 So, first let us we need to talk of informational inferiority or informational inferior games yeah, we had seen it before but I had to if you remember what I had to so far we saw it in the yes. So, we have seen this before we saw it in the case of the quantity competition where static Nash equilibrium was also a Nash equilibrium of this game right of the dynamic game that we saw it there. We also kind of saw it in this in the L1, L2, R1, R2 game and over there and it had the but over there what I had to do is I had to sort of introduce those fake actions because after player 1 played L1 there was nothing for player 2 to do. So, this was this we have seen elements of this before yes. I should also let us let us all another thing to point out here actually before I get to in informational inferiority can you tell me what are the what is the payoff in this in the in the starred equilibrium what are what are players getting. So, player 1 is getting 0 and player 2 is getting minus 1 right. So, whereas in the earlier game player 1 was getting minus 1 and player 2 was getting 0 right. So, you can see that these two firstly the second the first equilibrium the earlier equilibrium that we calculated we calculated in you know based on a certain logic and it gave it gave the player 1 this particular payoff. But there is also this other equilibrium which comes from the inferior game and in that game player 2 is actually getting is being is being better off okay. Now, this is this happened in an earlier example also this is it is not a general rule that you know being forgetting information or sort of ignoring in not forgetting ignoring information actually benefits a player there is no such rule. But here is again a case where such a thing is actually happening where lack of information or in other words not having this information actually is beneficial for that player okay because this is this is the game that player 2 would have player 2 would have been part of if he did not have this information it is should be the simultaneous movie. But the resulting with the payoff that he gets in the resulting game is actually is actually better because he is getting minus 1 here and in as compared to 0 here okay alright okay yeah no no no no so that is so he is not changing his strategy this is another equilibrium okay these are two potential outcomes of the game one outcome is that player 1 plays R1 and player 2 plays gamma 2 3 the other outcome is player 1 plays L1 and player 2 plays gamma 2 1 okay these are two different outcomes now about what Shashank said whether player 1's information has changed see there are two different pieces of information I mentioned this earlier also in the see one is information which is about the game okay about the structure of the game that has changed that if you look at this game compared and compare that with this one these are obviously the two games are different okay that is that is that is the case but the point is that this is an equilibrium of of this game okay so so it is not it is it also happens to be mathematically the case that it also can be like is basically mapped to an equilibrium from an inferior game that is that is one that is one point the other thing that so but as far as you know if you want to compare in information that is present to a that is available to a player during gameplay that is the one that is different in between the two games for player 2 for player 1 during the gameplay there is no change in information okay for player 1 he still starts with null information in both games player 2 just has lesser information here so when we talk of informational inferiority and so on right this is about what what players know intra game during the gameplay okay so first for this you need to suppose you have two partitions okay of a set so let us take a set S and suppose there are two partitions and let us call this part these partitions P1 and P2 these are partitions of S okay now when do we say that P1 is finer than P2 every set in P2 okay so so we say P1 is finer than P2 if every element of P1 is a subset of an element of P2 so let us take this set S and I have suppose partitioned it in in this kind of way to get let us say this is this black partition is say P2 then P1 then P1 is a finer version of this so which what this means is that if you take any element of P1 it is always contained completely in some element of P2 okay so which means something like this this here is a partition a finer partition clear now this is many people make this mistake you do not know if I want to just talk about it but you see many people make the mistake that of defining it the other way around if every element of P2 contains an element of P1 that is not the same same right so essentially what you need is when you want P1 to be finer than P2 it has to be that whatever information you get in P1 okay whenever you have whenever whatever information you have in P1 helps you also conclude the same information in P2 okay so it helps you also conclude what you had in P2 is this clear so which means that P1 is a more specific version as compared to P2 okay this gives you sharper finer information than P2 now if you take the important thing therefore is that you cannot have situations like this you cannot have situations where let us say I will just write this blue one is P2 P1 so if you cannot have a situation where a set like this of P1 is stretching across two sets of P2 that is not allowed because then there will be you know there I mean there is some information that was there in P2 which is not present necessarily in P1 okay so everything that you knew in P2 is also known to you in P1 okay so that is effectively the point okay so so now what you can do is so if you suppose you have a single act game suppose you have a single act game let us say denoted 1 okay now what you can do is create from