 So, to motivate today's concept, let me ask you, let me, let us think of a setting. Suppose, you are, you have, you are managing a airport security, okay. Now, you want to, there are, say, let us say, two ways, two roads to access the airport and you have some end guards that you can keep at those, at those roads. You have to decide who, how many guards at each airport, okay. Now, the thing that you, what you could do is, you could sort of just say, well, I am going to keep equal number of guards at each and so on. But you also have some idea that, let us say, terrorists are likely to approach the airport from one particular road, okay, more likely to approach. Or, or, or let us say, if a person is approaching the airport from a particular road, it is more, it is, it is like, there is a certain chance that that person is actually a terrorist, okay. So, the person that, any person that comes into the airport is, can be of one of three kinds. He can be, let us say, he can be a terrorist or he can be a contraband trafficker, truck trafficker or something like that and or he could be just an innocent traveler, okay. Now, what you do not know is, when you look at the person, you do not know which of these three categories he lies in, okay. But what you do know is, well, that in a sort of, in this, along this road, here is the chance that this is going to be a terrorist, on this road, there is, there is a certain chance that, there is some other chance that he is going to be a terrorist and so on. So, basically what the, the essential issue is that, you do not know when you are confronting a particular player, a particular opponent, you do not know whether that person is a terrorist or this or that, okay. And you have to come up with an allocation strategy for your troops, in such a way that works regardless of what the other, what the other guy is, right. So now, is this a game of, is there some, there is clearly some lack of information here, okay. The sense that, you do not know the, the identity of the person who is coming in or identity in the sense that you do not know which class he lies in, is he, is he a terrorist or drug trafficker or innocent traveler, but the fellow who is coming in, he knows his identity, right. Now, what can you, how can we model a game like this in what we have already studied and whether we can actually model a game, a situation like this in the frameworks that we have already studied. So, you, the main thing I want to point out is that, see is the, when we talked of imperfect information so far, what a player did not know is what was happening during gameplay. So, all the ignorance was something to do with what was happening during gameplay, he did not know the exact node you are on and therefore he did not know, he was on and therefore he did not know, say let us say what someone else has acted or what who has acted and so on and so forth, those kind of things were, so it was the ignorance of the player was about stuff that has happened during the, during the, during gameplay means when, during the time the actions were taken, ok. Now, this kind of uncertainty where you do not know the identity of the other player, whether the other player is a terrorist or a drug trafficker or innocent traveler, this is not something, this is not an uncertainty during gameplay. So, it would have been an uncertainty during gameplay, if to be a drug trafficker or to be a, or to be a terrorist or to be an innocent traveler was a part of that person's strategy and it was an action that he was choosing and it is an action and therefore and you did not observe that action, let us say, ok. But that is not the case here, right, the case is he is not choosing to be one or the other, he is one of the, one or the other, ok. So, what you see here is that there is actually a kind of a different type of lack of information that is going on here, that is a somewhere there is a, it is distinct from the, from the ignorance that you have during gameplay because and that arises because there is something that you cannot observe about what has happened during gameplay, right, whereas in this case the issue is the uncertainty is not about or the lack of knowledge is not about what has happened during the game, but rather about what the game actually is. You do not know whom you are playing against, you do not know if you are playing against a drug trafficker or you are playing against a terrorist or a playing against an innocent traveler, right. And the policy that you would have, how many troops you will have here or how many troops you will have there, etcetera, depend on that, right. But you have to decide eventually how to allocate your troops without that knowledge in any ways, is this clear, ok. So, there is a, this leads to a different form of uncertainty in the game or different form of lack of information in the game, ok. And that is, so we distinguish this from the earlier, earlier what we, the kind of lack of information we were referring to was imperfect information. So, imperfect information referred to the situation where there was ignorance about, so the ignorance was about events that have happened, that have happened during gameplay, ok. So, this kind of, the kind of ignorance that I am referring to right now is what is called incomplete information. So, incomplete information is referring to the setting where actually there could be one of several types of, several games or a player could be one of several types. And we, and a player does not know which of those