 games through certain examples, their players, actions, strategies, etc, it's the right time to make it a little more formal and developing the notation and the notions that we will be using in the rest of this course. So the first thing that we are going to discuss is about normal form representation of games. Sometimes it is also called the strategic form games. These two terms are used interchangeably in game theory literature and also in books. So strategic form or normal form games are most appropriate to represent games which are of one round. So the moment you pick your actions, the outcomes are realized and you get your utilities. So one example of such one shot game was the neighboring kingdom's dilemma that we have discussed in the very first lecture. So let's say we have a bunch of players which we are going to denote by this set N. This set N is enumerating all the identities of these players and here for simplicity we are using the numbers 1 to N to denote the identities of these players. So in all of this module and later on we will typically denote N as the set of players. Now each of these players, so if you pick one specific player I, what are the possible strategies that it can have? So this set of strategies is denoted by this set capital S I and one specific strategy living in this set of strategies is denoted by lowercase S I. So going back to our definition, our example of neighboring kingdom's dilemma, this set S where either S I was either agriculture or war for both the players and S I was either picking one of this strategy either A or W whichever this player was picking. Alright, so now the Cartesian product of all the strategy sets of all the players is what we are going to call the set of strategy profiles. Now before that we will also have to define what is a strategy profile. Strategy profile is nothing but a member of this set, the Cartesian product which means that when each of these players have picked certain strategies. So for instance player one has picked the strategy S I, player two has picked strategy S II and so on player N has picked the strategy SN then this the vector of all the strategies is called one strategy profile and that should live in this set capital S. In game theory we will also use a very special notation called this S subscript minus I which means that we are enumerating the same strategy profile as S but excluding agent I. So here we have all the strategies starting from S I to S I minus one and then S I plus one to the rest of the players. So I am just removing agent I strategy and looking at the strategies strategy profile of all the other players and that will be denoted by this notation S minus I. Okay, so why does that help? Then we can actually write the whole strategy profile S in a much more shorter notation which is given by S I which is the strategy of player I and also S minus I, so strategy of all the other players and we will be using this notation very often in our in our course. Now what is a we have also defined the the utilities or the payoffs because these players are rational we will have to define what are their what objective that they are trying to maximize and this objective is given by this utility function which is given by E1. Now once each of these players have chosen a specific strategy and therefore we have a strategy profile this utility function for player I is mapping that strategy profile to a real number. So imagine in the in the previous case we had so let's say player the kingdom A was the player so let's say we are looking at the utility of of that player A and if the if both these players are choosing let's say war comma war then we have defined the corresponding utility for player A. So suppose I think it was one in the in the previous example. Similarly you can define for all possible strategic profiles how this utility function is defined you can define it for all of them. So this is essentially the utility function for player I. Now in normal form or normal form game abbreviated as NFG the representation is succinctly given by these three tuples the tuple of three things the first one is the set of players the set N their strategies the strategy sets and their utilities these three things are essentially completely defining a normal form game and that is what we are going to use in the rest of the course. Now we are also going to assume that all the strategies strategy sets are finite for every player and therefore we are only considering finite games. We'll make certain comments about infinite games as well but we will see that for finite games we can actually show certain desirable properties which does not hold necessarily for infinite games. Okay so let us understand each of this notation that we have developed so far using a very specific example. So let's say we look at a game called the penalty shootout game. So this is referring to the game of football there is a so suppose I suppose that you know what a penalty shoot is there is a shooter who is shooting at the goal and there is a goalkeeper who tries to save it and there are three possible actions or in this case the strategy and action are the same because there is only one state of the game so they have this possible actions either shoot to the left shoot to the center or shoot to the right and similarly goalkeeper can also dive on the left stay at the center or dive on the right. So if both the players and the goalkeeper choose the same thing then it's a it's a loss for the for the shooter because then the goalkeeper will be we are assuming that the goalkeeper if the shooter also shoots to the left and goalkeeper also jumps to the left then the goalkeeper is going to save that and that will be a negative payoff for the shooter and possibly payoff for the goalkeeper. So in all the all the diagonal entries of this matrix you can see it is minus one comma one that is the shooter is getting a negative payoff and the goalkeeper is getting that positive payoff and for all other cases non-diagonal elements because the shooter is shooting at a direction where the goalkeeper is not jumping at therefore it is it is going to be scored the goal is going to be scored and shooter will get a positive payoff and the goalkeeper will get a negative payoff. So what is the the representation of this game in the normal form by the notation that we have developed? Clearly there are only two players the shooter and the goalkeeper so we are just defining let's say we are numbering them with one and two maybe shooter is player one and goalkeeper is player two. The strategy set in both this for both these players S1 and S2 are the same they comprise of these three entities LC and R left center and right those are the possible actions or strategies that they can take. Now if you look at a very specific player so let's say look at player one and a strategy profile L comma L so both these players are picking strategy L comma L then the utility for player one is going to be minus one similarly for the same player when the strategy profile is L comma C then it is getting a positive payoff one positive payoff of one and when the strategy profile is L comma R then then also it is