it suppose you have a single act game 1 okay now when would you say that another single act game 2 okay when would you say that when is a single act game 2 informationally inferior informationally inferior to 1 so which is the one that has fine so firstly so do you have say a single act game 1 and another single act game 2 when do you say that 2 is informationally inferior to 1 information set is an element of the partition it is just called the whole thing is a player set partitioned into information sets no so consider 2 so I had a notation for this I do not remember did I have n i just check n i or i i I think for information set of information sets of player i yeah yeah so we can do you can use that notation okay so consider 2 single act games I will just call them a and b defined on the same game 3 with the same player sets okay so you have 2 single act games a and b that are defined on the same game 3 and they have the same player sets okay if the player sets are different with it does not make sense to talk of comparing them informationally because the the nodes at which the players are playing are themselves going to be different right so the player sets are the same the tree is the same okay which and consequently payoffs and all that are also the same only thing that is going to be different is is the information sets across the players okay so let a i and i a b be the set of information sets player i in games a and b okay so then we say a is informationally inferior if for all for all i and n and for all eta i b in i i b there exists eta i a in i i a such that eta i b is a subset of eta i a okay so we say consider 2 single act games a and b defined on the same game 3 with same player sets and let these be let i a i a and i i b be the information sets of player i in n okay in games a and b we say a is informationally inferior to b if for all i in n that means for every player and you have that the information the the info the set of information sets as a partition of his of player i is player set is in b is finer than that in a okay b has a finer information partition okay all right so suppose now you have two games like this a and b is informationally inferior to b now what can you say about the strategy sets of a player which game would have more strategies the superior one right the richer one will have more strategies obviously because he has more information sets there he has more options he has more options there so play so so game b will always so for every player he will have more strategies in game b then he would have in then he would have in game a now one thing here also remember that when we are talking of another okay I forgot to mention this point see when we are talking of two games on the same game tree and one is you know inferior compared to the other what this means is that the inferior game you can say has in some in some sense being formed by joining together information sets of the richer game just like we did just like we had here right this information set was created out of two information sets from a richer game so but you may it may not always be possible you cannot just always just arbitrarily combine information sets because to begin with you need to have the same set of actions at each node in each information set otherwise it is not even feasible to combine so it is possible that you do not have infam inferior games at all so for example here in this case suppose just imagine there was imagine there was another you know another action here let us say at this node suppose there was another action to for this player then I could not put I cannot combine these two into one information set because then it would violate the definition of of an information set right I cannot put these two in the same information set so for me to get an informational inferior game I need to have nodes like that that can be combined into one information set okay so that is that is that is one requirement so the way we define therefore informational inferior games is we do not simply say that we you put together combine information sets we start with two games and we say one is inferior to the other if the information sets follow this property that there is this that the partition in one is richer than the partition in the other okay all right now if you now that so if B has is superior as compared to A then for every player there are more strategies and B than as compared to A okay can you say more than actually not only the number of strategies what can you say something more yeah yeah so every so the space of strategies is different now because after all strategies are going to be mappings from information sets to actions and information sets themselves have changed but because these information sets have gotten combined like this right you can basically do the following you can every strategy in their in the inferior game has an you can say has a equivalent strategy in the richer game so the inferior game basically involves ignoring information so this this particular one this strategy where you where the player 2 played L2 we said was essentially this strategy right where he was playing gamma 2 1 it was the constant strategy here here he was playing L2 as a as an action and there was only one information set here he was playing L2 and L2 at the two information set but they were they were equivalent in the sense that they led to the same game same path through the game tree okay so yeah so that is a so it should lead to the same history the same game so it should lead to the same path for assuming all for every strategy of the other players you should have it does the path that it generates should be the same then the two strategies are equivalent so so I mean we define so if you go to this sort of level right I mean you have to get physics into it essentially you have to talk of the history time