 Welcome to this session. In the previous session we have seen take away game and discuss the optimal play of players. We also introduced the formal definition of combinatorial games. So to recall in a combinatorial game there are 2 players, a combinatorial game there are 2 players make moves alternatively is the first point. The second point is that there are finite set of possible moves. Then we also have to specify the rules specifying this rules specify how players make what are the possibilities a player has from a given position or given a once a player makes a move the next player where he can lead to all that the rules are specified movements from a position other positions. So this is a very important thing and then we also need to say that the game ends when a player cannot make a move and then the fifth one is game eventually ends. So these 5 rules specify a combinatorial game. These are the definition for a combinatorial game. Now we need to understand these 5 rules does not say who is winning or anything. So the winning in general are specified by 2 things one is called normal play the other is called major play. In a normal play the last player to make a move winners he is the winner. In major play the last player to make a move is loser. So despite the fact that seemingly they look the negation of the other but they are different we will see them later. So we also discussed about the positions. So there is something called n position the other is called p position. A position in a game in a combinatorial game is called n position if the person who is going to make a move from that position he is going to be the winner whereas in a p position the person who had made the move the previous person who made the move he is the going to be the winner. So these are mainly for win-loss games. Here we are not considering the draw option of course it is possible to include the draw option. Now let us look at the simple game and we will discuss how this can be done. Let us look at the champ game I will consider a very simple champ game. So this is the poison. Now this is the initial position. Now from here what happens is that there are multiple possibilities that can happen. So from here it can go to the following thing this is this happens if this particular thing is taken by the player. Now it is also possible that it can go to this position if this happens if the player one takes this and of course there is another possibility is this one. Now from here it leads to only one this thing this is the terminal position and from here again it goes to this and from here it can go to either this or this and so this is known as a game tree. Of course even though I have written it like this putting this one but we can actually say that the game tree has the following thing there is a position here and from here there are several possibilities and then this gives you another position of the game and from here again go and like this goes and finally there will be terminal positions and in this terminal position it may be it will say whether player one is winner. So let us say let me put it as a plus 1 minus 1 or minus 1 plus 1 or 0 0 minus 1 plus 1. What I mean to say that here the player one or the left player or the blue player is winner in this case blue player is winner. Now in this case red player is winner and in this case it is a draw. So this game tree is a very very powerful idea in fact we will explore this game tree in a later course later part of the course. So right now we will present a very important result where we use this idea of game tree. So in fact if we recall we have not said anything about the following fact. Given a game how do we know that a person can win or not so is this possible is it really possible that the person one of the player will always win. So that result is actually given by what is known as a very famous result due to Jermelo. So we will use this we will discuss this Jermelo's theorem which actually tells that in any game of whatever we have been discussing a either the first player has a winning strategy or the second player has a winning strategy or there is a draw. So what it says is that in any game WLD so player 1 has a winning strategy or player 2 has a winning strategy or the game ends in a draw. So in any game one of the 3 holds. So in any commutative game with we are taking this win, loss, add draw all 3 possibilities are there in such games only one of these 3 will hold either 1 holds or 2 or 3. So this result is a very very bold result and Jermelo's actually has discussed this problem in connection with chess. So let us the chess is a quite an interesting game in the sense that of course it looks like that chess is a finite game but is it really finite game. See for example in chess you can actually repeat certain moves and if you repeat certain moves then what happens is that the chess is not going to be a finitely terminating game. So in order to make it finite people impose rules like for example if you repeat the position thrice the game ends in a draw with those restrictions the chess becomes a finitely terminating game and then the Jermelo's theorem tells you that in chess either white player has a winning strategy or black player has a winning strategy or the game ends in a draw. So that is the consequence of the Jermelo theorem. So let us try to prove this Jermelo's theorem how we will prove this the proof of Jermelo's theorem. So the proof is based on induction yesterday we have already seen the induction so once again we apply this induction and we prove it this Jermelo's theorem how we go about the proof. So in the induction step first it is based on the depth of the game tree. So what I mean by depth of the game tree suppose you take this so if this is the starting position now here there are 3 possibilities and then from here there are 2 possibilities and then it ends to this terminal node these are known as the terminal nodes from here again these are the moves and it goes and so here these nodes are terminal nodes these are the terminal nodes. Terminal nodes is where it is whether it is a win, loss or draw that is decided let us go back to the previous thing. So in this game let us rewrite this game once again here in the simple chump game to see it. So let us look at this chump game so the one choice as I said is it is it will be like this the other choice is the other is this. Now from here from here it comes to so now this is the terminal position this is the terminal position this is a terminal position. Now let us look at it if this is the first choices are by blue and after this red makes the decision and here again blue made the decision. So let us look at it in this case. So here it is a blue is the winner and here it is the red winner and here again the red winner. So let us look at it the blue has taken this one and therefore it has come and red has taken one of this