 A strategy xi in layer i strategy set is said to be strictly dominated for layer i if there exists a strategy xi dash in a xi such that this I am writing in the language of players minimizing. So, this is xi dash comma x minus i is strictly less than ui of xi comma x minus i and this has to be true for all x minus i in the set s minus i. What is their set s minus i? S minus i is simply the product over j of set sj j which is not equal to 1. So, a strategy xi is said to be strictly dominated for player i if there is another strategy that is better for this player regardless of what the other players play. xi dash is always better than xi and strictly so. Now, if the inequality above is weak that means it holds with equality for at least 1 x minus i then we say that then xi is said to be weakly dominated. So, that means if this inequality here is not strict for all x minus i that means that it is going to be at least one case where some other player the other players play some profile of strategies x minus i in which player i is indifferent between playing xi and xi dash. There is some case where these are the payoffs are going to be the same in that case we say it is weakly dominated. So, then you have a less than equal to here rather than a strict less than. Now, when you have a strictly dominated strategy then rationality directly tells you that that player will not play the strictly dominated one. The weakly dominated is trickier because weakly dominated just means that there is still going to be a possibility that he will play one of these two in some case. So, it is much trickier to say anything about weakly dominated strategy but if you have strict dominance like this then rationality directly tells you that player i will player i himself will not play that. Now, how to make use of this we will come to that in a moment. Yes, so I will come to that. So, in fact when you are so this elimination of dominated strategies if this has this property that so I was just defining this but essentially now what you can show is the following. Now, if you assume that then you can eliminate strictly dominated strategies recursively one after the other. So, long as they are available to be eliminated you can eliminate and it does not matter what order in which you eliminate. Regardless of the order in which you eliminate you will actually end up with the same outcome and finally if you end up with a unique strategy profile at the end of it all it turns out that profile has to be the natural. So, if you can eliminate strictly dominated if you find strictly dominated strategies in such and in such a way that you keep eliminating and finally you are left with just one strategy profile like it was in this case. If you go out here I can now eliminate with one more level of assumption I can eliminate also L for player 2 and then I am left with just 6 comma 2. You can check that 6 comma 2 is in fact a natural equilibrium and in fact the only natural equilibrium oh sorry sorry my mistake sorry 4 comma 3 u comma L is the minus 4 comma 3 you can eliminate R for the sake of recording I will repeat this. So, yeah so in this case for if you you can you can do one more level of assumption and eliminate eliminate the strategy R for player 2 and then what you are left with is then we will just the strategy L for player 2 and a payoff of 4 comma 3 and you can show that 4 comma 3 is in fact the natural equilibrium and in fact the only natural equilibrium of this game ok. So, this is the this was our definition of a strictly dominated strategy alongside this I also need a definition for for this for this this assumption that we made ok. Now the assumption that we made was that you remember here we said we assumed that each player each player knows that each player is rational and then we assume that each player knows that each player knows that each player is rational ok and you can continue this recursion further and further and actually go add infinite on this ok. So, and this is not just about rationality we can assume this we can make this assumption we can talk of this kind of recursive knowledge about any particular fact associated with the game all right. So, that is that is what I will define right now. So, a fact or an event an event is said to be common knowledge among players P 1 to P n if for any finite chain I 1 till I k. So, these guys are these are indices for players from these are from my player set the following holds. Player I 1 knows that player I 2 knows that dot dot dot player I k knows knows even. Now this is this is the definition of an event to be common knowledge. So, an event is said to be common knowledge if for every sequence of that you can take you can construct this sequence of players I 1 to I k you can repeat players also along that sequence you can the sequence can be of any length right and you should the following must be true that player I 1 knows that player I 2 knows that player I dot dot dot player I k knows that event ok. Now what does it mean to know is itself something I have not defined. So, that again needs a model we will we will come to that in maybe in the next lecture or something, but essentially this is what it means for something to be common knowledge. So, when something is common knowledge it is it is not only that the player