types is actually the true type, but each player does know his own type, right. So, when a terrorist comes he knows that he is a terrorist when and so on, ok. So, this, this requires a different way of modeling, ok. Essentially, this actually comes down to something that we studied at the start, which is, you know, the almond model of incomplete information was primarily about this. If you remember, there was this game of John and Paul, Paul was color blind or not color blind. It is about that, it is, that was, that was a situation of incomplete information because whether he is color blind or not is something that, that player knew, Paul was color blind or not is something that Paul knew, but John did not know, right. So, it is not a sort of a decision to be color blind or not, it was about whether he is color blind or not. And that essentially led to a situation where you did not know which, if we did not have a decision layer on top of it, but if they were say, let us say betting on which car won and so on, effectively it would have, the other player would have to bet, taking into account that he does not know whether Paul is color blind or not, right. So, this needs a framework of its own, ok. And so these, so what I will talk about today are these games, these are games of incomplete information. So, formally if the way we define this is you have a set of players, set of players and is your set of players, ok. Now the each player has what we call a type, ok and let T i be the set of types of player i. Now, what is a type, a type is basically his identity, ok. It can mean, you know, depending on the situation at hand type can be anything, it can be, let us say how much income he has, how much, how much he wants a particular item if he is a, let us say in an auction for example, a typical situation is an auction, a seller wants to sell an item, but he does not know how much the buyer, how much that item is valuable for that buyer, ok. Type is some, you can, most generally it is some form of private information, ok. It is something that is known to that player alone, but others do not know what it is, alright. So, each player has a type, player knows his type, right, but not the type of others, ok. He does not know the type of the other players, alright. Now, the way the game proceeds is that nature chooses a profile of types, ok, with let us say probability distribution P. So, nature will choose a profile of types that means one type for each player with a probability distribution P, this P we will take as common knowledge, ok, anything that you do. So, this P is something that is common knowledge, so all players agree on the probability with which various types are going to be generated. If there is any asymmetry in this also, then that can be modeled through another, you know, more another meta type essentially, which on which they disagree again, ok, on which there is some, they do not, with something else that they do not know, ok. So, this probability distribution P and this P is common knowledge, ok, so how does the game proceed? So, nature chooses T1 to Tn, which are the types of all players. Each player gets to see his own type, so P1, C is T1, P2, C is T2, etc. and Pn, C is Tn, ok. Now, because P is common knowledge, players do know the probability distribution with these, which these types are going to be realized and so they have effectively a belief about the types of other players, alright. So, the each player has a belief about the types of the other players and the belief comes from, let us say what is player i's belief that the type of the other players is T minus i. So, if we, so the answer to this question depends, so if I ask you what is player i's belief that the others are of type T minus i, now the answer to this question depends on when it is, when this belief is being computed. So, before the start of the game, before nature distributes the types, ok, or before the profile of types is realized, what is the belief that players have about, what is the belief that player i has about the types of the other players? No, no, it is still, it is P of T minus i, right. This is the probability with which I, player i put thing that the type, that the others have types given by T minus i, it is the probability, you know it is the marginal probability of T minus i, clear. But then after the type is realized, ok, once nature chooses the types that means once it defines, ok, who is the, you know, who has what talents or who has what endowments or who has what luck or whatever, once the types get realized, ok, what is, what is, what is going to be their belief, exactly. So, this is what we have learned, right, if you have information what you should be doing is conditioning on that information, you are, so that is what a Bayesian should do. So, a player once he gets the information of his type, his belief now gets updated, right, his belief now is that well, so this is the, let us say the x anti-belief, x anti means before the realization of T i. So, before the game starts, this is what we know, this is, it is going to be one of these, but once he, once T i, once the types are realized, he still has an uncertainty about the types of others, because he gets to observe only his type, right, so this, so after seeing his own type, this is going to be the belief, you can say let us say exposed after realization of, ok. So, now, the essentially as the theoretical question for us is what stage is the, at, you know, at what stage is actually this game being played? Actions