 We have discussed the computational complexity of several equilibrium concepts in the past. In particular, we have talked about the computational complexity of mixed strategy Nash equilibrium. And then we have seen that for correlated equilibrium, it is much easier. So, let us look at what is the complexity for finding a sub game perfect Nash equilibrium. So, here is the same game from the previous module and also the algorithm, the backward induction algorithm that finds once a sub game perfect Nash equilibrium for a for any PI EFG. This algorithm is nothing but a dynamic programming and the idea of a sub game perfection is essentially based on this idea of backward induction. So, if you think about it carefully, you are trying to find the sub game perfect Nash equilibrium, which is a Nash equilibrium at every possible sub game. So, therefore, the most natural thing is this backward induction which starts from the leaf nodes and finds the equilibrium at every sub game and goes towards the root. Now, there are several advantages for this kind of games, this perfect information extensive form games. The first thing that we observe is that sub game perfect Nash equilibrium is guaranteed to exist in finite PI EFGs. And you can argue it formally, but the intuition is that it is basically based on the fact that you have an algorithmic way of showing that the sub game perfect Nash equilibrium exists. And the algorithmic way is the backward induction algorithm. And you can show that this will always converge to a sub game perfect Nash equilibrium. Therefore, every finite PI EFG will have a have an SPNE. We have also seen that the sub game perfect Nash equilibrium is a pure strategy Nash equilibrium for the entire game. Because it is a Nash equilibrium at every sub game of this game, it is definitely going to be a PSNE. So, we know that the general games, normal form games may not have a pure strategy Nash equilibrium, but this extensive perfect information extensive form games are one class of games where pure strategy Nash equilibrium is guaranteed to exist. There are several other kinds of games where pure strategy Nash equilibrium exists for sure, this PI EFG is one of them. And the third advantage is this algorithm of backward induction to find SPNE is very simple. I mean, it is the most intuitive thing that you can think of. But the trouble here the disadvantage is that to find the SPNE will have to pass the whole tree. And for realistic games, the tree can really be very, very large. For instance, chess has a game tree which has 10 to the power 150 vertices. And this is a huge huge number. It is needless to say that this is more than the total number of molecules in the world. So, even if you have a very fast supercomputer, you would not be able to compute the sub game perfect Nash equilibrium for something, some game like chess. If it was possible by some means maybe some other kind of computational machinery which could find parts over all these 10 to the power 150 vertices and do the backward induction and find out the optimal strategy, then we would have solved the one of the very early questions that we have asked in this course, which is about the result by von Neumann, that which of these three statements for chess is true. So, we know that there exists a winning strategy for white, or there exists a winning strategy for black, or there exists a draw guaranteeing strategy for both these players. And one and exactly one of these three statements is true. But the that was not which was not answer is that we do not know which one them which of them is true. And if there exists some winning strategy or draw guaranteeing strategy, which strategy is that if we could find the SPNE, then we could have a conclusively given the answer of that. So, for small games like tic tac toe, we can actually list down. It does not have too many vertices in the game tree. We can do that and find the SPNE. And it will be a good exercise to do and find out that this SPNE is nothing but the draw guaranteeing strategy for both these players. So, we know that the there exists a strategy which guarantees the draw for both these players and that will that will emerge as the SPNE of this of this BIEFG. SPNE also has some other criticisms and these are mostly about the behavioral aspects. So, it asks about the cognitive limit of real players. So, let us give an example. This game is called the centipede game, particularly because of the structure of this game. It looks like a centipede. So, in every round of this game or every stage of this game, every player, players take turns and play this game. And they have two options to pick one action play across or play down. So, these are the two possibilities. So, these two actions. And if the previous player has played across, then the next player gets a chance to play again. And it also has the same set of actions available to it. And if any player picks the action D, the down, then the game ends and that gives a specific outcome, specific utility to both these players. So, let us look at this five stages of this centipede game. Now, what is the sub game perfect Nash equilibrium of this game? If you look at, if you use the backward induction, then at this node, you will apply the try to find out the Nash equilibrium pure strategy Nash equilibrium for player one. You can see that this number is larger. So, therefore, playing D is a better response. So, if you go back now one step and ask what player two should play, it knows if it plays A, one will play D and it will get an utility of three. While if it plays D, then it gets a utility of four. So, therefore, playing D is also a better option for player two. And you can continue this and you will see that for all the players playing down is the best response. So, therefore, the sub game perfect Nash equilibrium is the case where all the players are playing D. Now, what is the problem with this prediction? There has been extensive studies of real world real people playing this centipede game and the populations where random participants, university students or even grandmasters of chess almost all of them have played it at least for few rounds. So, it never happened that it ends at the very beginning. So, there had been various reasons that has been claimed for this. Maybe players are altruistic. They also care about the utilities of the other players or they have limited computational capability so that they cannot compute the sub game perfect Nash equilibrium and therefore do not pick that action. Or there are certain kind of developments or variations of this game. So, it has also been the same game has also been experimented with a larger difference in the payoffs. So, for instance, if all these numbers all the utility numbers were multiplied by 10 or maybe 100, then does it make any difference to the way people play? It turns out that it does really make a difference and the experiments have actually exhibited that. So, when the players have much larger stake, much larger utility when they play one thing versus other, then they do think a little more carefully and try to give a better choice and that leads to a closer to SPNE performance. So, that was the qualitative criticism about the idea of SPNE that it does not work so well in practice but there are certain other criticism from the theoretical aspect of SPNE as well. So, sub game perfect Nash equilibrium is actually talking about what action will you play when the game reaches that history. So, for instance, in the in our previous example, we have seen that what action should player one will play if it reaches this particular node. But the sub game perfect Nash equilibrium, so we have seen this AG comma CF to be the sub game perfect Nash equilibrium. And if this player the first player is playing A, then there is no reason to think that it will ever end up in this node. So, a player who is playing the action A at the first round that is taking a completely different path in the game tree and it will never reach a state which is which will give the opportunity to the same player to play G again. So, then what's the point of giving this kind of a guarantee which itself says that you cannot reach that particular node. So, these are the two main criticisms about SPNE. However, for various kinds of games strategic extensive form games, we can actually use SPNE. This is quite a well known equilibrium concept. So, we will now go and extend this idea for sub game perfect Nash equilibrium to a game where we cannot really observe all the states of this game. So, so far we have discussed only games like chess or tic-tac-toe where you can actually observe the actions and the current state of the game perfectly. This is the perfect information game. But there are certain games and we have discussed this earlier games like cards where you cannot really observe the whole current state of the game. Certain part of that game is certainly visible, but certain parts are not visible. So, in that case the players use certain kinds of beliefs that is what we are going to use for this extension of this idea of SPNE.