evolution what is the what is the exact sequence of events and then talk in terms of that so that because the game tree records the entire history of the game so you have to you know go in that kind of almost thermodynamic language type language so notice that if A is informationally inferior to B then we have then for all players I we will have this property that if I take the set of strategies of player I in game A I will just put this in quotes it is it is a subset of the set of strategies that player I has in game B okay and this this thing this is in quotes because I you know I technically you cannot but you can you can map them as an equivalent strategy in player in game B okay so that is that is why this is in code okay alright so so now let us write out the theorem let A be an n person so let A be an n person single act game that is informationally inferior to another single act game B okay so what is being presumed here therefore is that these two games are on the same game tree they have the same players at same payoffs and so on okay except that now A is informationally inferior to B then first any Nash equilibrium of A also constitutes a Nash equilibrium of B and second second is this observation that we had here see when we looked at this you remember we I just looked at what was this matrix where did we get this matrix from this matrix just came from the first two columns of from here right so now if if I if I look at a if I if I see this this game and I find there is this equilibrium here and I look at this equilibrium and say well well this equilibrium actually involves player 1 playing L1 and player 2 playing a constant strategy L2 at each information set well if in that case I can actually say that well this strategy L2 is actually implementable also in an inferior game right this one gamma 2 3 is not implementable in the inferior game because it requires player 2 to change its action based on the information whereas this strategy is implementable in the inferior game from this itself I should be able to conclude that this is in fact an an equilibrium of the inferior game okay so if you have a strategy like this if you have a Nash equilibrium like this in which the strategies of the players are actually subsets are are actually elements of the strategy set of an inferior game okay so just you have a you have a richer game you find this Nash equilibrium and that Nash equilibrium has the property that the strategies are in fact strategies of that are implementable in an inferior game then this equilibrium is also an equilibrium of the inferior game so this is essentially a converse result okay so every equilibrium of the inferior game does carry over conversely if you find an equilibrium in the richer game whose strategies are implementable in the inferior game for every player then it is also an equilibrium in the inferior in the inferior game okay so if gamma 1 star to gamma n star is a Nash equilibrium of the richer game B such that gamma i star belongs to gamma i a for all i then gamma 1 star to gamma n star is an Nash equilibrium of a okay all right so so we will prove this now today so can you tell me what would be the logic why is it that an equilibrium of an inferior game carries over as an equilibrium in the richer game no no no see remember it is it is B so that will help you prove 2 okay but not 1 so 2 is trivial in that sense because B has fewer strategy has B has more strategies than so the richer game has more strategies okay because of this subset relation that I wrote here it has to be so B is B is every player has more strategies in B so if a strategy is optimal for that player over a larger set it will be optimal also over a smaller set only thing we need to check is if it is available in the smaller set okay this is something we have seen before also you know in the dominance and so on okay but why is the why is one true see we basically what you so the idea is essentially this see player in in the in the inferior game player is ignoring information okay and he is picking some strategy and that turns out to be the best response given what the others are playing okay now if the question is this so the entire profile of strategies is available in the richer game also the question is is this is there a better response available in the richer game if so if there was a better response in the richer game then this one would not be the best response in the richer game is this clear so you take so let us take 2 players for simplicity player 1 player 2 player 2 is playing a strategy in the inferior game okay in response to what player 1 player 1 has played okay now move to now both these strategies can be ported to the richer game okay they are available as strategies in the richer game now in the richer game player 2 say has has now player 2 has potentially more strategies in the richer game now with these additional options does he have a better response than what he was playing earlier that is the essentially what we need to show because if he has a better response then this equilibrium property is lost right so can you argue that there cannot be a better response yeah so for again it is a response so it is for given what the others are playing why but why so with so he has more options now why cannot he do better yeah yeah so there is always a that yeah of course this gets copied that is correct so but the point is see the question is this stuff here this matrix comes up here right question is see player 2 now has more options here so yeah yeah this additional stuff why is it useless okay so good good so so actually this is this is this is actually the hint see basically you what you are seeing here is that this stuff gets copied here the entire matrix does not get entire two columns do not get copied but if you see these two get repeated what does this mean that suppose there was