one and therefore it has come and blue is the last person to take a move and then red is forced to take the poisonous chocolate and hence the blue person is winner. Remember we are looking at a normal play here. So these are the terminal nodes. So now going back to this so as I said we are going to the use this depth of this game tree. Depth is basically the what it means is that so for example in this game the depth is here 1, 2, 3, 4 basically this entire length length of this there are three steps it is going down. So we are looking at this thing and if when there is a zero depth means we are at terminal nodes. Depth is 1 means 1 position above terminal. So like this we can we have the definition of depth. Now once the depth is there so we will use this induction. Induction is basically we use induction depth is basically is what we use and first thing is that we need to show to prove the induction to use the induction we need to the first we need to prove this if depth is 0 it automatically it means win or loss or draw is already determined. Therefore if the depth is 0 means it is already determined. Now what we do is that induction hypothesis we assume that the result the theorem is true for all game trees with depth less than n. If the depth is less than n we are assuming that the theorem holds true. Then next in the next step we need to prove the theorem for game trees with depth equals to n. So let us look at it it becomes very easy to see the proof. So now the depth is n. So that means let us assume the position is here now from here let us assume whoever the player there are certain choices that he is making how many it is whatever it is there will be some certain thing here and again here something here again something here again something here something. Let me call this as t1, t2, t3, t4 whatever tk. So what here k is basically the number of moves available from this position. Now look at this each of these sub trees t1, t2 this tk have depth less than n that is the interesting part here. Because already this position and from there they make the move and from this position we know that from this position we know that the depth is n but already we made a move. So therefore the depth has to be less than n. So that each t1 each of what essentially let me write it here each t1, tk have depth less than n. Therefore by induction hypothesis viewing this t1, tk as games on their own one of the three holds. So what do we mean by one of the three? Let us go back to the statement either player one has a winning strategy or player two has a winning strategy or game ends in a draw one of these three holds. So that means it is possible that t1 may be a win of one some player like that t2 same like this tk the same thing happens. So whatever it is what it says is that in t1 we know that let us say one of the player is winning or t2 is there. So now what the player at this position can see is that he will look at in t1 his winner or not or draw which is the best possibility. Again in t2 is he winning or not? Who is winning? See remember let us say if this position here let us say in these things let us say blue has made the move. That means if I say that t1 the game tree t1 is a winning game let us say if t1 is that is a t1 is winning position for player one implies. So this position let me call this as a star, star is winning position for player two. It is quite obvious to see this fact because remember one point which I would like to stress here is that in all these games it is not important who the player is. The important point is who the player who is making the move who is not making the move the player to make a move. So in t1 is a winning position for player one means what? The person who is going to make a move at the position t1 he is the winner. If the person who is making the move at t1 is going to winner in fact if you recall that the notation n this t1 is called n, t1 is n position. So if t1 is n position star is going to be p position. So therefore t1 is a winning position for player one that is t1 is an n position star is going to be a p position. Of course there is a small thing I do not want to use this n position and p positions here this is just to illustrate this idea only I have used but because there is a draw here when n and p positions we are using is that draw is not taken care here. So let us we will not use this n and p positions further in this proof but this is just to illustrate. If t1 is a winning position for player one star is going to be winning position for player two. So in other sense by that induction hypothesis we know this t1 can be replaced by let us say player two winning position maybe t1 could be for player two winner here it could be player one here it could be draw whatever it is. Now this blue player who is making the decision at this place so he is going to be decide which of these choices are giving him the best. So there are finitely many choices here there are finitely many choices and which one of them he is going to choose therefore if all of them are losing positions if t1 is a losing position for player one t2 is also losing position for player two t3 is also losing position for player all of them are losing positions then there is no it becomes a this blue player is going to be the loser here and whatever it is he will choose one of them. In this sense what you are what we are really saying is that the induction hypothesis says that with depth is equals to n implies the three points that may recall one is player one has winning strategy second is player two has winning strategy third is draw game ends in draw. So one of these three holds for the t1, t2, tk and hence for depth is equals to n theorem holds for depth is equals to n this now induction completes the proof of the theorem. So to summarize this theorem what we have really used is just a very simple mathematical principle of induction. What we have said is that if the depth is 0 then obviously the winners are winner winning either it is a winning position or a losing position or draw for the players that is determined automatically. If the depth is less than n we are assuming that the theorem is true and then we are showing that if the depth is equals to n then also one of these three holds. So that proof is very very simple and which we have gone through this one. So this theorem actually has introduced us the very crucial point in sense. So like yesterday we have used this take away game there it became it was little easier so how we can say whether a game has a winner or not. So we can use this theorem to say that in any comminutral game with win last draw one of these three always will hold. So that is the essence of this theorem we keep using this theorem again and again in this comminutral theory part as well as in the other parts of this course. So we end this session with this and we will continue it in the next session.