getting a strategy then also getting at the utility of one for the same so this is essentially denoting these three numbers. Similarly you can write down the utility for player two so for the same set of strategy profiles L, L, L, L, C and L, R they will be these three numbers respectively right and similarly for you can fill the rest of the things you can write the same things UIs for C comma L and so on C comma C, C comma R and the numbers will be as given in this matrix. Alright so we know exactly how to represent this game in the normal form and when we are going to analyze we already said that we are going to make two assumptions we are going to assume that players are rational and intelligent and rationality just means that they are they are picking each of these agents are going to pick actions or strategies that maximize their utility the utilities as given in this matrix. Intelligence is a as we have seen already it is a it is a little circular in its definition we call a player to be intelligent if she knows the rules of the game perfectly and picks the action considering that there are other rational and intelligent players in the game so what are the implications of that we will see as we discuss more examples in particular an intelligent player will think and pick actions like a game theorist and that is one of the implications of these two assumptions. Now one very important property that we are going to use quite often not not only for for game theory but several other applications where you are dealing with information percolation the this particular notion of common knowledge will be useful in all of those contexts so what do we mean by a common knowledge a fact it is defined in the following way a fact is a common knowledge if all the players know this fact all players know that all the other players are also knowing knows this fact all other players know this fact all players know that all other players know that all other players know this fact and so on you can just keep on repeating this as many times and that will mean that will make a fact to be a common knowledge now what is the implication even though the definition sounds a little funny we'll soon see an example where we'll figure out that this there is a very profound implication of this notion of common knowledge let us look at that example so here is one very specific example if this is an example of an isolated island where there exist three blue white people so eye color can either be blue or black let's assume that and these three people all of them have blue eyes and there is suppose there does not exist any reflecting medium in this island so that they can see their eye colors and also they do not talk about their eye colors so what that means is that they can look at the eye colors of the other people other players but they cannot see their own eye color now suppose one day a sage comes to this island and says makes the following statement it says that the sage says that blue white people are bad for this island and they must leave and there is at least one blue white person in this island so this is the end of that statement now we are going to assume that this sage is a person who cannot be disputed this person is like a oracle whatever he says is his truth so if someone realizes that his or her eye color is blue on this island then he or she just leaves the island at the end of the day so that is the setting let us assume that now maybe you can just take a minute pause the video and think about it what are the implications of this statement whether the these people will leave immediately whether they will wait for a few days to understand what their eye color is or whether they will never be able to find out their own eye colors because there is no reflecting media they don't talk about each other they can only see other side colors but cannot see their own eye color so let me answer this question in step in steps so suppose I mean how does this information of common knowledge percolate when the sage made this remark so if there was only one blue white person then what would have happened you can clearly understand that that the moment sage makes this statement on the first day itself the person who has blue eyes he looks at the other two person's eyes and finds that those eye colors were black and because sage cannot be disputed then that person must be the only person who has blue eyes and therefore at the end of the day that person should leave and looking at the fact that that person has actually left after the after day one the other people will not leave because now they know that he was the only person who had blue eyes and if if he was not the only person he wouldn't he would not have understood that he has the blue eyes on the end of day one now let's look at the the next level if there were two people who had blue eyes so on the first day what would have happened these two blue eyed people will see that there exists one blue eyed and one black eyed person so he could have still thought that this this person the other blue eyed person is the only blue eyed person in this island and maybe at the end of day one he must leave the fact that he also looks in the same way and sees that there is one blue and one black eyed person and he also thinks in argues in the same way he will wait until the second day so the fact that after day one none of this blue eyed person has actually left that makes both of this blue eyed person understand that both of their eye colors are blue and at end of day two then both these people should leave and looking at the fact that both of these people have actually left after day two the black eyed person will not leave because he sees that there are two blue eyed person if he was a blue eyed person then this two people would not have left on day two all right so now we can actually see I think you you have all already started to see the answer that if there were three people all of them had blue eyed all of them were blue eyed then all of them see that there are two other people so maybe they think that his eye color is black so they will wait until day two and they will see whether this two blue eyed person actually leaves the island on on end of day two if they don't in leave then he will be he will confirm that his eye color is also blue then he will also all these three people will actually leave the island simultaneously on day three so that's the implication of of common knowledge it's a very interesting idea the definition is slightly funny but but that is that is how it is so one of the assumptions that we are going to make in the context of game theory is that the fact that all the players are rational and intelligent it's a common knowledge so you are not going to assume that the other players are not going to maximize their utilities or they are not understanding enough the rules of the game and cannot think about the game in the way a game says things about it rather they will actually assume that they know it the rationality and intelligence is is known to them known to every other people and they also know that the all the other players are also rational and intelligent and so on.