 Welcome to this NPTEL course on game theory. Game theory is a mathematical discipline which models the conflicting behavior. The subject originated during the world war and the polymathematician Von Neumann is considered to be the founder of this subject. The subject has some earlier results by M. L. Borel and other people, but the Von Neumann is considered as the main person who created this branch of subject. Their Von Neumann's book with economist Morgenstern, the games and economic behavior is the first book on the subject which laid the foundations. So in this subject there are two or multiple people who make their decisions simultaneously and accordingly the each player gets a benefit or payoff. Now the objective is to decide how to choose their decision depending on other choices as well. Of course the major problem here is that when a player is making his decision he does not know what the other players are making. So these games are known as economic games, classical games and etc. There is another class of games known as combinatorial games. The combinatorial games are the games that we have played as a kid. The examples are tic-tac-toe, chess and other games like that. These games are this, the theory of community such games is developed mainly by John Conway, Richard Guy and Erwin Balkam. In fact they have written a very famous book called Winning Ways. Right now it has 4 volumes. As mentioned in the introductory video of this course, this course is divided into 4 parts. The first part is about combinatorial games. In second and third we will look at non-cooperative games, zero sum and non-zero sum game and in the final part we will look at the cooperative games. Now we will concentrate on combinatorial games. We will walk through some examples. Through these examples we will introduce some of the terminology here. We start with tic-tac-toe. Tic-tac-toe is a game played by 2 players on a 3 by 3 grid. Players make alternative moves. So starts with first players who makes or marks a row or column or diagonal is the winner. So let us see what it means. Suppose let us say the player 1, let us say places a mark here. Then player 2, let us say he chooses a mark here. Then player 1 again makes his move, let us say he makes his move here. Then player 2, if he does not place a mark here, player 1 is going to make a 3 in this, along this diagonal. Therefore player 1 will, so he will try to place his marker here. Then player 1 will be forced to play here. Then player 2 will play here. In the next player 1 whatever, wherever he places, so let us say he places here. Then player 2 will be forced again here. And then player 1 finally left with this mark. Now in this instance no player could make 3 marks in a row. So therefore this game has ended in a draw. But let us look at another instance. Let us say player 1 has started making mark here. Then let us say player 2 has made mark here. Then player 1 let us say has put here. Then let us say player 2 has put here. Now of course obviously player 1 is forced to play here. And in the next player 2 let us say he makes a mark here. But then player 1 puts here and he wins. So in this setup player 1 wins. So this is a tic-tac-tac-toe game. Of course in this game we also have an option of draw. But in most common combinatorial games we always assume that the game ends either in a win or a loss. But the tic-tac-toe since this is well known to us we have start use this to illustrate the ideas of combinatorial games. Even though in a classical sense tic-tac-toe is not a combinatorial game. But the arguments that we develop in combinatorial games are used to analyze such games. Now we will see another game. So this is called a dominating game. The dominating game consists of a square cell where the players again make moves alternately. Player 1 chooses two consecutive cells horizontally. Player 2 chooses two cells of course consecutively and vertically. So in a sense player 1 is going to choose two cells like this and player 2 will choose something like this. Now when they do like this the alternatively start choosing it whoever makes their last move he is going to be the winner. So let us say already the player 1 has chosen this and player 2 has chosen this. Now let us say the player 1 has chosen this two consecutive cells then let us assume player 2 now has chosen this. Player 1 now can choose then player 2 let us choose this. Then player 1 is let us say he goes to choose this. Then player 2 let us say he chooses this. Then player 1 now goes back to this he chooses this. Then player 2 has these are all free for him. So let us say he chooses this. Then player 1 is going to choose this then again player 2 can choose let us say this. Now player 1 has one choice here that is it. So therefore he has only one choice here he will choose this then player 2 can choose this. Now player 1 has no move here. So therefore player 2 is the winner. Player 2 is winner in this setup. So this is a dominating game. Note that in this instance of dominating game player 1 has no more moves to make. Of course player 2 is the person to make the last move and hence player 2 is the winner. So typically in this combinatorial games the last person to make the move is the winner. There