 We will start today with a more formal study of what of the examples that we had shown so far. So the examples we had discussed were both dynamic games but we had never formally written out what a dynamic game is. So what I will do today is tell you exactly that. So a dynamic game is basically described using what can be what is called an extensive form. And here is an example of an extensive form so here player 1 is the one plays first he starts at this node he has 2 actions L1 and R1 then player 2 if player 1 plays L1 then player 2 then it is turned for player 2 to play and he has let us say 3 actions L2, M2 and R2 and if player 1 plays R1 then player 2 again has 3 actions L2, M2 and R2 and the this is a zero sum game so player and so the payoffs are written like this 0, 0, 3 and 1, 3 and 0 and 6, 2 and 7. So what this means is that if player 1 plays L1 and then player 2 plays L2 then player 1 will get 1, player 1 will get minus 1 and player 2 will get 1. So player 1 is looking to get the least possible number here and player 2 is trying to get the maximum possible number. So whatever is the number here player 1 is looking to minimize it so you can think of it this way that player 1 gets 1 player 2 gets minus 1. So player 1 is the minimizing player and player 2 is the maximizing player this is a zero sum game in which player 2 can observe exactly what player 1 has played. Now this way of describing the game is called an extensive form so a tree like this in which you demarcate what the actions of the players are and what whose turn it is to play. I have not yet mentioned what players know at each stage but that we will come to in this case just let us take it for simplicity we are taking the case that player 2 knows what player 1 has played and then at the end of the tree you write out the payoffs. In this case this is a zero sum game so all I need to write out is these just one number. If this was a non-zero sum game then I would have to write a pair of numbers one for player 1 and one for player 2. Now this way of describing the game what it does is it actually tells you exactly what is happening during the gameplay so how the information is flowing during the gameplay. So player 1 has played L1 then player 2 has seen it and he has options L2, M2, R2 etc. The earlier way where the way we were writing games earlier where we were writing them as a table that is what is called a normal form. So a normal form is one where you just list out the strategies for all the players and you ask and you solve for the game from there. Every extensive form can also be written as a normal form and vice versa it is just that the extensive form tends to be more explicit in terms of the interactions that are happening during the gameplay. The normal form is this table is the way we write this out as a table so I will explain that. So let us list out the strategies for the two players. So what are the strategies for player 1 so how many strategies for player 1, two strategies for player 1 and the strategy is the first strategy is let us say to play is to simply play L1 and the second strategy for player 1 is to play R1. So here my notation is that the superscript will stand for the players index this is the player and this here is will the subscript stands for the strategy number. So to play L1 is to play the constant strategy L1 is player 1's first strategy to play the constant strategy R1 is player 1's second strategy. How many strategies does player 2 have by 9 yeah so the player 2's strategy is now remember we are in a dynamic game so player 2 can observe what player 1 has done so he has to pick his action as a function of what player 1 has done right. So if player 1 has played L1 he has to pick L2, M2 or R2 at L1 and then if player 2 has player 1 has played R1 he plays L2, M2 or R2 at this node so he has to effectively play a function which is a function of the action that player 1 has played right. So how many choices does he have at this node he has three choices here so he can take any either L2, M2 or R2 at this node and he has three choices again at the second node right. So every combination here is a possible strategy for player 2 right. So the player 2 has 9 strategies and the reason it is 9 is because it is 3 here times 3 here okay. So let us list these out so let us write out player 2 strategy 1 and now this is a function of what player 2 knows and what he knows is basically the action of player 1 okay. So player 2 let us write it as a function of the action of player 1 and so I am going to write U1, U1 here so little bit of a control theoretic notation here so U1 is the action of player 1 okay. So gamma 2 of U1, gamma 2 1 of U1 could be something like this for example it could be that he plays L2 if U1 is L1 that means if player 1 has played L1 he responds with L2 and if player 1 has played R1 he responds with, he responds with R2. Another strategy is let us say gamma 2 2 this strategy is to simply regardless of what player 1 has played he simply plays L2 so irrespective of what player 1 has played he is going to play L2 so which means that at this node he plays L2 and at this node he again also he plays L2 okay. Another strategy is irrespective of what player 1 has played he plays R2 and likewise the third one is that regardless of what he player 1 has played he plays M2 this is clear okay. Let me write this in this correct order M2 R2 so this will be the third one this