 Welcome back to the NPTEL course on game theory. So, in the previous sessions we have seen combinatorial games. Now we will see the classical game theory which is developed by Von Neumann. So, in this subject the game theory is what exactly is game theory? We have already seen in the combinatorial games that there are two players who make decisions alternatively. In this classical game theory which is sometimes also called economic games, in this the two players instead of choosing their actions alternatively they choose simultaneously. Of course, in this classical game theory the games where the players choose their decisions or actions alternatively they are known as extensive form games. We will come back to that at a later stage. As I said in this games the players make decisions alternatively. So, to start with we will consider the following example which is known as matching pennies. There are two players and both the players have two strategies H and T. Now if both the players choose same strategy then player one gets a unit of money from player two. If they differ then player one has to pay to player two. So, let me write it as a pictorial representation. So, there is a player one here and player two and player one has two choices H and T and player two also has two choices H and T. So, as I said if the both the choices are same the player one gets a unit and player two has to pay to player one. So, that is the meaning of this 1 minus 1. Similarly, if the when both players choose T and T it is the same thing player one gets one unit player two has to pay one unit to player one. And if their choices are different like player one chooses T and player two chooses H player one has to pay to player two therefore his utility will be minus 1 and player two is getting the money one unit. So, one here. Similarly, if player one chooses H and player two chooses T it is the same thing player one has to pay one unit to player two. So, this is a simple example of a two player zero sum game. So, let me introduce the terminology. The two players is mainly because there are only two players here. Why is this zero sum? Now, look at it player one if player one receives one unit then player two is paying that much to player one. So, the sum of these two is always zero. You look at it. So, this sum is zero this sum is also zero this sum is also zero this is also zero. So, therefore, this is known as zero sum game. So, in fact the way this game is played by children is the following thing. They have two coins separately individually and both of them toss the coin and if the outcomes match then player one gets one unit from player two and if the outcomes do not match the player two gets one unit from player one. So, this is the matching pennies example. So, now let us go to the next game which is known as a rock paper scissors. Again, it is a two player game and there are three strategies now three choices. So, the choices are rock paper scissors and once again we draw a pictorial representation. So, the player one player two player two has three choices RPS player one also has three choices RPS. So, the rules are the following thing. So, the interpretation of this game helps rock can be covered by paper paper can be cut by scissors scissors can be broken by rock that is the intuition behind this game. What is mean is that if a player one chooses rock player two chooses paper because player two can hide this rock. So, therefore player two gets one unit from player one. So, let me start when player one chooses rock player two chooses paper. So, player two gets one unit and player one gets minus one. Similarly, if player one chooses rock player two chooses scissors the scissors can be broken by rock. So, therefore player one gets one unit and player two has to pay one unit to player one. Similarly, like paper and scissors rock player one chooses paper and player two chooses rock then player one gets one unit and player two has to pay that one unit to player one paper and scissors. So, scissors can be used to cut the paper. So, minus one and one here when scissors and rock already discussed about this. So, minus one and one and when player one chooses scissors player to chooses paper then it will be one minus one. Now, what remains here is the diagonal entries where both players choose the same thing in which case we assume that both of them are getting nothing. So, if we really look at it this game is once again a zero sum game and stoop player game. So, now let us look at another game. So, this is known as a coordination game there are again two players. So, actually the story helps in understanding this game. So, there are two friends let us say assume there are several ways to tell this and one friend like going to movies the other friend likes to go to watch sports. So, therefore there are two choices for them. So, for the sake of simplicity I will only choice one choice two only I will put. So, once again draw the pictures the choice one choice two. So, let us say C1 is basically movie C2 is sports. So, player one let us say he prefers movie and player two prefers sports. If player one goes to a movie and player two goes to sports that means they are not going together to a place. So, there in which case they will not get any utility. So, here it will be 0 0 and similarly in this place also it is 0 0. Now, let us look at the this entry here in this entry both the players are choosing to go to movies. We know that the player one prefers movie whereas player two prefers sports. So, player one is getting maximum benefit. But at the same time player two is also with player one. So, therefore he also gets some utility may not be the greatest. So, player one gets four and player two gets two. Now, if both of them decide to go to C2 that means that is the player two's best choice. So, in which case player two gets the maximum benefit and player one also gets some benefit. So, it will be 2 4. So, now you look at it the sum of these entries is not equals to 0. Of course, this sum of these two are 0. So, this is known as non-zero sum game. So, once we see these games we can there are several examples which we will discuss throughout this course. Now, let us understand this games more formally. So, how do we define it formally? There are two players and player one has a choice set. He has some set of choices. Let me write it as x and player two also has some set of choices. Let me call this as y. Now, player one has utility. Let me denote it by pi 1 which will be a function from x cross y to r. And similarly player 2 has a utility pi 2 which is a function again from x cross y to r. So, this is the question here is what are their optimal choices. So, this is the basic question that game theory tries to understand. And of course, this is a basic and also the first question. So, here we need to understand several terms. The most important thing here is optimal. What do you mean by optimal? What is an optimal choice? Before going into this discussion about optimal, let us understand the previous examples what is this x and y. So, let us go back to previous examples. So, here in this example x is same as y which is same as h t. Then pi 1 h h is 1 which is same as pi 1 t t. Similarly, pi 2 is nothing pi 2 of sorry pi 1 of h comma t is minus 1 which is same as pi 1 of t comma h. And pi 2 is nothing but minus of pi 1. So, this is exactly in the same way as we introduced. Now, let us look at the next example. Here there are 3 choices. This is basically x which is also same as y and pi 1 r r is 0 pi 2 r r is 0. So, we can write it pi 1 r r is 0 