is another version where the last person to make a move is going to be the loser which are known as miser games. But the common version is the normal play where the last person to make the move is winner. So let us explain another game which is known as chomp. So this has again kind of rectangular cell and each cell is considered as a chocolate here and this cell consists of poison. All the cells are considered as a poison. So whenever a player as usual the players make moves they alternatively if a player makes let us say a move here let us say player 1 has decided to take here then he not only takes this particular cell he also takes the all the cells above and right to it. So that means this all these cells he will choose. Next player 2 will choose another cell let us say player 2 has chosen this. Then he will choose all the things above and right to this. Then next player 1 let us say he chooses this then he will choose all this. Then player 2 let us he will choose this. So here in this setup player 2 is winner. So this chomp is another game where the in this instance player 2 is winner. So now we will look at another game this game is known as hex. There are 2 players here one is a blue player and another is let us say red player letter. So the goal of this game is to make a path from one side to the another side. So the blue player wishes to mark these hexagonal cells such that he makes a path from this side to this side. And the red player would like to make a mark from this side to this side and they will be alternatively making their moves. So in each move the players will pick one of this hexagon cell and mark them with their respective color. So let us play this game. So let us say blue player starts here let us say he makes it blue then the red player comes. So let us say red player next blue player again then the red player next blue player let us say he will put this here and the red player now chooses let us say here next blue player will come and he will mark here and the next red player will do here and then the blue player comes here and now we have a path from the blue path from here here here this. So in this instance the blue player won this game. Next we will see one more game which is known as a take away game. So in this take away game there are some set of coins right now we will put it as a sticks here. So let us say let us say ten sticks again two players alternatively make the moves and then the in each move players can pick one or two or three sticks again last player to pick his winner. So there are certain number of sticks in this picture we have ten sticks and the players are making their moves alternatively and each player can pick either one stick or two sticks or three sticks. Now the last player to pick is a winner is the winner here. So now let us look at this one let us say the first player let us say he picks three first player. Now let us say second player in his setup let us say he picked two then the first player will pick only one then the second player whatever he picks here and in the last player will pick the last stick here. So this is a game where in this instant the blue player has won. Now the whole idea of this communal game is to understand such games and see whether a player has a winning strategy or not particular who has a winning strategy. For example in this game can we say decisively say that the player one always has a winning strategy or is it true that the player to can enforce the win. So in the following sessions we will discuss about some of these games and understand their winning strategies but before concluding this session would like to point out one very interesting application of these communal games. So we have seen in the previous game this hex game. Using this hex game we can actually prove what is known as a Brauer fixed point theorem. So I will state what Brauer fixed point theorem is and then we will come back to this later. So let us take a function f from let us say interval 01 to interval 01. We assume that this is a continuous function then a point x in 01 is called a fixed point of f if ff is equals to x. Now the question is does there exist a fixed point. This is a question where Brauer has proved that let us say kb convex and compact set. Then every continuous f from k into k has a fixed point. So this is a very very deep theorem and in this course we will see how this hex game can be used to prove this fixed point theorem. But before concluding this session let me just give you a hint about proving one dimensional case. So f is from 01 to 01. Consider g of x to be fx minus x. Now note that g of 0 is greater than equals to 0, g of 1 is negative. Therefore there exists x in 01 such that g of x is equals to 0 which implies fx is equals to x. Of course small issues one needs to check. Here the main thing is that how am I guaranteeing this existence of 0. This I will leave it to you as an exercise. So this is the kind of mathematics that we require to follow this course and we will assume that level of mathematics. Barring certain results in this course we will prove almost all and all results and we will not leave any theorem unproved. With this I will conclude this session. We will meet again in the next session.