is the fourth one. How many more strategies so I have listed out 4 you said there are 9 can you so let us take one more so this is the fifth strategy for player 2 yes so in this case yes that is what it would be it is a Cartesian product of the action so that is actually the one way of counting the number of strategies yeah yeah where are you asking here yeah this whole this function is one strategy so I have to write this function right and it is a function of what it is a function of what player 1 has played okay so if player 1 has played L1 the action for player 2 is L2 in this strategy and if player 1 has played R1 the action of player 2 is R2 this is one such strategy. Another strategy is to is regardless of what player 1 has played you just play L2 regardless of what player 1 has played you play R2 and then likewise regardless you play M2 okay so you so these are the strategies for the players are functions okay for the first player they are always trivial functions because he has he does not have any information for the second player it is a function of what he knows in this case he knows what player 1 has played okay so here is another here is another strategy so if player 1 plays L1 then you respond with L2 if player 1 plays R1 then you respond with M2 then player 2 responds with M2 now here is another one strategy number 7 is to play M2 if player 1 plays L1 and R2 if player 1 plays R1 strategy number 8 is M2 R2 U1 equal to L1 and U1 equal to sorry my mistake yeah L2 and here is the last one which is R2 M2 U1 equals L1 and U1 equals R1 okay so this these here are my nine strategies for player 2 1 2 3 4 5 6 7 8 9 so like we did in the in the in the last class what we can do is we can now write out this game as if it is a game between two players player 1 has two strategies player 2 has nine strategies right and right and just express this whole thing as a table and ask okay what is what is the what are the payoffs that the players are getting so that that way of expressing a game is what is called a normal form game so we could write out a table like this this is player 1 player 1 has two strategies let me just write them as L1 and R1 those are his strategies player 2 has nine strategies 1 3 4 now can someone help me fill out the the what what are the going what are going to be the payoffs so player 1 if player 1 plays L1 for and player 2 responds with gamma 2 1 okay the first strategy gamma 2 1 here so I here I get 1 let us stick to L1 for player 1 let us go horizontally okay so player 2 responds with 2 okay fine where is it so what is 0 so player 1 plays player 2 plays 1 okay so that will be then no I think it is by easier to go horizontally okay so let us let us stick to L1 for so then you can go branch wise sticking to this branch so let us stick to L1 first for player 1 and player 2 is playing now the his second strategy which is gamma 2 2 which means he is going to respond with L2 what what does he get 1 then now gamma 2 3 which means he is responding with M2 3 okay gamma 2 4 0 very good gamma 2 5 1 again gamma 2 6 gamma 2 7 3 gamma 2 8 3 again very good and gamma 2 9 will be 0 okay next stroke 7 then then 6 then 7 6 7 right yeah then 6 then 2 is that right okay all right so this as I said is what is called a normal form of the game so you take a normal form like this this is a way this game could have been can the normal form has the problem that or has the has the you can say the shortcoming that it actually conceals from you what is the happening within the game right during the gameplay you just know that play the players have these many strategies and these are the pair we do not know what is the interaction that is happening during gameplay like player 1 as we just know player 1 has 2 strategies L1 and R1 player 2 has these 9 strategies the fact that player 2 is responding to player 1 and and that he has such an information is actually not revealed through the normal form whereas the extensive form makes this whole thing very explicit the only the but the the while this it makes things very explicit the extensive form has is a little if you wanted to solve it directly from without writing the normal form then you would need to develop your own set of techniques okay because you would need you need a fresh set of techniques because we really never solved this directly the other problem that happens with the extensive form and this you have seen yourself that 1 tends to sort of there is a possibility of making mistakes in in the sense that you can because you know you know that player 1 has this information you tend to often think that well player 1 is going to act as if he is responding in each case giving a best response to player for every for every action that player 1 sorry because player 2 has this information he plays as if he is going to respond with a best response to every action that player 1 plays and what we saw was that actually hides a few some set of equilibria so if you want to actually find all the equilibria of the game the best the the most sort of short way of doing that is to convert an extensive form eventually to a normal form and then find find all the equilibria the extensive form gives rise gives you gives you very explicitly what is going on inside the game but you know we it it is limited in the sense that it does not let you explore all the equilibria very problem you know in a very systematic way now I will explain how this