something better here okay suppose there was a strategy which was better what would that strategy do that mean what it would do is add some information set okay a player would take a different action than he was taking at the in the inferior game right in the inferior game he was taking some action at each information set now at there is now those informations in the richer game the information sets are split up he has some additional information and with the in the basis of that additional information he is now taking a different action somewhere but the point is that this is a single act game okay so what happens therefore is that if he was taking a different action there along that particular path then he could have committed to that action even with even in the inferior game because that that taking that constant action was available to him over there also so he could have if the in whatever you know new strategy you have could have been implemented as a constant strategy in the in the inferior game and if he and he would he could and he would then you know in retrospect basically have played what what would have you know whatever would have been more beneficial to him in retrospect he could have just simply played that in the in the inferior game okay so let us just let us just do this formally so so suppose so actually the second one is trivial so I will just focus on the first one the first we need to show this right so let gamma 1 star to gamma n star the Nash equilibrium of a yeah of course of course in general that is exactly the point so richer game will have many more Nash equilibrium okay in general it will have so it will have all these things that it inherits from inferior games but it will have Nash equilibria of its own which are not present in inferior games okay not for sure obviously yeah but in general that it will because you can always have you know pathological cases like that but in general it would okay so in fact this is one of the arguments people use for eliminating you know threat type equilibria saying that threat equilibria are actually just you know fake equilibria they are coming up from then they are in fact equilibria of an inferior game that just happened to show up here okay so because see the essentially the the you can say the sort of the quintessential equilibrium of the dynamic game is the one that uses all its information everything else is an equilibrium in fact of a inferior game which is just happening it is just showing up here because of the you know way we have defined the equilibrium yeah you can say threatening I mean yeah depending on the cost function that is right yeah so that is what is called sub game perfectness so that is essentially then game is called an equilibrium is called sub game perfect if it is if it induces an equilibrium on every sub game okay so this one here this equilibrium here is sub game perfect because what it does is on every sub game it is in fact an equilibrium so it is an equilibrium on this sub game and also on this sub game so that is a way of refining equilibria that is also one argument yeah sub game needs to be defined I have to first define a sub extensive form and a sub extensive form is one where the you know the information sets do not cross across across sub trees so you take sub trees and but the information sets should not cross across them then it is a sub game a sub extensive form and then that comprises of sub games and so on okay so let gamma 1 star to gamma n star be an ash equilibrium of a and so we we have of course that gamma i star belongs to capital gamma i b because it is a it is present in the it is present as a strategy in the richer game okay now assume let us go by contradiction so now assume that gamma 1 star to gamma n star is not an ash equilibrium of of p which means there exists player i in n okay and a gamma i hat let us say in gamma i b such that if you look at this fellow gamma hat i comma gamma minus i star this is better than gamma i star comma gamma minus i star okay so gamma hat i is better than gamma gamma i star which is what he was playing in the ash equilibrium of of the inferior game okay if you remember what I told you if I give you an n tuple of strategies or give you a strategy profile it defines for me a unique path in the tree it is starting from the root to a particular leaf node so this now is a defines a path for me right this this profile defines for me a path okay so this gamma hat i comma gamma minus i star defines a path and it is a path in in the richer game b now so since this defines for me a path and I said this is a single act game which means that this path has to intersect the information the player set of each player at most once okay now in this case player i is actually shifting and benefiting so he is played along this path he has in fact played along this path if he was not playing along this path there is no way he is his his shift would have mattered right so which means that there exists a unique information set the i b in i i b intersecting so there is this particular path that that is that is passing and it has to intersect player i is information set so some information set of player i and it and in fact exactly one information set of player i because it is a single act game okay so b is the richer game right so if every information set in b is a subset of some information set in game a right the richer one is a subset of some something in the in the coarser one in right so which means that there is an information set in game game a which is a superset of eta i b since a is inferior to b inferior to b there exists an eta i a in i i a so there is an information set of player i in game a such that eta i b is a is a subset of this right so there is a larger set so basically eta i b is so you have this situation where you have information set eta i b and that is in fact