it chooses different kinds of normal form games given the type profile. So, the strategy both pure and mixed has the very similar structure, similar definition, just the difference being that now it is a mapping from this type set to the set of actions for pure strategies. So, if you are having a specific type, player i has a specific type, then the strategy corresponding to that type is a specific action in that set. So, this is what we are going to call as the strategy in the Bayesian game setting and it will be written as si of theta i. Similarly, the mixed strategy will be a probability distribution over the same action set ai for that particular theta i. So, sigma i theta i is the mixed strategy when the type of player i is theta i and that is living in the simplex. So, the simplex is nothing but the collection of all probability distributions over the elements of capital ai. Now, we will use the following two terms. Actually, we will be using three terms and that term is essentially is showing that how a player experiences its utility in multiple stages. So, in game theory, typically all these terms of when a player experiences its utility is denoted by this Latin names. So, x ante is somewhat, so, let us say the nature is actually choosing theta of i for player i. That is according to the definition of Bayesian game, this common prior realizes this theta i. Now, before this theta i has realized, player i can take a decision and that decision is called the x ante decision. So, if you are trying to calculate what is the utility, if you are taking an action, all these things are called the x ante utility or x ante action. I am just defining the terminology in this case. So, now that you have realized, so theta i and player i knows what this theta i is, but it does not know what theta minus i is. So, theta minus i not known in that situation, what we call that as x interie. So, interie because you can see your own type, but you do not know other types. There is a third stage after you have realized your theta i, you know your theta i and theta minus i is also known. So, maybe it is at the end of the game when everyone has their types realized and you have also seen everyone's type, then what is, then those situations are those utilities and the actions are called exposed. So, exposed utilities are exposed actions. So, far that we have discussed the kind of situations are actually exposed situations because we are assuming that there is no uncertainty with these types. So, you already know what types of the players other players are. So, in this Bayesian game setting, if you want to look at the classical game setting that we have discussed so far, the classical normal form games, it is an exposed situation where types have been realized and everybody knows it, the type becomes a common knowledge for all these players. Then whatever you can guarantee Nash equilibrium and all those concepts, that those are exposed. So, we have already discussed this. So, here we will be interested only in this x-interim and the x-ante situations. So, what is an x-ante utility? So, this is remember that it is the expected utility for a specific player even before observing its own type, it does not know what its own theta i is. Therefore, it will have to take an expectation with respect to this common prior. So, it does not know what realization theta i has happened of theta i has happened, but it certainly knows what is the probability with which it can be picked. So, it actually knows the common prior that is a common knowledge. So, therefore, it can look at the utility the mixed strategy for all the players. So, theta sigma of theta is nothing but a shorthand notation of sigma i theta i. So, sigma 1 theta 1 let us say sigma 2 theta 2 and so on sigma m theta n. This whole vector we have used this shorthand to denote that. And of course, this is at a specific theta. So, the type profile is given by the theta. Now, the player does not know it. In fact, player 1 does not know any component of it deterministically. It can only take an expectation with respect to that. So, the utility of player i when it is expected over that p theta and that is given by this ui of sigma. So, this is the anti-utility of that player. And if you want to expand that out, this is just doing the same thing. So, this is the product because all these decisions, all these actions were taken independently by each of these players. This will come out as a product. You take the corresponding utility for those actions and that type profile. You first take the summation over all the actions and then you also take the expectation with respect to the thetas because the thetas has not realized yet. So, this is the x anti-utility for player i. Now, the once you are in the x-interim state that is you have observed your own type, but you have not observed other types, what you can do because you have a common prior, you can somewhat guess what the other player's types could be. In some sense, we can compute this posterior distribution according to the Bayes rule which says that what is the probability that the other players are going to choose theta minus i given my own type is theta i which is deterministically known to me. Now, here you can see that in order to define this posterior, we need that the denominator is positive and that is the reason why we needed the positive marginal assumption to be very crucial. Now, the x-interim utility is very simple to define. Now, the difference is that player i knows its own type. Therefore, it can take the posterior distribution, it can take the expectation with respect to that posterior distribution of all the other player's types. So, here at least one component is now known, the other components unknown and therefore it is taking the expectation with respect to that. So, the left hand side is essentially the x-interim utility just to make sure that this utility, the left hand side is nothing but ui of sigma i theta i because this is deterministically known. For all the other players, so it is sigma minus i theta minus i, it is actually not a it is not a function of theta minus i because it has been already expected over. So, it is definitely a function of theta i because that is that is perfectly known to player i. So, at some point