you know what exactly all happens here but that is always the assumption yeah yeah so anyway let us let us try to solve for this can you tell me what would be so now this is a zero sum game so if we are effectively then looking for a saddle point so can you tell me what is the saddle point for this player 1 wants to minimize player 2 wants to maximize l 1 comma l 1 and 7 yeah so this is this is a saddle point why clearly player 1 does not want to if player 2 is playing 7 player 1 does not want to shift from l 1 and player 1 is playing l 1 player player 2 is for player 2 it is optimal to play 7 this is a saddle point for sure no no no that is exactly what I said see these are now these strategies are being chosen simultaneously and the strategies are that these are the strategies these are the 9 strategies for player 2 these are the 9 strategies for player 1 sorry 2 strategies for player 1 these are being chosen simultaneously the actions are being chosen in sequence okay so we can look for a saddle point now in this in the space of strategies consider knowing that the strategies are being chosen before the start of the game although actions will be chosen during gameplay okay so l 1 comma 8 is also a saddle point so these two are saddle points and because these are saddle points you will see you will have the usual property that they would they would all have the same value okay so the the both both of them have to have the same value they are 3 now let us try to understand in some way what this let us see if we can sorry you can you can and I explained that last time see the reason we are looking for a Nash equilibrium or a saddle point in this in for this game here is because it is effectively now a simultaneous move game but at the level of the strategies although the actions are being are being chosen dynamically the strategies are being chosen before the start of the of the game as a whole okay so therefore we are we we can so in this space we are if it is effectively any other any other simultaneous move game right where these so long as we interpret the space of strategies correctly the the all the other theory continues to apply okay so the point is that the strategies are being chosen knowing that there is going to be you know information that is going to come along and that is why the space of strategies and so on is has been enlarged to allow for all of them okay okay let us so this is this is just an example of how to convert this this into a normal form let me let us do one more example and this will be now what I this this is some more interesting so suppose you have now player 1 who can who has 3 actions okay l 1 m 1 r 1 player 2 then has let us say 2 actions l 2 and l 2 r 2 at every node okay again this is also 0 sum 3 1 minus 1 1 0 and now here is the important thing player 2 here so the nodes where player 2 act but player 2 can now observe whether player 1 has played r 1 or has not played r 1 when he has not played r 1 he cannot distinguish between whether player 1 has played l 1 or m 1 so what player 2 can tell is if either l 1 or m 1 has been played by player 1 or r 1 has been played okay but not whether l 1 specifically or m 1 specifically has been played okay so the way we depict this in an extensive form is that we put these 2 nodes in one bubble like this or one box like this now what does this mean what this means is see when player 1 plays l 1 the game would reach this node okay if player 1 plays m 1 the game would reach this node at either node it is the turn of player 2 to play but what player 2 does not know is whether he is at this node or he is at this node okay so he can only know that it is his turn to play because it is one of these 2 but does not know which now the payoff that that would get realized is realized at the end here so after he plays the action after the fact he will know okay all right well if he suppose he plays l 2 if he plays l 2 and he gets a payoff of 3 then he will know after the fact that player 1 had played l 1 right or if he gets minus 1 then he would know that player 1 had played m 1 but at this node he does not know which of the 2 has actually actually happened okay so this is you can think of any game of cards or and so on where such you know where some information is hidden this is exactly how these things play out right so this is this is a case in which player 2 we say has imperfect information so imperfect information means is is referring to the situation where player 2 is ignorant about something that would happen during gameplay the game is known to him but he knows that there are certain elements of the game during gameplay which will be not accessible to him okay all right so now can you tell me what are the strategies for the 2 players firstly let us count how many strategies how many strategies for player 1 3 for player 1 okay how many strategies for player 2 4 why 4 yeah so the point is strategies are not mapping from nodes to actions okay they are mapping from information to action the information at these 2 nodes these 2 nodes here is the same for the player he has the same information he has no way of distinguishing between these 2 okay he has a way of distinguishing after the fact after the game is over but by that time it is too late okay so at the time of choosing his action he does not have a way of distinguishing between these 2 so he is therefore compelled to play the same strategy regardless