 Let us take stock of where we are. We introduced dynamic games three lectures ago and particular extensive form games and our main conclusion was that in the previous lecture was that these such games will inherit equilibrium from informationally inferior games. So, a dynamic game inherits equilibria from informationally inferior games. So, what this implies is that these such games will in general have a large number of equilibria because all the equilibria of inferior games will also show up as equilibria here. Now, if you want to find all equilibria, to find all equilibria of a game, all equilibria of a game, what we need to do is go to the normal form which means we have to list out all the strategies of all the players, make it into a table and then find the Nash equilibrium treating that as a simultaneous move game in this space of strategy. The big challenge with this sort of approach is that it obviously takes a lot of effort because you have to list out a large number all these strategies. Usually the strategies are exponentially many in the number of actions and number of information sets. So, this is not going to be an easy computationally or even analytically an easy task. So, another alternative is to analyze the structure of the game. If you really want to find all equilibria you have to go down to the normal form. But if you want to find some equilibrium, at least one equilibrium or then what you could do is analyze the structure of the game and from the structure of the game try to assess if you can argue what the likely mode of play is going to be. So, one of these we had done earlier also if you remember in the first game that we saw we had this L1, R1, L2, R2 type of game and what we did was we started arguing about this game by saying so this was player 1, this was player 2 and the way we argued about this game is that we said let well if the game comes to this node then player 2 would play a particular strategy then player 1 has to decide whether he takes the game to the left node or to the right node and then based on that you decide player 1 strategy. Now this way of reasoning through the game actually gave us an equilibrium of the game. So, now this fact is actually more generally true so that is the thing I want to point out to you it is not that hard to show but we will skip the proof. So, if you have a game of perfect information if G is a game of perfect information then its equilibrium can be found its equilibrium or then an equilibrium can be found by backward induction. Now what do I mean by backward induction? By backward induction basically I mean this process that you start with the nodes that are the parents of the leaf nodes. So, these are these 2 are your leaf nodes take the parent of the leaf node there will be some player acting at that node and since this is a game of perfect information that player has knows exactly whichever is that player he knows exactly that that is where he would be that is the node he would be at and so then in that case you can ask what is the best action that the player should take at that node then what you do is remove if you once you once you know that what action is to be taken at that node you can just remove the rest of the this part of the tree the part of the tree that starts from this node and replace it with the payoff that the players that that player and all that the players would get when this player takes the optimal action at this node. So, let us just take a so the zero sum game was that this is player 1 2 actions L 1 R 1 then player 2 3 actions L 2 M 2 R 2 now and here are the these are the payoffs this is for player 2 player 2 is a maximizing clear he is looking for the largest value player 1 is the minimizing clear looking for the least value. So, what you do is you start with as I said leaf but the parent of the leaf node. So, take so these three leaf nodes have this node as the parent. So, I started so assume you are here so assume that the game is here if the game is here what would player 2 play player 2 wants to maximize the player 2 would play M 2 right. So, player 2 would play M 2. So, then what this means is I can now say well this tree has been condensed to something like this this part of the tree has been condensed to something like this where player 2 has already played M 2 and the resulting payoff has been accrued. So, that means this the tree now is something like this R 1 to R 2 now at this what about at this node now here again play it is player 2 is turn to play what would player 2 play if you would play R 2 you would R 2 to get 7 right. So, then what this means is this further reduces in the following way in this form and now player 1 is minimizing what would player 1 play he would play L 1 right. So, at so technically we should actually be going from here directly here but the so long as you are careful it does it should not matter the point you start with all the leaves first go and then go to their parent then you take the next the leaves of the of the next tree that you get go to their parent and so on and so forth. So, this is how you so by doing this what you have effectively found is an equilibrium of the game and how do I understand this as an equilibrium of the game well. So, this is sorry player 1 how do I understand this as the equilibrium of the game well the equilibrium is that player 1 is playing L 1 and player 2 is playing like this player 2 strategy is that if player 1 plays L 1 he would play M 2 and if player 2 player 1 plays R 1 he would play R 2 this clear. So, that is how I am interpret this strategy. So, essentially what this what this has done is it has told me what to do at every information set right it is told me that at this information set player 2 should be playing M 2 at this information set player 2 