 So in the last two modules, we have given an introduction to game theory and mechanism design. So from this lecture onwards, we'll essentially try to look at the theory, the game theory part in detail and we'll develop some of the definitions that we have done earlier little informally, we'll make those things more formal. Now as we said last time, game theory is an analytical approach for predicting reasonable outcome in a game, which is a strategic interaction between multiple players. And there were several terms, the building blocks where the set of players or agents, these players had their strategies and also they had certain utilities or payoffs. Now we also made a distinction last time a little informally about between the terms action and strategy, we'll also make that a little more formal this time. However, in all these cases, we are going to assume that these players are going to be rational and intelligent and something that we have already told in the previous lecture. So the best way to start discussing about it is to look at a very specific example and we are going to illustrate these ideas of all the terms that has been used using the game of chess. And this is a result due to Von Neumann and Oscar Morgenstern, one of the founding figures in the area of game theory. So I believe all of you know what a game of chess is. You might have played or you have seen it playing. So there are two types of players. One is one player plays with the white pieces and the other player plays with the black pieces. And we're just going to use the same term white player and the black player to mean that which kind of pieces they're playing with. And each of these, there are 16 pieces for each of these players. Now, every piece has a certain legal move. So for instance, the queen has a very specific set of moves that it can do. The horse has a horse on the night has a very specific move and so on. Now, this moves we are going to call as actions at every point in time, every position of the species on this board, the chess board. They can take a very specific action and that is what we're going to call by this term action. And this is the same type of action that we have discussed in the previous example of the neighboring kingdom still. Now, the game of chess starts with the white player and then player takes turns and it can end only in three different situations. Either it can be a win for the white player, if the white player captures the black king, or it can be a black win in which case the black player captures the white king, or it can be a draw and draw in chess can happen in various ways. You can see a complete listing in the official chess games page. But here we are making things a little more informal. The nobody has a legal move, but the kings are not in a check. Both players agree to a draw or the board position is such that nobody can win and there are many more. So all those things we are going to call as draw where neither white wins or black wins. Now, what can happen in this kind of a situation? As a game theoretic point of view, we are going to ask the following questions. And all these questions are quite interesting to us. So does the white player have a winning strategy? So what is a winning strategy? A plan of actions, a sequence of actions such that it wins irrespective of the moves of B. So this is very important that whether player W wins no matter what the action has been picked by B, then we are going to call it a winning strategy for the white player. Similarly, you can define the black player's winning strategy. And the question is, does black have a winning strategy? Or do they have a strategy where they can at least guarantee a draw? Maybe none of them win, but at least they can guarantee a draw. You can contrast this with smaller games, smaller such kind of sequential games like tic-tac-toe, but you know that how you can ensure a draw. Maybe you might not win, you might not ensure a winning strategy, but you can at least guarantee a draw. Now, I would like to emphasize that there is a difference between a winning strategy. So something apart from these three outcomes, there are some other outcomes also can happen. None of this is true. So white might not have a winning strategy, B might not have a winning strategy, and they might not even have a strategy to guarantee a draw. Now, initially it might be a little puzzling how this can happen. And I want to focus on the fact that having a winning strategy does not change the outcome of the game. The outcome of the game will still be either a win for white or a win for B or a draw for each of them. But what a winning strategy is saying is that you can ensure that no matter whatever the other player is playing. So this is much more restrictive than just having an outcome. So that is what is being captured by the fourth condition, that is fourth possibility, that neither of these three things might be possible. This brings us to a more formal definition of what a strategy is. So strategy, we have earlier said that it is a mapping from the game situation or maybe the state of the game to the action. So it's a mapping. Whenever you are giving a specific game situation or a state of the game, it is going to give you a pop out one action. And this is what a strategy is. It's a complete contingency plan which tells you whenever you are in this particular state of this game, you are going to take this action. That is a strategy. So in the context of chess, let us look at what these things are. So I'm just going to distinguish two terminologies. One is a board position. If you just take a snapshot, suppose the game of chess is being played, you are just taking a picture. You are taking a snapshot of that unfinished game. Then that is the thing that you get is a board position. While game situation is something like tracing the history. So when you take the snapshot, you just get that snapshot. You do not have any information how you actually reach that state. There might be multiple ways how you can play multiple histories which will also lead to the same board position. But they will all have a different game situation. Game situation is essentially the history. So let's make it a little more formal with our notation. So we are going to denote a board position by this notation xk. So maybe after k stages, you have reached this particular board position, which is nothing but a snapshot. The positions of each of these pieces, which has not been eliminated on this board, and that is going to be a board position. And game situation, therefore, is the sequence of this board position starting from the initial. So x0 is the first board position, which is always the same in this case. But then based on whichever action has been chosen by each of these players, first white chooses an action and it goes to a board position x1. Then black chooses another action which leads to the board position of x2 and so on. Finally, if k is even, then you know that it's a turn for white to play. And if it is odd, then you know it's an action. The next action will be taken by b, the black player. So to represent this pictorially, you can think of this as a tree. And we are also going to call this a game tree. So different board positions are just the nodes of this tree. And you can imagine that there might be, so because the same board position can be obtained in multiple ways, there could be reputations of nodes in this game tree. So x0 is the initial board position and because of the rules or the norms of chess, the first move will be made by the white player. So this different ages are showing what are the feasible moves that white can give in the initial board position. And this tree is showing that if white player is picking this particular