 So far we have discussed different kinds of games. So it might be a good time to stop and look back What are the things that we have discussed and what are the things that we are going to discuss next? So games broadly can be divided into two types non cooperative games and cooperative games So you have we haven't discussed anything about cooperative games and we will not do that in this course So these are those kind of games where the utilities are defined not only on individual players Of course, there are utilities on on players, but also there are utilities or values defined on a coalition so coalition is a collection of individuals and because the it is assumed that multiple players can cooperate the The properties that we really need for cooperative games is slightly different from the non cooperative setting so in this course, we are going to only focus on the non cooperative games and That is what we have started with we have first look looked at the the types of games Which we call the complete information game. I'm coming in a minute what we mean by complete and what is the difference within complete information and Within that complete information setup. We have looked at two different representations of games One is the normal form the other one is the extensive form the normal form Representation is more appropriate for simultaneous move games and the equilibrium concepts are different while for Extensive form the form games. It is more appropriate for multiple stage games and the equilibrium are a slightly different now From this module onwards, we are going to discuss what is known as incomplete information game so even though in the in the previous examples Be it perfect information or imperfect information the players actually knew which game they are playing They might not observe certain states of the game, but they They were sure they were at least knew what is the game tree or the game matrix, etc In the incomplete information setting. We are actually going to relax that we are going to say that players do not Deterministically know which game they are playing. They might have a probabilistic belief about which game they are playing But this this being probabilistic that is actually bringing in the incomplete part in their information There are several other types of games like repeated games stochastic games, which we are not going to discuss here So as we said the the complete information setting that we have discussed so far players Deterministically know that the game that they are playing But in the incomplete information setting they do not really know which game they are playing So we will soon give an example, which will make this make this clear Sometimes we are going to refer to some of this information. So let's say and Why it is incomplete? Of course, there is some information that is available to each of these players and this information is essentially private information which we are going to call their types and in particular in this in this Discussion will be focusing only on one special subclass of this incomplete information games, which are known as Bayesian games As the term Bayesian says that you have some sort of a prior distribution and Based on that and your own information about your own type you have a Bayesian belief about the other players types So let's look at the example, which will make things much more easy to follow So let's look at one soccer game and there are two competing teams which are planning to play against each other Now each of this game This teams can choose a game plan. So game plan is something like their plan How they are going to play whether they're going to play aggressively. So that means they are aiming to win Let's call. This is an aggressive style of playing Aggressive type of playing or they're aiming to draw which is a little passive type of game plan and Both these things whether they are they are planning to play aggressively or planning to play passively is Essentially what their type is. So this player has an aggressive type or this player has a passive type Where the players at this competing teams Now this what is known to them is whether they are playing aggressively or passively So this is this means that these private signals are Non-deterministically to them and this could be often caused by certain external factors For instance, if the weather condition is favorable, maybe they are playing aggressive if they have player Some of these key players who are injured then possibly they are forced to play passively But all this information is privately known only to them and that Decides how they are going to play in the actual game. So The the utilities as you will see will be actually dependent on which kind of types they choose So let's use this shorthand for for aiming to win as W that is the aggressive type as W and the passive type as D that is they are aiming to draw and Based on their types this Two different types. These two players can have four different type profiles and which are given by WWWDW and DD which and the the meanings are quite evident now once This players have chosen their types The the game actually is dependent on their choices of types So, for instance, if they both of them are playing aggressively then maybe a specific Utility structure is true So they are playing a different game when both of them are playing aggressively under that aggressive mode if they are They choose this action of attacking and defending. They might get different utilities So I would not be very much worried about the actual numbers and how you can justify that The whole point that I'm trying to make is based on their type profile There might be different kinds of utility matrices. That is the whole point of this patient game now Based on this type profile since the the game is changing and for a specific player Let's say player one can only Deterministically know whether they are what they are typing. So whether they are W or they are D That is known. So only this component the first component is known to player one and the second component is Deterministically known only to player two and not to player one and vice versa so therefore This this game is incompletely known by each of these players They do not deterministically know which game they are playing This was quite different from the games that we have discussed before where the game matrix or the game tree was Known beforehand for both both these players. So this is the incomplete information part So we are going to make a couple of assumptions in this setting We are going to assume that the probabilities of choosing choosing these different games This is chosen by let's say nature and this is coming from a common prior distribution So you can imagine that this Whether this will be a game where both players are actually choosing aggressive strategies or one player is choosing aggressive the other player is choosing Defensive strategies that is given by some probability distribution over these four type profiles So let's make the things a little formal So a Bayesian game is represented by as before certain a couple of few things The first entry is the set of players as as usual The second Entry is what we what we are going to call the set of types of player i so theta capital theta i Is nothing but the set of types of player i And this is as we have seen in this example that can take certain values and it could be different for different players Now the third entry is what we just said This is the common prior distribution over the entire type profile So capital theta is the Cartesian product of this individual capital theta i's Which means that it is putting some probability Distribution over the entire type profile. So it is going to in if you go back to the previous example It is giving this probability with what probability This particular type profile is chosen for instance We we are also going to put another restriction for this common prior We are going to assume that if you take the marginal So if you just take the sum over this sum over the theta minus i's Then what you will get so this sum after doing this sum you will get this marginal p of theta i We are going to assume that this p of theta i is always positive What does that mean it just says that the probability of a Of choosing of occurring a specific theta i is positive If it is zero, then it means that the the probability the nature Chooses that theta i with probability zero And therefore you can also look at the game by just pruning that particular type from your typeset from player i's typeset And nothing will change. No properties will change in the in the game So without loss of generality, we can assume that this marginals for every theta i Is going to be positive Now the last entry in this description of Bayesian game Is nothing but a collection of normal form games So because we have seen that once we choose a specific type profile There is a unique game for that that unique normal form game is represented by gamma of gamma of theta So what does that mean it's a normal form game for the type profile theta and this by the standard notation we can Show I mean we can write that this gamma of theta is Has the same set of players as before It's action sets for each of these players could be dependent on theta Even though in in this context in this discussion and and later We are all always going to assume that the action set is the same for all type profiles theta So this is this is a simplification However, the utility for every player is very much dependent on this theta. So this you can see that So suppose the the theta is actually d comma d In this context Then the gamma of that theta is nothing but this game that we have written here So this is this is that normal form game And we know that if we look at a specific utility Of a specific player, let's say player one And at that theta which we have already assumed and if if it is attack comma defense For these two players, we know exactly what is the What is the utility? So in this case, it is going to be this utility for player one So that is represented by this utility Function ui it is a mapping from that set of all action action profiles set of action profiles for all these players And also the type profile to the set of real numbers Okay, so that is the that is the description of the game. I hope that it is it makes the the description clear So if we look at the the Bayesian game Stage by stage So why it is Bayesian because you have a Bayesian prior and we'll very soon see that what kind of properties We are expecting over this and that has a kind of a implication From the from the Bayes rule So in the Bayesian game, which is an incomplete information game So there are certain stages that Happen one by one. So the first thing is that the type profile is chosen randomly according to that common prior This is chosen by the nature Now once this is chosen each player observes her own type So this whole vector is realized, but a player i only observes its own type the component i And then player i picks its action, which is ai so which is coming from this set capital ai And once that is chosen So player i chooses its action ai the other players are choosing their actions a minus i And that type profile is given by theta i theta minus i Player i gets an utility and every player gets this utility Their corresponding utilities. So that is the the description of the Bayesian game