should be playing R 2 this is clear now this is very easy to do in this kind of structure because there is every node is a separate information set every node is a singleton information set the game is of perfect information. Now, imagine you had something like this I will give you a more complicated example. So, suppose you had a game like this with three players this is player 1 this is these are nodes of player 2 ok player 2 cannot tell between these two nodes I do not know I am not going to give them names to the actions with the important thing to see is the structure then this is player 3 ok and player for player 3 the information sets are like this. Now, how would you do this how would you apply extend this logic. So, the first immediate trouble you encounter is this that if you start from leaf and go to the parent of the leaf you can you cannot look at this sub tree in isolation you cannot look at just this tree in isolation just mark this here look at this yellow tree can you look at player 3's decision on just the yellow tree in isolation it is not possible because player 3 does not know that he is on the yellow tree he is confused about whether he is on the yellow tree or on the blue tree player 3 does not know whether he is on the yellow tree or the blue tree. So, the decision for player 3 you cannot decompose it like this that you say ok well what is he going to do at this node the point is at this node is not a well defined question because player does not know that he is at this node he just knows he is at one of these two nodes. So, whatever action you take has to be a common action that would be prescribed at either of these two nodes either on that will be that it should be the action that he will take on the blue tree as well as the yellow tree is this clear. So, then how do we proceed further then which part can be made ok. So, this has can be made into a of P2 and P3 see there is a trouble here also right. So, you may think that well what I can do is let me take a simpler example I think I before I jump one second let us take this example first. This is a simpler problem now what would you how would you proceed no we are not this is we are not playing security players are playing optimally you can the way of the numbers do not matter the logic is that is the point is how are we going to decompose the game. I am not asking which specific action is to be chosen but how do we decompose the game is the issue a specific instance you may be able to solve once I put in the numbers that is okay that is a different matter. So, now well player 3 cannot you for player 3 you cannot decide between these two no you cannot basically reason separately for these this node and this node. Now you can but you can do the following you can okay let us go one step further into the tree from this node onwards this particular this part of the tree let us let me call it something color it with something else let us call this the green tree this part of the tree can I reason separately about this I can because this tree is sort of a game in itself it starts with player 2 and player 2 knows that he is at this node because because this is a singleton information set for player 2 player 2 knows that he is at this node and then player 3 what does player 3 know at this this information set p2 has played one of those two actions what else does you know so if I call this l1 and r1 so l1 and r1 so what does player 3 know at this at this information set he knows that p1 has played r1 right that information is there see what what player 3 knows is that he is at one of these two nodes and the only way the game history could have come to one of these two nodes is if player 1 has played r1 clear so from this the although he does not know what has happened after player 1 has played r1 he he does know that player 1 has played r1 okay so essentially we are now getting to the question of what exactly do players know and how does that knowledge of what players know how does that help us decompose the game so here player 1 knows r1 has been played he also knows that p1 p2 has played but he does not know that p2 what p2 has played this clear he knows that p2 has played but he does not know what p2 has played okay so therefore he for the therefore this green game is a is a game in itself it is p2 has played something okay and now it is p3 is turned to play all right likewise p2 essentially knowing that he is at this node knows that this is going to become essentially a simultaneous move game between p2 and p3 from here onwards because he will not p2 p3 will not know what p1 is what p2 has played and p2 obviously p3 has not sorry p3 does not know what p2 p2 is going to play or has played and p2 does not know what p3 has played because p3 has not played yet clear so therefore this is effectively a simultaneous move game from here onwards so this green tree therefore can be analyzed as a simultaneous move game okay likewise there is a green tree here on the left also this green tree also can be analyzed as a simultaneous move game okay so now these are now games in their own right you can analyze them simultaneous move games find their equilibrium assuming an equilibrium exists and all that okay this once that is done you can then come back to this level where p1 has to decide between L1 and R1 okay and the payoff that is coming from at each of these nodes the payoff that comes from this red node on the left and the orange node on the right what is this payoff going to come from it is going to come from the simultaneous the equilibrium of the simultaneous move games the red one will come from the equilibrium of this left green tree and orange one will come from the equilibrium of this okay and then now you player one can now say okay which of these two is better for