action, then it will lead to this board position which is x1. Similarly, after that, suppose player b makes this move, suppose it makes this move, then it will go to the next board position which is let's say x2. And as we said, there could be some other places. For instance, x2 might also appear somewhere else in this game tree in a different way, different sequence of actions. They might end up having the same board position, but the game situation will be different because that will actually trace the whole path starting from the root to that node. So in this context, so when the board position is x2, this particular x2 has been reached by this particular history. This particular game situation x0, x1 to x2, that is the game situation. While there might be some other x2, there might be some other node which also has the same board position, but that might have been traced by a different game situation. So now let us look at what strategy is. So strategy is the mapping from the game situation. So because game situation itself is telling us the entire game history to a specific action. So if a specific game situation has been reached, which is a unique node in this game tree, because every node in this game tree is a unique game situation, the strategy will map that to an action. So if this particular game situation has been reached, what will be the action of the player that is going to play here? That is going to be the strategy. And maybe you can think of this as a complete contingency plan for each of these players. You might be wondering how does it matter? How does it differ in any way? If I actually end up in a specific board position, why should my strategy not be the same? The answer is yes. For games like chess, it might be easier to look at just the board position and not the game situation. But this is a more general description because there could be certain games. So for instance, games of card where you cannot really observe each of the actions that has been played by each of the player because you are in an uncertain situation. You might take a look at your own cards. You might also see the cards that has been played so far, but you cannot see the other player's hands. So there is always an uncertainty. So based on what actions you have already seen, you might wonder that if this player has played this many things, you might infer that this might not be possible and therefore the entire history of the game is important to make a choice of action. And that is where this game situation, the strategy depending on the game situation becomes more important, not just the current state of the game. Therefore, we are going to consider the mapping from this game situation to the set of actions. A strategy for this white player, we are going to denote that as a function sw that associates every game situation x0 to xk, which sometimes also called the history of the game. And if the k is even, with the board position xk plus 1 such that this move xk plus 1 to xk to xk plus 1 is a single valid move of the white player. So if you want to write this out, so s of w takes as input a specific history. So let's say x0 to xk. This is the history of the game situation and it just pops out one action which takes this xk to xk plus 1. So this is the meaning of the strategy. Okay, so and similarly you can define a very similar function for the strategy of player b. Now, once you have decided what is the sw, so sw is giving you a mapping from each history to a specific action which is feasible for that player. Once you have picked a pair of a strategy pair like sw and sb, you are certain that you can reach an outcome because it gives you So what player will do is already being given by sw at every stage of the game. Similarly, what black will do is given by sw at every stage of the game. So the moment you are giving this sw and sb, it will give you one path through this game tree and you will end at an outcome. Now this outcome is also sometimes called the one play of the game. So you are going via this game tree. So game tree is enumerating all possible paths in this game. But once you have chosen your sw and sb, you are going to trace out one particular path. So this game tree and we'll end up in a leaf node, which is just the end of that game. Now the question that might be interesting in this case because this is a finite game. Where does it end? Does it end in a win for white or win for black in a draw? That should be an interesting question to ask. And can a particular player guarantee an outcome? So for instance, can the white players say that I have a strategy through which no matter what the other player is playing, I am always going to win. Can they ensure such kind of stuff? Now I just want to remind again that the game ends only in three possible outcomes. So all the leaf nodes in this finite game tree will have one of these three possibilities. Either white wins or the black wins or it ends in a draw. Now winning strategy, so because we are making it formal, the winning strategy for W is a specific strategy. Let's say ASW star, such that for every ASB, mind this word, for every ASB, this strategy profile ASW star, ASB ends in a win for the white player. This is what we mean by a winning strategy. And if you want to consider a strategy which guarantees at least a draw for white player, then that is going to be defined as ASW prime. Such that for every ASB, again for every ASB, no matter what the other player is playing, ASW prime, ASB either ends in a draw or a win for W. I mean, both of these two things can happen based on what ASB you have chosen. So it may be possible that the strategy that you have chosen might end up in a draw if the other player chooses a very specific ASB, which is very competitive. But if it is a foolish strategy by player B, then in the same strategy, you might end up in a win. But analogous definition, you can also do for player B, winning strategy for player B or draw guaranteeing strategy for player B. Now it's not obvious as we have discussed that such strategies exist, whether there exists some kind of a thing, we do not know. So there comes one of the most classic results, one of the very early results due to von Neumann in 1928. It says that in chess, it essentially demystifies this question that I have just asked, one and only one of the following statement is true. And what are these statements? Either W has a winning strategy or Black has a winning strategy or each player has a strategy guaranteeing at least a draw. So it says that one of these three things are possible and exactly one of these three things is possible. And that fourth option that we have just said in the beginning that maybe none of this is true is not true. So exactly one of these things will be possible. Now what it does not say is that which one of these three is true. So there is a slight amount of open space still. It does not say which one of these three things is true and also it does not say which that strategy is. It just says that it has a winning strategy or Black has a winning strategy, but it does not say which strategy is that. So if that was known, for instance, if you look at very simple games like tic-tac-toe, you exactly know that this third thing is true. In fact, you can argue in a very similar way for the game of tic-tac-toe. And you can show that there exists a draw guaranteeing strategy by both players and you exactly know what that draw guaranteeing strategy is. That is just because the game of tic-tac-toe is not very large. If you exhaustively list down the game tree, that will not be very large. And if we knew which one of these three things is true and if we knew what is that strategy, then chess would have been as boring as the other tic-tac-toe.