 So, last time we wrote out this theorem which is let G hat be obtained from G by the elimination a weakly dominated. So, let G hat be obtained from G by the elimination of a weakly dominated strategy S hat J for a player J every equilibrium of G hat is an equilibrium of G. So, recall what the setting was we had G which was a game comprising of n players these were the strategy sets and these were their utilities. This is when I put these curly brackets around ESI means that I am looking at S1 to SN similarly curly brackets around UI is U1 to UN. So, this was your game G and then we considered another game G hat which is formed from the same set of players and these strategy sets S hat I and the same utilities UI and what was the connection between S hat and S hat I. So, S hat I was actually equal to SI for all I except for a specific player J and for player J it was equal to X SJ minus X hat J that chain my notations this was I use the notation X hat J. So, for this is for I equal to J right. So, in other words we all other player strategy sets were held the same it was player J strategy set that we change and how did we change it we remove one particular strategy X hat J from this and what was special about X hat J? X hat J was so X hat J is a weakly dominated strategy and what does the theorem say the theorem basically says that every equilibrium of G is G hat is an equilibrium of G which means that you look at the smaller game the reduced game that is formed from after elimination then all the equilibria that are there of this game are also equilibria of the original game and the we summarize this by saying that no that essentially if you eliminate weakly dominated strategies no new equilibria get created ok. So, if you eliminate weakly dominated strategies then no new equilibria get created ok. So, now let us see so this only says that no new no new equilibria get created all the equilibrium of your of G hat were already present in G, but you could have potentially lost equilibrium right. So, which now when you are losing equilibria it means that there is some possible strategic outcome of the game which has now which has been eliminated because you decided to ignore the particular strategy for a particular player right or you do not ignore I would say overlook a particular strategy for a particular player right. So, now question is when can we say that even that no equilibria are lost as well the non-equilibria are created, but can we ensure that no new no equilibria are lost in this process. So, that is essentially the main thing that we are building towards theorem. So, if x hat j if the above x hat j is strictly dominated then the set of equilibria of G is equal to the set of equilibria of G hat. So, this is the this is what you what we wanted to this is the main result of this particular exercise that if you eliminate a strictly dominated strategy for any player then the smaller game will have the same set of equilibria as the original game. So, neither are the no new equilibria are formed and no equilibria of the original game are lost in the process. So, let us prove this. So, as far as proving this is concerned we already know that no new equilibria are created. So, it suffices to show that no new no equilibria are lost. So, it suffices to show that if x star is an equilibrium of which one of G the original one then x star. Now, I want to show that it is an equilibrium of G hat, but remember we also showed another small lemma which said that if you take if you have a big game and I consider a smaller game and if the equilibrium of the origin of the bigger game is present in the smaller game then it is necessarily an equilibrium of the smaller game. So, it suffices to show that that if x star is an equilibrium of G then x star just has to be present in G hat then that is enough then it follows from there that x star is an equilibrium of G hat then x star is a strategy profile in G hat. So, in other words what do we need to argue that we need to argue that if x star is an equilibrium and x hat j is the strategy that you eliminated and x the eliminated strategy could not have been part of any equilibrium. If that is that can be argued then you are done because then you are sure that the equilibrium is present after even after elimination. So, which implies that it suffices to show that is not an eliminated strategy. So, what strategy profiles get eliminated as a result of elimination of x hat j? Now x hat j is a strategy of player j. So, x hat j gets eliminated for that player, but profiles strategy profiles that means for n players what gets eliminated? Yeah x hat j comma x minus j for any x minus j all other combinations the others can play whatever they want all of that gets eliminated. So, when you are removing that particular row for that player every combination with every other every column or whatever you however you want to think of it all of that gets eliminated. Now, let t j be a strategy dominates x hat j. What does this mean? That means regardless of what the others play t j is better for player j than x hat j. So, that means if I look at u j of t j comma x minus j this is strictly better than u j of x hat j comma x minus j and this is true for all x minus j in s minus j. So, now what do I do here? I just put in now this has to be true for every x minus j. So, I can put in my favorite x minus j and in this case the choice is natural I just put x minus j as x minus j star the equilibrium strategy the part of the profile from the equilibrium. So, put x minus j as x minus j star and that gives you that u j of t j comma x minus j star is strictly less than u j of x hat j x minus j star. Let t j equal to no we cannot take t j equal to x j star because x star is an equilibrium and t j is just a strategy that we know it dominates x hat j. So, we have taken that x star is an equilibrium of g. So, and t j where is t j again remember t j is a strategy for in the original game because it was dominated x hat j from that original. So, now x star is being an equilibrium since x star is an equilibrium is a Nash equilibrium of g implies what u j of x star j comma x minus j star is less than equal to u j of t j comma x minus j star. And this in turn is we know that this in turn is from here we know that this is actually less than equal to u j of x hat j comma x minus j star. What does this mean? So, we now have that the left hand side here is strictly less than the right hand what is the meaning of that? This means that basically you are this x see what is common here x star minus j is here x minus j star is here as well. So, the inequality is