them and then play that once again you are you this would again specify for you strategies because what you would get is that at each information set you would know what player is doing so player 3 what he is doing in this information set will come from this sub game and what he is going to do at this information set is going to come from this sub likewise player 2 also okay and then player 1 is this clear so this is how you would analyze this sort of game so now let us come to the earlier game that I that I had drawn and let us see how if there is a way to work around this so now suppose if you had this structure what could you do so okay so again let us ask who knows what what does player 3 know here what does player 3 know here at this information set p2 has played yes p1 has played okay so if you want I will give names here so what does p3 know at this information set that he knows that p2 has played he knows that p1 has played anything else he knows p1 has played r1 right he knows that p1 has played r1 because he knows that he is at one of these two nodes okay when he is at one of these two nodes it has to be that it has come from this part of the tree so he knows that p1 has played r1 p2 does not know however that p1 has played r1 although p1 has played before him p2 does not know that p1 has played r1 but p3 knows this is clear so now it turns out that this cannot be decomposed further because of this issue okay this is this is this this thing this one cannot cannot be decomposed further because of this particular problem okay now let me show you another one which which also has this problem okay we need to think clearly about what each player knows L2 R2 L2 R2 L1 R1 and now I will draw an information set which stretches like this so let us call this let us call this the orange information set and the other one is a green information set okay this is for player player 3 is turned to play here now tell me what does player 3 know at the green information set to begin with he knows that p1 has played r1 it is a singleton information set so he knows the exact sequence of actions that led to that okay so he knows that p1 has played r1 and then p2 has played r2 okay what does he know at the orange information set so he knows p1 has played of course p2 has played okay what else so what Ashwin was saying what what were you saying p2 has not played R2 right p2 could have played R2 but he could have played R2 here right so you have to be careful what it what he knows is that r1 r2 has that that particular sequence has not occurred every other possibility is there means player 1 has played R1 and player 2 has played R2 that the end of these two has not happened okay it could be that player 1 has not played R1 which means L1 has happened and something else has happened or player 1 has played R1 and player 2 has not played R2 which means in this case player 2 has played L2 is this clear so this is so the extensive form basic is essentially capturing for you the entire history of the game the once you know which node you are at you have perfect history of everything that has happened to bring you to that node okay when you have in perfect information like this it means that there is some loss of state information essentially the node of the game is the state the node that the game is at is the state of the game it tells you where you you know think of this as a game of chess for example you know players have various moves the game keeps moving from one stage to another game chess is a game of perfect information because players can see where the what the where the game is right the exact configuration of the game okay so that gives you a game of perfect information and being at a particular node you can you node means not a node just not just does not just mean the exact position of the pieces okay the same position could have a reason for multiple histories node means the exact history that has got you to that okay players are observing all of that that entire history is recorded so the players know are aware of that but there are versions of you know more fancy versions like there is blind chess or and so on right where where players are where certain only certain things are revealed to players then players may not know which what is the exact which of the several histories that could have led to a certain configuration have actually occurred is it clear okay so now let me ask you a trick question here so let us take this okay the way we defined information sets in in extensive form games we said they are a partition of the players player set right nodes of the tree are divided into player sets player sets of other partitioned into information sets now this information set is of player 3 so we know we were talking of what player 3 knows of at this information set suppose I ask you a question what does player 1 know at this information set if you think about it the way we modeled an extensive form game we said we players have some information and that information is at their information sets and information sets are subsets of are subsets they are formed from nodes that where it is the turn of the player to play right now this means that I am not even defining for you what the information is of another player whose know whose turn it is not to play right so at this node it is the node it is the turn of player 3 to play I am not even telling you what is the information of other players at this node so player 3's turn has come to play now but you might want to know what do other players know at this juncture right so this is actually not modeled in the game okay the question is does it matter does it matter what player 1 knows at this node yeah exactly what matters is what he knows when his time when it is his turn to act it is of no use