 We have seen an example of a PI-EFG, the perfect information extensive form game, transformed into a normal form game and we have also seen how we can find the usual concepts of equilibrium like the Nash equilibrium, pure strategy Nash equilibrium in that transformed NFG. But we have also concluded that there are certain outcomes in that PSNE which does not really is a credible threat. So let us look at one more example and this will be the running example for this module which will also show some of those phenomena and we will see how we can mitigate that problem. So here this is the game tree. So there are two players, player one and player two. So player one plays in the first round, player two plays in the second round. If it goes in this branch of the tree then after two's play the game ends. If it goes to this branch and if two plays F then there is another turn for player one to play and that ends in this following way. At the terminal nodes you can see the corresponding utilities of all these players. Now what are the strategies as we have defined in the previous module for the extensive form game? It is the complete contingency plan. What player one can do at every possible state of the game where that is a player, every possible history of the game, non-terminal history where this player is playing. So player one plays at the origin and also at this history which is BF. So here, so therefore at these two points it can take actions like AG. So the strategy of this player one could be written as A-G-A-H. So this means that at the first history where it is a player it is choosing the action A and when the second occasion where that player is again playing it is picking the action H. Similarly BG and BH for player one. Similarly player two can play either at this node or at this node at these two non-terminal histories and they are based on whatever actions they have picked. So it can be C, E, C, F, D, E and DF. So you can do a very similar exercise that we did in the last module transform this game into a normal form game and also try to find out what are the PSNs. And let me give you the answer here. There are three PSNs in this case. AG, CF, AHCF and PHCE. These are the three possible outcomes. Now like the previous example there is certain things which are not really very credible. So if you look at these two strategy profiles player one is playing in the second round here this player is playing the action H. So what does it mean? If the player so player one can play H only when it is at this node and if this node is ever reached in this game do you think player one will play H? Because if player one plays H it gets an utility of one while by playing cheat gets a utility of two. So this goes against the idea of rationality that we have defined that whenever there is an option wherever a player can pick an action which maximizes its utility it will always pick that. So this is not consistent with that notion of rationality. So therefore even though this turns out to be a pure strategy Nash equilibrium this threat or this outcome is not really credible. And the problem here is that when we are transforming the game we are actually losing this information that this player can actually observe the outcome here. Player one can observe and then pick its action. So our equilibrium notion should also change based on which kind of games we are choosing. So we should look at a notion of rational outcome where it considers the history and ensures utility maximization at every possible history. And that is exactly where we bring in the concept of sub game. So by the name you can kind of guess what sub games are. So it is a game that is rooted at an intermediate vertex or a non-terminal history. So the sub game of PIEFGG rooted at a history H is the restriction of this game G to the descendants of H. So suppose we are at a history of B. So let us say this is one history. So this history is nothing but B because there is only one action that has been picked by player one. So at that history the subtree rooted at this history, this subtree we are going to call that as the sub game rooted at this history H. And we are also going to define the set of all sub games of G which is the collection of all sub games at some history of G. So if we collect together all possible non-terminal histories and look at the corresponding sub games then the list of sub games we are going to call that the set of sub games of G. Now as we have already hinted that our equilibrium concept should look at every sub game and essentially try to find out an equilibrium that is appropriate for that sub game. So that is the notion of sub game perfection. The sub game perfection idea is essentially talking about the best response of the player who is at that sub game and this should hold for every sub game of this game. So what is the definition of a sub game perfect natural equilibrium? So this is a refinement of the equilibrium notion that we have defined for the normal form game. So the sub game perfect natural equilibrium of an PIEFG are all such strategy profiles as such that for any sub game of G prime of G the restriction of S to G prime is a PSNE of G prime. So notice that this should be a PSNE of that sub game and by restriction you just prune all the other strategies which is not which is not relevant for G prime. So removing all those strategies whatever remains that is the restriction of S to G prime. So we can look at the corresponding example the same example that we have started with. So we have seen that there are three PSNEs but are they all of are all of them sub game perfect natural equilibrium and let's see one by one. So if we look at this natural equilibrium pure strategy natural equilibrium and we restrict that to this particular set this particular sub game of this game then we can see that the restriction of this strategy. So let's say this is S1 and this the second strategy is S2. So then this profile if you look at S1 is to this profile's restriction to G prime if you want to call it that way so where this is G prime. So this will only be H because there is only one player remaining and its action will be H and the other player has no a nothing