of which node he is at whether he is at L whether he is at this node or whether he is at this node or he is at this node the action that he would end up having to play is the same okay so he is compelled to play the same action essentially without the knowledge of which node he is in in this bubble right so consequently we have to in our definition of a strategy we have to take into account that the same action is being chosen at these 2 that is equivalent to saying essentially that we have collapsed these 2 to 1 to 1 unit right the strategy of a player is a mapping from its information to its actions okay and the information here being is simply that L1 or M1 has been played okay here the information is that R1 has been played is clear okay so let us quickly write out the 3 strategies for player 1 first L1 M1 R1 and there are 4 strategies for player 2 and those are so now you know it starts getting a little tedious to start writing this because you have to represent his information see earlier the information was equal to the action okay earlier players information was equal to the action of player 1 so he knew exactly what player 1 had played now if I want to write something like this I have to write okay if player 1 has played L1 or M1 okay so you have to so the point is you have to find a way of representing it information can be represented in multiple equivalent ways okay so you can represent this for example you can simply say that either let us call this for example node X this box you will call it X and here it is Y okay so at X so gamma 1 is so this at X so he at X he plays L2 so at X and at Y at Y he plays M sorry R2 sorry R2 for example another possibility is that he plays R2 at X and L2 at Y and then there are the other strategies where he always plays L2 or always plays R2 regardless of where whether he is at X or Y clear so now once again I can create a table a normal form like this from for this game this is again a zero sum game clear so tell me what are the what should I fill in so if player 1 plays L1 player 2 follows it up and player 2 plays gamma 2 1 so let us write out the payoffs properly so player 1 plays L1 and then player 2 is going to play gamma 2 1 which means that at in this in this set X he is going to play L2 here regardless of which node he is at okay so he is so which means that if player 1 has played L1 then this guy is going to get to node X and he is going to then respond with L2 right so you get then what do I need to put here 3 okay then player 1 no no no no no so this is you need to be clear about this see player 1 has played L1 player 1 has played L1 but player 2 does not know that player 1 has played L1 the game has reached this node but player 2 does not know that the game has reached this node he is just playing L2 without that knowledge all right he is going to play L2 even at if the game had reached this node okay in this in this strategy gamma 2 1 okay so player 2 plays L2 then the game reaches this node this leaf node and the then the payoff is 3 okay let us do for gamma 2 2 L1 followed by gamma 2 2 gamma 2 2 is which means that at node X he is going to play R2 at at X whatever this bubble X he is going to play R2 so the sequence of actions then is L1 followed by R2 and that gets you to 1 okay then L1 followed by L1 and then gamma 2 3 which means again L1 followed by L2 that again gives you 3 L1 followed by a gamma 2 4 which is again L1 followed by R2 and that is 1 okay now now tell me for M1 minus 1 minus 1 see the thing so player 1 is playing M1 player 2 is still playing L2 here okay so again you get to minus 1 player 1 plays M1 player 2 plays gamma 2 2 that means he is going to play R2 at X then it is going to be 1 then again L2 here which means he is going to get minus 1 here and then there is a 1 correct okay now player 1 plays R1 so then I need to look at what player 2 is going to do at Y at Y he is going to play he is he is playing R2 so it is 0 in this he is going to play L2 at Y so that gives him 2 in gamma 2 3 he is going to play L2 throughout so that again gives him 2 and gamma 2 4 he is going to play R2 throughout so that gives him 0 okay so now tell me what is the saddle point here player 1 must play M1 okay and okay so this is and what is player 2 playing no there is no R2 the gamma what gamma 2 2 okay so that means let me see let us check this so if player 2 is playing gamma 2 2 then player 1 wants to play M1 okay player 1 is playing M1 then player 2 can play gamma 2 2 yeah that is okay so this is this is one saddle point any other any other saddle point no let us check let us be sure yeah there is no other saddle point this is the only saddle so what we can see here so firstly there is no there is no other saddle point the other is he has an interesting there is actually something interesting about we can think about this saddle point in a sort of interesting way so if you think about the game from the point of view of player 1 okay player 1 has these three actions L1 M1 R1 he also knows that player 2 cannot tell the difference between L1 and M1 he knows that player 2 will not know if he has played either L1 or M1 so in other words if player 1 plays either L1 or M1 player 2 will not know which of those has been played but he if he plays R1 he knows that it will he knows that player 2 will know that R1 has been played is this clear okay so the choice for player 1 in some sense is the following the choice is he