knowing so what someone else knows when it is not your turn to act right okay so what matters is what you know when it is your turn to act okay now here suppose further after this it was player 1's turn to act at that time we can ask okay what does he know I mean this general knowledge about what is going on in the game is of no is of no consequence okay so that is why the way we modeled this is what play what the player whose turn it is to act knows at that in at the time of acting is this clear okay so one one thing which you have realized now is that the whether you can decompose or not really depends on how the information is spreading through the or how it is structured through this through the game right so if your information sets are such that they can help they can they are sort of they allow for this kind of decomposition then you can do some decomposition based arguments okay so this leads to this whole theory of what is called information structures so information structure the information structure is essentially a description of who knows what okay in the game okay who essentially describes who knows what okay so in particular here as I said the here P3 knows that P1 has played R1 but does not know what P2 has played etc here P2 knows that P1 has played but does not know what he has played etc etc okay so whether you can decompose or simplify games really depends on what the what the information structure of the problem is so the information structure actually tells you whether the game has a any kind of admits any kind of algorithmic type of solution it means where you can once you can decompose you can try to make an algorithm out of it right otherwise you have to write a big normal form and so compute directly where if you can do what the information structure decides whether you can do something analogous to backward induction in a game okay now backward induction for those of you who know a little bit of dynamic programming would immediately realize that this is actually nothing but dynamic program right you are starting from the last time instant you see what we would do there come take that as the cost to go then go to the time instant before and so on and so forth okay so this is essentially a multiplayer analog of dynamic programming now what this also means is that even if you had a single player problem whether you can do dynamic programming or not has something to do with the information structure of the problem okay for instance this could very well have been although I have written here player one player two player three these could very well have been you know player one followed by player one again right it could be player one whose turn is this to play first but then he forgets what is played then he has two actions and then he has some partial memory well if he had played R1 and R2 then he remembers otherwise he does not now whether you can actually compute a policy recursively like in Bellman's equation dynamic programming and so on depends on these information structure of the problem okay so information structures are fundamental essentially to any kind of dynamic decision making problem information how information from the previous time step is passed on to the next time step is fundamental okay this is also related to many other phenomena I mean loss of information for example in thermodynamics physics and so on all of this is somehow closely related to this closely related to you know whenever dissipation happens for example you lose information from one time step to the other and therefore there is fundamentally an uncertainty about you know what the configuration is okay all right so this this is so this so information structures are fundamental now what we will what I will quickly tell you is generalize this particular fact here which is you know backward induction so we were we can do backward induction when the game is of is a game of perfect information when the game is of perfect information of course you can do backward induction but what about the structure allows for backward induction is the question okay so for example we just saw that this kind of structure gives you some form of backward induction though not exactly but some kind of decomposition is available okay this one does not this one does not and so on so we will just I will give you an overview of this about when it is possible when it is not possible then so on okay so at the heart of it what we want to do is we want to take a general dynamic game and decompose it into either you have perfect in games of perfect information okay in which case you can they can be decomposed further or you get to a stage get to a single turn node from where a simultaneous move game begins is this clear so we want to get to this kind of a structure so either you have a game of perfect information say something like this or a single turn node from which if you go down take the subtree starting from there that subtree is a game of is a simultaneous move game once it is a simultaneous move game we know how to analyze and there is nothing more to decompose and because there is a single turn node to start off with it means that that is a well defined subtree because the starting player knows that he is at that node is this clear so this is the decomposition and frankly there is nothing much beyond this this is the max we can do in terms of decomposing because information structures can be very wild here you as you can see this this is just one example you can have some some crazy information structure like this stretching from one level to the next very very complicated information structures can be there so in a given this kind of the amount of variety there is in information structures best we can do is you know in this in these kind of settings okay.