are being chosen after the types get realized, ok, that is correct and that is understandable, but is the game being played before the types get realized or is the game being played after the types are realized? Each player knows his type, correct, but he could still play in, so if you remember the way, how did we think of, how did we think of dynamic games, we said extensive form games is as if player has to pick a strategy, which is a function which maps every information set to action and his, this strategy is chosen even before the game begins, right, so before the realization of any information, this strategy is to be chosen and we are playing this game in the space of strategies before the start of the game, no, so the question is, should I be playing this game before the realization of the types knowing and come up with a plan for what I would do in each type or should I be playing this game after the realization of the type? See, the reason, the reason this is a slightly tricky question is because in each case your belief is different, before the realize you, you know, your payoff or whatever, etc, those functions are already well defined for you, but the belief is different in each case. In the first case, you have a belief of, which is the ex ante belief about the types of everyone, of all players, whereas after the realization of the type your belief is conditioned on the knowledge that you know your own type, okay, so this is not that straightforward a question actually and in fact, this is one of the main contributions of this line of, this line of research that gives, you know, it gives a lot of clarity to, to this particular thing, to this question, yeah, so, okay, so what happens after the types get realized, after the types get realized, players now play, it's, it's, it can be let's say, it's, let's say it's a static game after that, okay, so they have some set of actions, actions, let's say static, okay, it could be dynamic also, but you can take it to be static, simple, let's keep things simple, so, so those are, that's also there, but let's, for the moment, let's just take it to be static, okay, so in short, they have some finitely many actions that need to be chosen and you can even have the actions dependent on the types, okay, so in each type, the player could have a different set of actions, like for example, the actions that are available to a terrorist is not the same as actions available to a, to a innocent traveler and so on, right, so each, you could have, but the point is you would have a set of actions for each type and then after that there is, once the types get realized, you know, it's a usual game, so actually let me write out the full chronology, the types get realized, okay, so he, now his players believe after the type gets realized is P of t, t minus i given t i, alright, then players select actions, actions, let's say ui in ui of ti, okay, so this is the set of actions, set of actions of player i in type ti, alright, and the players would get a player i gets payoff j i or cost, let's say incur the cost, j of t comma a, where t is the vector of types, t1 to tn, oh sorry, t comma u, u is the vector of actions, so this is the, this is the cost to player i in type profile p and action profile u, is this clear, so if the vector of types is t and they take actions, if the profile of types is t and the profile of actions is u, then this is the payoff that player i gets, alright, now what are the, what are the strategies of the player, well you can think of the strategies of the player now, for player i is a function gamma i that maps his type, this union of actions, overall types and you have to make sure that you take a feasible action in each this thing, right, so you have to gamma i of ti has to belong to ui of ti for all, for all i, so he has to take an action from what is available, alright, we can even allow randomization, so let's take a behavioral strategy, behavioral strategy B, where he chooses an action ui in type ti, this is going to be a probability distribution on ui of ti, this is your, this is a, this here is a pure strategy and this is a behavioral strategy, so you can see here that is, once I write this out in this way, it is, there is a sort of resemblance emerging with games of imperfect information, it is as if the type is the information set of the player, okay, and in fact that is, that is, that was one of the main insights, actually people did not know how to approach such games of incomplete information and that they could be, you know, if you think, if you introduce nature into the picture, okay, and if there is a common knowledge about the probability and you know, a bunch of other axioms give you common knowledge about the probability etc, what that does is, it basically reduces a game of incomplete information to one of imperfect information, okay, where the lack of information is essentially about the types of other players, okay, so suppose the, either are two players, let's say player one and player two, player one has let's say two types, I am going to write them as one, one one and one two, and player two has only one type, okay, and let's say these types one and two get decided with probability half each, okay, so probability of I1, 11, 2 is half, probability of I2, 2 is also half, and after the types are realized, the players have the actions that players have are, so the action of set of player one is let's say top and he can choose let's say up and down, up, down is his action set, and for, this