has to say well should I play R1 and then player 1 and then player 2 will know that I have played R1 and then he will respond to that or should I not disclose or what I have played by playing either L1 or M1 so if he is going to play either L1 or M1 then for then this part of the tree this part of the tree is effectively for in that sub game what is called a sub game I will make all these things more formal but just look at this this part of the tree for the two players is effectively like a simultaneous move game because player 1 has not revealed what action is taken player 2 has not seen the action okay player 2 cannot see the action that player 1 has taken and so therefore for that sub game is no different form a simultaneous move game so the choice for player 1 is effectively should I engage with player 1 in by playing R1 and revealing in which case he would know that I have played R1 or should I engage with him in this simultaneous move game right so effectively that has that is the sort of choice that is that is that is the strategic choice so should so there are and there could be pluses and minuses in both of this here there is disclosure and therefore player 2 will play in a certain way here but the payoffs could be different in both cases here the player 2 does not know what player 1 has played so therefore he could play in a different way okay so if you just looked at this particular sub matrix the the sub game formed from this okay from just this part of the tree from just this part of the tree that is effectively just a simultaneous move simultaneous move game so we can I will just write that sub matrix here in this in that sub matrix player 1 has two actions L1 or M1 player 2 has two actions L2 or R2 right and what are the payoffs L1 followed by L2 gives you 3, L1 followed by R2 gives you 1, M1 followed by L2 gives you minus 1 and this is 1 okay so this is player 2 this is player 1 now in this game okay is there a saddle point in this game right so the saddle point is M1 comma R2 this is the saddle point right so what effect what has effectively happened is that well you can say well player 1 can now say that well if I get into this sub game here then effectively what I have to do be doing in this game is to play M1 and then player 1 player 2 would then be responding with with R2 or I can say I just do not get into this sub game at all and instead play R1 and then player 2 will respond will come to know and then he will respond with whatever okay he would respond with what here in this case I do not know we will have to check whatever it is yeah so player 2 will yeah player 2 will respond with L2 okay so if player 1 gets into this this simultaneous move game so engages player 2 in a simultaneous move game here then the rational outcome of that is that player 1 will is for player 1 to play M1 and player 2 to play R2 and then they would player 1 would get 1 okay if player 1 gets into this into this game where player 2 is going to observe what he has played then player 2 is going to respond with L2 and then he is going to get 2 so effectively then player 1 has the choice of either playing this simultaneous move game or playing this dynamic game or this you know this game where where player 2 can see what player 1 has done in this game he is going to get 1 in this game he is going to get 2 and effectively what therefore what he is doing is well he is choosing that 1 is better than 2 for player 1 he is the minimizing player so he is therefore he decides to go with this you can interpret the solution of this game in this sort of way that essentially that is what has happened is the player 1 has basically broken down this this game into these sub games and then try to say well let me analyze this one separately or this one separately and let us say what where should I be concentrating effectively right so this is this is so it turns out that we can actually do this sort of analysis for several games provided they are they have this kind of decomposability structure okay the only thing that you do miss out on is the is the is possibilities that that that that various sub games could get interlinked for example it is possible for player 2 to say well now it does not happen in this particular game but it is possible for player 2 to do the following so it is possible that player 2 says I ignore this information about x and y and regardless of what has been played by player 1 I am going to commit to playing either L2 or R2 throughout now what that does is it interlinks the two parts of the tree so then you cannot reason about that game you know subtree wise okay but in many cases as I said one of the ways of finding a solution can be done by doing this kind of a decomposition okay alright so alright so the normal form does not reveal anything like that right the normal form this could have been the normal form of a static game as well it says nothing in terms of the so the normal form is a way of finding all solutions right it does you can it does mean something so you can look at look at some part of it like for example this I think is the is that here okay but you have to be careful if you need to be able to number your strategies appropriately for you to get a clean sub matrix and so on out of it otherwise you know they could be jumbled up it really depends on how you have enumerated the strategies with this now I can define for you what an extensive form actually is