later we will use this fact that it is it is a it is a function of theta i and that theta i is deterministically known to player i. So, this is the left hand side here. So, now let us look at the relationship between these two utilities. So, of course, the x-interim state is giving more information to this player. So, if you take the expectation with respect to p of theta i for player i over the, so this term inside is nothing but the x-interim utility. Take the expectation with respect to p theta i what you will get is the x-ante utility. Let us look at a few examples to understand what this means. So, suppose we have the first example is that you have two players who are bargaining over a cell of a single indivisible item. Let us see what trying to buy an item and there is one player who is a seller. Its type is given by the price at which this player is willing to sell. So, let us say that price is the minimum price below which it would not be able to sell is something like the cost price or the purchase price. The other one is the other player is the buyer. Its type is the maximum price at which it is willing to buy. If the seller asks for a larger price, then the buyer would not be able to buy it. So, for simplicity, let us assume that the type of all these players, so type set for both these players is a set of integers from 1 to 100. And similarly, their action sets are also the same set which we are going to call their bids. So, they can have a specific type. So, their type, so as we said the type is the price. So, for seller it is the maximum minimum price at which it can sell, for buyer it is the maximum price at which it can buy and they can actually report this, but reported bids could be something different. And we know that this is from experience that if the bid of the seller is smaller than or equal to that of the buyer, then the trade will happen at a price and let us assume that the trade happens at the price average of these two bids. Let us say the seller is saying, I am happy to get the, happy to sell this object at any price greater than equal to 10. And the buyer says that I am willing to buy it at any price less than or equal to 20, 20 rupees, then the trade will happen at 15 rupees, that is the assumption here. If the other thing happens if the seller says I would not sell it below 25 rupees and the buyer says what it was saying 20 rupees and less, then the trade will not happen that we know. So, suppose the type generation in this case is independent and uniform over theta 1 and theta 2. So, it means that if you are looking at the probability distribution of theta 2 given theta i, it is just the unconditional probability theta 2 which is 1 over 100 because there are 100 such entries here and it is picked uniformly at random. Similarly, for P of theta 1, it is 1 over 100. Now the utility, you can see it very clearly. If player 1 is choosing the action A1 and player 2 is choosing this action A2, then the trade will happen only if assuming that the second player is the buyer and the first player is the seller, the trade will only happen if A2 is greater than or equal to A1. Otherwise, the trade will not happen and both these agents will get zero utility. So, for player 1 who is the seller, theta 1 is its cost price and the item is getting sold at this price which is the average of these two numbers. The difference is essentially its utility. Similarly, for the buyer, theta 2 is the value that it gets whenever it purchases that item and it purchases it at this price so that this difference is the native utility for that player. And in this case, the common prior is given by, so this theta 1, theta 2 is nothing but the marginal. So, it is a product of the marginals P theta 1, theta 2 because we have assumed this to be independent and that is 1 by 10000. So, let us look at the second example here. The second example is that of a sealed grid auction. So, the setting is almost similar. There are two players and they are both willing to buy an object. Now, this is not a seller and both of these players are essentially buyers and there is some auction going on. There is an auctioneer who is selling this object. It is not a player. This auctioneer is not part of the game. So, these two players can report, I mean they have their values and their bids which is living in this interval 0 to 1. So, you can normalize the whole space and say that they can pick any real number between 0 and 1 as their values and their bids are also the same. Now, the allocation function is such that, so here is the deal. So, this is saying that the first player is going to be the winner if its bid is at least as much as the second player's bid. So, this is the case where the tie is broken in favor of player 1. So, this O of 1 is nothing but the winner function. So, the indicator function that player 1 is going to be the winner. It becomes a winner if b1 is greater than equal to b2 otherwise it loses. If it loses then player 2 becomes the winner. Now, you can think of the beliefs, the patient belief in this case is coming from the common prior distribution where it is uniform over all theta 1, theta 2. So, theta 1, theta 2 is living in this two-dimensional square 0, 1 square and that is giving rise to this posterior distribution of theta 2 given theta 1 and similarly for theta 1 given theta 2. The utility is essentially given by this expression. So, player 1 is choosing the bid of b1, player 2 is choosing b2 and if their types are such then if a player wins. So, here is the assumption that if a player wins then it gets that item but it pays its own bid. So, let us say winner pays its bid in that case whatever it has bid bi is going to be the exactly the payment and it values that item as theta i. So, this difference is the net utility when it wins and this is just the indicator function that it wins or not. If it wins then it gets this otherwise it gets 0. So, in some in the literature we will see that this kind of an auction where you are paying your own bid is known as the first price auction, first price still bid auction. We will analyze this game much in further detail later, later in the next module and also later in the course but this is what the game is and here is the Bayesian game representation of that first price auction.