is the same for every type, and in for player two the action set is left, comma right, okay, and this is what we are just seeing is that he has only one, so now can you tell me how do I, can I write this as a, as a game of imperfect information, so let's go through the chronology, first nature is playing, nature picks the types for everyone, so the game begins actually at this node where nature is going to play, we will denote nature by a player 0, okay, so nature picks a type profile, how many type profiles are there, there are two possible type profiles, right, 1, 1, 1, 1, 1 and 2, and 1, 2 and 2, okay, so one profile is this which is 1, 1 and 2, this is one type profile, the other is that there is 1, 2, comma 2, okay, now player one, each player gets to observe his own type, so now after this the game is let's static game, okay, I won't write out the matrix and all that, but the game is a static game, so now play, so let's say player one is playing here, so player one had two choices up and down, and after that player two has two choices left and right, okay, so now can you tell me what are the information sets of the players, for let's start with player one, this is for player one, this is for player two, what does player one know, player one knows his own type, right, so he can distinguish between this node and this node, right, so for player one, this is one information set, this is another information set, okay, what about player two, for player two he again knows his own type but does not know the type of the other player, okay, so for him what are the information sets, all four in one information set, is this clear, now why are all four in the same information set, so because I have that two reasons here, one is that he cannot see, so this is because player two, okay, cannot distinguish between the types of player one, okay, that is one, second is he cannot observe, I assumed that the game is static after the types get realized, so he cannot distinguish between the actions of player one, okay, now suppose if he could distinguish between the actions, okay, so let's say he could tell whether player one has played up or down, then what would be the information set, everyone agrees with that, what would be the information set in that case, suppose player two could observe what player one is doing but does not know the type of player one, so he then what could I put in, tell me what I should put in which, so both the use would be in one information set, so this would be one information set and both the Ds would be in one information set, is this clear, still he does not know the type of the player but he does observe the action, he does not know the type of player one but he does observe his action, is this clear, okay, so this is the sort of game of incomplete information, these are also popularly known as Bayesian games, okay, so there is the term used for them called Bayesian games and the reason for that will become clear soon, now I will just give you a window into what kind of issues that arise in this, see for example, you know suppose you are in this situation like we have here, player can observe the action but he does not know what the type of the player is, right, so just imagine a security situation, you know, you can observe someone's actions, you can observe that, okay, this is what he did, this is what he, this is what he reported to me, this is what he ticked on the form, etc, etc, the question then for the security officer is, he wants to know what is the type, right, he wants to know from there, from those, from the reported whatever actions he can observe, from there he wants to know what the type of the other player is, is this fellow a terrorist, is this fellow this, is this fellow, is this fellow a you know drug trafficker or is he just innocent, right and from the point of view of the terrorist, the problem is that what kind of actions should I take that do not reveal my type, right, I should, so the terrorist would always want to take an action that makes him no different from the innocent traveler, makes him look no different from the innocent traveler, the innocent traveler would want to take an action that makes confirms him as an innocent traveler, okay, so you can see that there is a very interesting soup here of decision making and information that is both going on here because the kind of action that you take reveals something about what you know, right, you reveal something in this case what you know means what you know in terms of your information there, what type were you born with, okay, so actually there are very interesting questions that come up exactly because of this that what you know, so now the challenge in some sense for this, for the security this protocol is to come up with a protocol that distinguishes between the types as finally as possible, so that you know can you get players to take actions that will eventually reveal some signature about their type, is it clear, okay, so we will come to all of this subsequently as we build through this because this is essentially becomes goes into more and more into information and into games and the informational aspects of games, okay, alright, so, okay, so alright, so now let us come back to this thing, so players select actions these are the payoffs that they get and these are the strategies that you need today, okay, so now let us write out the two the payoffs as I said there are two stages at which we can describe the game, alright, so first let us write out there are first