contained completely in another information set eta eta i a okay so maybe I will just draw this here so this is and this is enough to this guy here is eta i b and this is eta i a so now the game has come to this path okay come through this path in fact it has come through a specific node here right and at that node and and in particular at that information set player player i chose to take a chose to take a different action than what gamma i star was telling him to take right and that is how you have got gamma i hat okay so now what I can do is I can define a gamma i tilde which is same as gamma i star everywhere else but on this information set that means on eta i a yeah sorry here he has more information so what he yeah so yeah so what I yes so what I can do is define a gamma i uh uh gamma tilde so define a gamma i tilde gamma i tilde which is going to be the same as gamma i star everywhere else but on eta i a okay it replicates the action that I took on in gamma i hat correct so here suppose at this node I was taking an action suppose I was taking an action here suppose you know an action l suppose here okay in in gamma so gamma hat i tells me to take action gamma hat i in of eta i b this tells me to take action l whereas gamma star was asking me to tell take action something else let us say okay so gamma i star was asking me to take this action gamma i star of now what I can do is this this action l is also available at eta i a because eta i a is a larger information set than eta i b so all the actions that are available at every node in eta i b are also available as actions everywhere else in eta i eta i a okay because it is form it is an inferior one so r is also available there so I can define a gamma tilde that basically does this it is the same as gamma i star on the rest of the tree all these other parts here it just does same as what gamma i star was doing but when it comes to this one it sees that oh r was better so let me just play r okay so on this information set here basically on this information set it plays r at every node in this information set now look at the path that gamma tilde i along with gamma minus i star is that what is the path that I am going to get with gamma tilde i and gamma minus i star basically this is a single act game so all the players that have played before me are present in minus i so I will still come to this same node the game will still come to this node itself at this node earlier instead of where I was taking l now I will take r right and thereafter that is my turn to play after that I do not get to play it is a single act game and the game then proceeds again based on what minus i star is doing and so what will happen therefore I will get the same payoff as I would get in gamma hat i from gamma tilde but gamma tilde is present as a strategy in the inferior game gamma hat i was a strategy in the richer game okay it was making use of this information and refining the action but that refined action I can just take blindly across the entire information set eta i okay and it leads to the same path and it will give me the same cost player i gets the same cost so in short what we get is a contradiction that i the gamma star could not have also been an equilibrium because there is a better strategy even in the earlier game gamma tilde there is a better strategy gamma tilde i for the player i in even in the even in the game where we started off and that is your contradiction yeah so no because then you will have to make them into a team the the point is the no no no no it is it is not the same because you have to respect the so a team of n players each having different information is not the same as one player it will it will you can potentially but it will become a very messy thing essentially every player acting distinctly will become a different player but then they have so it becomes a team versus team type of setting okay so where is therefore the so let us write out the construction of okay gamma so gamma tilde i okay so let me write this node right this node where we reach let us call this node let us call this node x okay x is the node intersecting this path this is that node okay so eta i a is so x actually belongs to eta i okay so now gamma tilde i is defined in this way so gamma tilde i is equal to for eta i is equal to gamma star i for all eta i not equal to eta i a and tilde i at eta i a actually takes the same action as gamma hat i took at eta i b this implies the if you take the cost of gamma tilde comma gamma minus i gamma minus i star that is equal to the cost to player i from gamma hat i comma gamma minus i star and this guy is in fact strictly less than gamma i star comma gamma minus i star contradiction to gamma so if this contradicts that this is an ash equilibrium of a which means that this is also an ash equilibrium of b okay so that is effectively the idea okay so once again the point is upon reaching x if you had a better action with additional information then you could have taken that action even without the information and you would have had the same path and the same cost that is basically the point so this thing right if where this thing where the player in retrospect is essentially if we hired the information I would have played differently that is in fact being incorporated as a constant action even when you did not have the information and because it is a single ad game the whole thing follows through actually a fact like this is also true for multi ad games it is just much more messier to show okay and there you have to you know you have to allow for all kinds of calculations where you know players forget and this and that so we will not do that but this is this this is true in general that the equilibria of inferior games will carry over as equilibria of richer games regardless of what the you know how many times players play okay all right