because of this one here we are still looking at u j the inequality is because of this x star j and x hat j means that these two cannot have could not have been equal because if they were equal then you would have got you could not have gotten a strict inequality here which means that x star j is not equal to x hat j. So, which means x star j is not equal to x hat j which means what? Which means actually x star is present in the as a strategy as a profile in g hat because x star j is not x hat j it is not the eliminated one. So, it is definitely present in the in the game that you get after elimination. So, consequently you get we have that the set of equilibria remain of g is equal to the set of equilibria of g hat. So, what this is what this is told us is that now that the elimination of strictly dominated strategies is with you know there is no punishment for that you there is no nothing you lose you not you do not create fictitious new equilibria nor do you lose any equilibria as a result of this. Is this clear? So, this basically justifies for us the whole process of iteratively eliminated eliminating strictly dominated strategies the exactly the thing that we use to solve for solve for the solve the game the in our previous class this justifies it that essentially after doing this elimination if when you are left with only one only one particular strategy that necessarily has to be the Nash equilibrium of that of the original game right. So, actually let us just write this down formally. So, if the so corollary if the process of elimination of strictly dominated strategies results in single strategy profile profile x star then x star is a is a Nash equilibrium of the original game. So, if you I should I should have made use the word iterated elimination if that is implicit here that if you keep doing this elimination again and again and you are left with a single strategy at the end then that has to be a Nash equilibrium of the original game. Why is that the case? Because we just saw that every time you eliminate you are neither losing equilibria nor creating equilibria and this happens at every step right and so finally if you are left with just one single strategy profile right one strategy for each player then it has to be that that itself that particular one that was left is necessarily the Nash equilibrium of the previous the immediate previous game and also of subsequent all the previous predecessors of that right you in short the original game that you started yes. So, I will tell you so if this happens then it is a Nash equilibrium of the original game. So, this automatically guarantees also that if you have if you end up with one then this is also a proof of existence that is definitely a Nash equilibrium of the original game right because when you are left with just one strategy that one strategy profile you have a trivial game and that that one point then is the Nash equilibrium of that G hat of the reduced game that you have found and that is by the theorem is equal to the Nash equilibrium of the yes. So, that is what I just so I can show you examples where I will come to that. So, stick less is required if you eliminate weakly dominated strategies this result is not true anymore okay you will lose equilibrium in the process yeah. So, if you end up with one it is it is it is necessarily an equilibrium then it is not then the statement is not true anymore then it is not true that no it is a Nash equilibrium of the original game okay, but it is what all equilibria are not retained. So, I will there is in fact a stronger version of this since Shashank has bought this up let me mention this. So, can I stand in this by saying something more is a unique Nash equilibrium right is the unique is rather than a is the unique Nash equilibrium of the original game. So, if you if after weekly the elimination of weakly dominated strategies if you are left with a single one then that is necessarily a Nash equilibrium of the original game correct that much holds, but that it is the only Nash equilibrium is not true anymore okay. So, it is the so when you are left with a single profile you have effectively then by this process discovered the Nash equilibrium of the game okay. Now the another corollary of this is actually which is implicit in the way I have stated this theorem see if it so happens that you are getting eventually to one strategy profile and that is the unique that then turns out to be the unique Nash equilibrium of the original game. Now suppose you eliminated strategies in a different order okay and then ended up again at this x star right what would that that would again tell you that it is the it is the Nash equilibrium it is actually the unique Nash equilibrium of the original game what this means is essentially the if you are ending up at a unique strategy profile then the order of elimination does not matter okay because even if you in fact forget ending up with a unique strategy forget about ending up with a unique strategy profile the very fact that you neither lose equilibrium or gain equilibrium means that you can eliminate in any order the smaller game will always have the same set of equilibria as the original game okay. Now the order this invariance of with the order of elimination is again not true for weakly dominated strategies once you have if you are eliminating weakly dominated strategies the equilibria that you end up with right because you lose you potentially lose some equilibria the equilibria that you end up with is sensitive to the order in which you have in which you have eliminated okay so let us actually do one example I yes but they would they would all have the same set of equilibria as the original one yeah so I have another example okay so layer 2 layer 1 top middle bottom and so okay so player 1 has so this is the game player 1 has 3 strategies top middle and bottom player 2 has 3 strategies left center and right okay and the payoffs are this 1 comma 2 2 comma 3 0 comma 3 2 comma 2 2 comma 1 3 comma 2 2 comma 1 0 comma 0 1 comma 0 and again both are both players are maximizing yeah so there are there are several options so let us let us there are strategies here in this game that are weakly dominating other strategies okay so and what I the point I want to make is that actually the order in which you eliminate will matter so let us let us go through one or two different orders okay so first let us first first observe let us take this so t is actually dom weakly dominated by m right so I will just write it like this you know t is weakly dominated by m for player 1 is that clear why because you are comparing 2 