let us write out the payoff after all the types are realized, okay, there is a payoff that is of so we or we have a cost function here which is a function of the which is a function of the types and the actions, alright, now and we have allowed for players to choose their actions randomly, you know, do randomization in as a behavioral strategy, okay, so now let us write out first this thing which is this is J i of B given T, so this is B 1 action 1 given T 1 all the way B n action u n given T n, okay, so this here the random this is the randomization that occurs due to the random choice of the action due to the players randomizing, okay, there is no natural randomization here from nature this is a type profile has been realized players are playing randomly and that randomization is results in an this is the expectation that is due to over that randomization, okay, so why should be summing up here over all these UIs, okay, each Ui in capital Ui of T i for all i n for all i n, clear. Now, this is the payoff that would get real that would be observable to what player or rather at what time sorry at what stage, so this is the payoff for player i when does player i get to see this payoff, so nature has chosen the type players see their own see their own types then take their actions, okay, or to do some randomization randomly choose their actions the expectation over all that has been taken, so when do they get when do this player i get to see this the answer is never the reason is because to know to a certain to see this payoff you need to know the types of the entire type profile this player this payoff is never seen because no player gets to see all types rather the types of all players, okay, so this is just a notional thing this is an intermediate calculation related object, okay, so now the if this is so in short if this were the type that type profile that nature pick and this is what the players played B1 to Bn is what the players played then this would be the payoff, alright, now, so now you can average over what the nature would pick and say well take expectation of the above, so that is this, okay, so this here is the payoff is the is what this now tell me when is this payoff seen when is j of B seen, sorry, yeah, this is P of T what am I writing, yeah, so when is this payoff seen this is the payoff that is seen before the start of the game before the types get realized, right, because before the types get realized I do not know what the types are I am taking the expectation over all possible scenarios, okay, assuming players are playing these behavioral strategies one for you know for each type, okay, so this is the payoff before the types are realized this one here is the payoff to after essentially is kind of a payoff that is seen only to an observer of the game you know sees all the types, okay, alright, now there is therefore an interim payoff which is seen by each player, okay, that is this quantity which is j of B i given T i, so the interim payoff is now that I condition on what I know, so this and I take expectation over what I do not know, okay, and what is P of T minus i given T i well this is P of T divided by, okay, and clearly these two are related you can see these there is a relation between these two, yeah, and the relation is that J i of B is equal to P of T i, fine, alright, so actually so in the theory of these kind of games of incomplete information the basic question one has to ask is where are we are analyzing the game, so are we analyzing the game here as I said which is at the start of the game the way we analyze dynamic games in general and or are we analyzing the game here where players are you know kind of are aware of their type and are now optimizing this, okay, this is the cost of payoff that they are faced with and essentially this is a question of some form of delayed commitment where is that T i is T i, T minus i, T i, T minus i, right, so essentially the it is a this is a similar essentially a question of delayed commitment because we whether you whether you play here or whether you play here, okay, so one is delayed commitment, the second is the you know does it I mean you can is it so actually I should correct myself it is delayed it is a form of delayed commitment type question but that also needs to be established because does playing at this stage before the start of the game can that include in it as a special case delaying until we see this until you see your own type, right, so the broad question is is that some way by in which the are these do these two ways of looking at the game lead to do different answers, if these two lead to two different answers then we are in trouble because then it means that we need to we need to decide which of the two is really our game is this the game or is this the game, okay, so now the incredible thing actually is that this it turns out that these two are in fact you know equivalent you can you can view the game in this way and look for a Nash equilibrium in the space of bees or you can view the game in this way and again look for the Nash Nash equilibrium in the space of bees they are actually equivalent, okay, now this this here this particular Nash equilibrium a Nash equilibrium in which we are looked where players act after the knowledge of their types sorry are sort of choosing their strategies after the knowledge of their types this this this sort of thing is what is called a Bayesian Nash equilibrium, okay, whereas this a game at this level is what is called a Nash equilibrium, okay