to play because this player does not play so it's like empty. Now you can clearly see that this cannot be a pure strategy natural equilibrium in this reduced game G prime because in this game this player can player 1 can clearly improve if it plays G. So H is not a natural equilibrium national equilibrium strategy for this player here. So similarly you can rule out this outcome as well but if you look at this AG, CF you can see that every for every possible such sub game. So if you look at this G comma so this sub game G prime that we have looked at if you restrict it to here then all that you have is G comma F of course that is a strategy pure strategy natural equilibrium for player for both the players. Similarly if you look at this sub game there also it's it's best because their player 2 is going to play C which is the best response for this player because this utility is larger than this one and finally if you look at the whole sub game rooted at the root node then also you will find that AG comma CF is a natural equilibrium from that case. So summarizing this AG comma CF survives this test of being the optimal thing being a PSNA at all possible sub games of this game G. So therefore this is a SPNE sub game perfect national equilibrium the other two other two equilibrium are not SPNAs. Now the natural question is how should we compute this SPN sub game perfect national equilibrium. So we have already seen that this starts with so this idea of sub game perfection starts with all possible sub games of this game. So therefore we should actually look at the sub games from a bottom up fashion because we already know that if there exists some sort of a equilibrium pure strategy national equilibrium at the most lower part of this game. So for instance in this part of the game in the games in the games which considers this as a sub game this strategy should certainly be consistent with that. So the idea here is to essentially use something like a backward induction algorithm. So we start from the the leaf nodes and we try to find out what are the pure strategy national equilibrium in this sub game and then move upwards keeping that as one of the outcomes in that sub game. So let us illustrate this with this example so that we will be able to understand. So let us look at this particular node first. So here player one is the player and you are trying to find out what is the pure strategy national equilibrium at that sub game and you can find that that is G. So fair enough you can note that down for player one and we are going from backward induction. So we are writing everything from right to left. Now if you now come back at this point where player two is a player and we see that now if player two plays E it gets a utility of 5 but if it plays F then it knows already because it has already solved the sub game below player one is going to play G and it will get a utility of 10. So therefore for player two which is trying to maximize its utility it is going to pick this outcome F. Similarly at this node where player two is also considering which action to play it looks at these two actions C and D and it sees that it gets a higher utility when it plays C. So therefore it will pick C and that will be written here. This is the strategy for player two and looking at all these things now at the very root player one when it considers playing these two strategies it can either pick A or B. If it picks A it knows that two is going to pick C and therefore the player one is going to get an utility of three while if it plays B then it knows that two is going to play F and then one will play again G and it will get an utility of two. So since three is larger than two it is beneficial for player one to pick this action A and AG will be the final strategy for player one. So AG, CF you can see that this is essentially the same strategy profile that we had actually found to be SPNE by eliminating the PSNEs. But that is so rather than doing this rather than finding all PSNEs and ruling out things which are not SPNE this is a much better algorithm because it will save a lot of time. It is computationally efficient. So here we have the backward induction algorithm written in the form of a pseudo code. What it does is that essentially it is doing some sort of a recursive call. So let us go over this step by step. So this is a function backward induction which takes as input specific history H and it will return the utility of the player who is the player at that H and also the action that is picked by that player. Now we know that if this is a terminal node then there is no action to pick it will just return the utility of that player of the player who is a player at H and also the second entry that is the action will be a null thing. If it is not so if this if statement is false then you start with the initializing the best utility of that player. So notice that this is the player who is the player at H initialize it to a very large negative number let us say minus infinity and then you iterate over all possible actions of that player at that history. So this is going over all possible actions available to that player at that history and you are just appending that action to the current history. So you have this current history here you append that action that you are considering to play and call the backward induction again and whatever you get back only the first component that is the utility component you are going to consider that as the utility at the child of that pH of that player that is playing at H and now it is very simple if the utility at that child or the current utility is greater than the currently best utility then you are going to replace you are going to consider that to be the to be the best utility and also A to be the best action at that stage. Right so that is what we are going to use and then finally one once you are done with this whole for loop and you have found the the most optimal action there then you are going to return the the best utility and also the best action from this player. So this is the backward induction algorithm and we have just used the same algorithm in this form but a little more informally and pictorially.