which is same as pi 1 s s which is same as pi 1 t t. Then pi 1 r s is 1 which is also same as pi 1 p r which is also same as pi 1 s p. Like wise we can write down the other things. So, now if you go back to and of course in this example once again pi 2 is same as minus pi 1. In the next example there are 2 choices x is equals to y is equals to c 1 c 2 and pi 1 c 1 c 1 is 4 pi 1 c 2 c 2 is 2 and pi 1 c 1 c 2 pi 1 c 2 c 1 is 0. Similarly, we can write pi 2. And here as I said pi 2 is certainly not equals to minus pi 1. That is the reason in other words pi 1 plus pi 2 is not equals to 0. This is the reason why it is called non-zero sum. So, now coming back to the definition as I said there are 2 players. The player 1 has a set of choices x, player 2 has a set of choices y, player 1's utility is given by this function and player 2's utility is given by this function. And because I am saying utility therefore each player's object is to maximize their utility. So, both the players have to make maximize their utility. How do they do it? So, we need to understand how they will do it. So, first we need to understand they are different from optimization problems. In an optimization problem there is a decision maker there will be a single player typically is called as a decision maker. He has some set of choices available and he has to choose the best choice among them. And accordingly there is a utility function. If it is a maximization problem it you will have an utility if it is a minimization problem there is a cost. So, so far we are writing everything in as a utility maximization problems. We can also write it as a minimization. So, that is a standard way we can look at the minimization. Now, look at this interesting thing here. The player 1 he cannot just simply optimize over his choices. The reason is his utility depends on the choices made by the second player. In the same way the first second player cannot simply maximize over his choices because his pay of utility function depends on the first player's choices. So, the game theory is basically deals with such situations where there are multiple people and each person has their own utility and their utility depends not only on their choice, but on the choices of other players. So, here even though we have mentioned about a two player we can actually extend this notation to multiple players. So, let us look at that. So, there are n players and the players i's choice set let me call it as a xi and player i's pay of is given by pi i which is a function of x 1 cross x 2 cross x n to r. So, of course i becomes 1 to n any of this. So, now here if you look at it there are n players and each player has their own set of choices and their own payoff functions and each player's object to maximize pi i. As I said the player's objective depends on the choices of others. So, how do they maximize? Requires to introduce notion of equilibrium. So, the games what exactly it means is we need to introduce this notion of equilibrium and what is this equilibrium? So, that forms the basic question what is equilibrium? So, before going further let us understand few things. So, the few points which I have not said earlier let us look at it. So, how are they making decisions? If you recall in a communitarian games for example, tic-tac-toe how this decision making process is there? So, player 1 first makes his decision after observing player 1's decision player 2 makes his decision and it goes on alternatively and here it is completely different. Look at recall this matching pennies. So, there are two players there are two choices. Now, if player 1 makes let us say h then if player 2 somehow knows that player 1 has made h then player 2 will certainly choose this t and then there is no game here it is very simple. So, this is not what we want here in this in these games players do not know what others have done this is important. In other word what it means is that players make their decisions that means player 1 makes his decision and player 2 simultaneously he makes his decision without knowing what the other player is making. So, this is the very important this thing. So, they make their decisions simultaneously. So, they make their decisions simultaneously and they do not know what the other person have done and they will not let their decision known to others. So, this example itself will tell you. So, if player 1 lets his decision known to player 2 before he makes his decision then player 1 is losing it and he does not want to do that. So, what that means is rationality what means players are selfish what it means that every player wants to make the best no player would like to lose. So, whatever decision a player makes he makes in such a way that his utility is maximized irrespective of what the other player is doing it. So, the selfishness or in the language of economics it is called rationality. So, this is a very important concept then there is another thing which is known as intelligent. The players are intelligent enough to understand what decision they should make they are and most importantly we assume that both players are equally intelligent and rational. That means whatever player 1 can think about this game player 2 can also think about this same way and they also know that the player 1 is let us say player 1 knows that player 2 is rational and intelligent. And player 2 also knows that player 1 is rational and intelligent and not only that one. Player 1 knows that that player 2 knows that player 1 is rational. Like this you can think about an information aspects and we assume that all these informations are known to them. Everyone knows that the other player, other players are equally intelligent, rational and everything. So this is a very very important aspect here. Now once this independent, this intelligent and rationality is there and in the previous thing we said they are making their decisions simultaneously, independent of the other players. So now that the basic setup of this games are. Now we still need to introduce this notion of equilibrium. So what is the equilibrium notion? So let us look at the following thing. Now let us say player 1, somehow player 1 knows that he is playing let us say y in y. We are looking at the 2 player game and let us assume that somehow player 1 knows that player 2 is playing y in x. What will player 1 do? Because we have assumed that he is intelligent he will choose x which maximizes his utility, his utility is given by pi 1 x, y over x in x. So for a fixed y, if I know the player 2 has fixed y then player 1 will choose x which maximizes this quantity. That means he solves x in x pi 1 x, y. And whatever maximizes this quantity that he will choose. Let us look at the similar thing. Suppose player 2 knows that player 1 is playing x. Then like previous player 1 to solves max over y in y of pi 2 x, y. Whatever y maximizes this quantity player 2 is going to play that one. So this gives the following notion. This introduces the following notion. A pair x star, y star. A pair of player 1's choice here and player 2's choice y star such that given that given y star x star maximizes pi 1 x, y star. Similarly given x star y star maximizes pi 2 x star y. So once player 1, player 2 fixes his strategy y star, player 1's choice x star should maximizes this. And similarly if player 1's strategy is fixed here then player 2 strategy y star should maximizes this quantity. So this is basically the equilibrium notion. In fact this is known as Nash equilibrium. So we will discuss more about this equilibrium concept and several examples in the next session.