strictly greater than 1 2 equal to 2 3 greater than strictly greater than 0 okay so t is weakly dominated by m so but there is weak dominance so because in this case you get the t and m give you the same thing okay okay so t weakly dominates m now so suppose I then suppose I did eliminate t okay suppose I did eliminate t then in that case what would I what would I be left with I would be left with this smaller box here this one okay so now in this anything dominates on anything else yeah r actually dominates l again weakly 2 equals 2 sorry l dominates r sorry minus 2 equals 2 and 1 is greater than 0 so l dominates r okay so l dominates r I can remove I can suppose I remove r here now I am left with this smaller box where now m weakly dominates b so you can again remove this okay and then c is dominated by l and so you are left with an equilibrium which is followed by r dominated by l followed by b dominated by m followed by c which is dominated by l and that is that leaves us with the equilibrium ml okay so ml is a nice equilibrium of the original game this that give that came about from this order now let us see if we can go through another order so now let us suppose we take the let us take another order let us take the order b is is actually dominated by by m right so I can eliminate b so b is dominated by m so get rid of b first okay I am not going to keep making marks now so let us let us remove yeah b is removed so you are left with just the upper box okay now in this anything dominates anything else r dominates c let us see r dominates c okay let us let us see what happens r dominates c so c less than equal to r okay r dominates c so I have removed you can remove c then and you are left with then p m and and l l l and r okay r actually then dominates l okay so then l is gone so actually r dominates both in this case r dominates c as well as l right after having removed b r dominates both c and l right so you can get rid of both so l okay so l and c are both been removed and now you can come you have to just compare t and m in the in column r so then obviously t is worse than m you are left with so you can remove that gives you then m r is a Nash equilibrium okay so you can see here what happened was when we eliminated l actually the previous Nash equilibrium which was m l previous means the one from the earlier earlier order of elimination that Nash equilibrium is now got eliminated when you eliminated l whereas and and and you are left with m r as a Nash equilibrium is clear so so what does this mean essentially it means that the reason this has happened is because you are eliminating weakly dominated strategies there are these ties here right so for for if you take m m l and m r for example for the for the for player 2 they both they both give payoff 2 right but you said that r dominates l and you removed removed l but actually but it was it was strictly dominating only when player 1 was playing t then player 1 was playing m they were equal and he eventually ended up playing m in the equilibrium and that is why you lost out on one equilibrium right okay so in other words the set of equilibria the set of equilibria of the reduced game obtained by elimination of weakly dominated strategies is dependent on the order of elimination does the remainder set of profile yes it could it could change based on the order because you may yeah you could that could happen but of course if by by elimination of strictly dominated strategies I I mean you follow the same process you means you eliminate strictly dominated strategies of course you if you eliminate the exact same strategies regardless of the order you will get the same profile but if you are so in the in other words you eliminate strictly dominated strategies but you all eliminate them in any particular order and therefore you are not eliminating the same set okay then you will you you could get yeah could be could be I am not sure but I think one can construct an example where you eliminate something prematurely that could happen I think yeah yeah so I mean what a player we are not analyzing this as players right so we have as observers of the game we have eliminated this on behalf of the players okay yeah so all of these are Nash equilibria there are actually more you can play around with this game a little bit more you will see there are more Nash equilibria also you can come up with other orders of elimination and so all of them are Nash equilibria it is our our effort at trying to arrive at one of them by elimination which is the problem so you you basically we had no we did not have as much justification to eliminate weakly dominated strategies and as a result we have not properly analyzed the game that is what the conclusion is yeah no not at all not at all so if if you have a unique Nash equation is if there is a unique Nash equilibrium can you necessarily find it by elimination of you know dominated strategies no so you will you will in general end up with a game you will often have a game where there is in fact generically you will have a game that has no dominated strategies but does have a Nash equilibrium some dominating strategies but an Nash equilibrium does not exist yes yeah that is possible okay so what this is basically done is now it has helped us in some cases in certain types of problems wherever this process actually converges to a eventually one unique strategy profile in that case we have basically through a logical elimination you know solved for the solution of the game and it turns out that actually coincides with the Nash equilibrium itself okay so this this way of solving a game you know like a puzzle trying to say okay this would happen then okay then that would happen and so on and so forth almost you know trying to like a logical logical puzzle this way you can you can do in up to this up to this point when the game has the property that it ends up eventually this process of strictly eliminating strictly dominant strategies ends up end you with one strategy profile okay now in general what you would be left with is some is a set of strategy is basically a reduced game from which you have removed all these dominated strategies and but that is it you cannot analyze that any further with this logic okay and that is where you need you need a leap of logic which is which is what the Nash equilibrium does that it tries to learn reason about okay what are the communication requirements in the game and so on and so forth okay so you go into that you need to get into that whole line of thinking once you get once you have done this kind of basic elimination okay all right okay