know each player knows about that particular event, but it is that also the other players have witnessed him knowing it ok. So, the way events turn out end up being common knowledge is when there is a say a public announcement ok. If there is like a like a traffic signal for example, traffic signal everyone watches the traffic signal and everyone else also knows that everyone else is watching. So, whatever is the is the is the content of that signal it is it is known to everybody and not only individually known to everyone, everyone knows that everyone knows and everyone knows that everyone knows that everyone knows and so on ok. So, events for events basically become common knowledge when there is a public witness of those events. So, things that appear in the newspaper, public announcements etcetera those are those become common knowledge. Personalized notifications on your mobile phone are not common knowledge ok. Things that come to you alone you the same thing may have come to others ok, but it is not common knowledge because they are just knowledge of your of individuals, but not common knowledge is this clear ok. So, this is actually that an important distinction it leads to one of these many of these sort of logical puzzles that you may have heard of about you know muddy children puzzles and so on you can build up about this on the internet about the distinction the fine distinction between each person each player knowing a thing versus that thing being common knowledge. So, and that so that we often the term used for this is that each when each player knows a certain thing we say that it is mutual knowledge where it each player knows that thing, but it is not common knowledge until you know all of these condition that you know ok. So, now what what can be so now what we since we have we are talking of games of you know that could have basically an arbitrary size any number of strategies if you want to start using the properties of dominance and to eliminate strategies effectively what we need to assume is that rationality is not only is players are rational, but actually that rationality itself is common knowledge ok. When I told you also that you know when the prisoners were told this particular thing they were they were each told this thing by by the judge effectively that matrix was also common knowledge amongst them each player knows knew that knew this matrix the other knew that they know this matrix etcetera etcetera etcetera ok. So, the standard assumption that we make in games is that that payoffs strategy sets are all common knowledge. So, we those those things are common knowledge rationality whether we assume it to be common knowledge or not really is depends on the situation at hand it is not a universal assumption that we always make in game alright. Now, however if rationality is assumed to be common knowledge then you can use the properties of dominance in some cases to try and solve for the game that is what we just see. So, I will now let us let me just state that in the form of a an actual theorem ok. So, here is the theorem. So, let let G be a game comprised of the following. So, you have n players you have a set of strategies SI for those for the players you also have payoffs UI for these players ok. And now suppose let G hat be this game which is formed from the same set of players, but as different strategy set and I will tell you how these are related S hat i for the player and the same payoff functions UI ok. Now, S hat i is like this. So, S hat i is just the relation between S hat i and S i is that S hat i is a subset of S i and this is true for all then. So, essentially G hat now is a game that is formed from fewer strategies than the original one. Some strategies have been removed or eliminated. Then if X star equal to X star 1 till X star n is a Nash equilibrium of the original game G such that X star i belongs to S hat i for all i then X star is a Nash equilibrium also of G hat. So, what is this theorem basically saying? So, suppose let G be your G be a game and then you construct another game G hat by eliminating some strategies from G. So, the elimination is captured here S hat i is a subset of S i and now suppose X star is a Nash equilibrium of G. If X star is a Nash equilibrium of G such and with the property with this property this is this is important with the property that X star i is in S hat i. So, X star is actually feasible for the for this the game that you have formed after elimination then X star is also a Nash equilibrium of G hat. So, you have a Nash you have your original game you have a Nash equilibrium of that game you have another game that is formed from eliminating some strategies but after elimination this Nash equilibrium is still present in the in the in the smaller game. Then the claim is that that is also a Nash equilibrium of the smaller game. And the proof is very simple. So, let us just quickly do the proof. So, we clearly have so X star is a Nash equilibrium. So, U of X star U i of X star i comma X star minus i is less than equal to U i of X i comma X star minus i for all X i and for all i in N. Now, since this is true for all X i in S i this is naturally also true for all X i in S hat i because because S hat i is a is a subset of of S i. So, therefore, this we have that this is X star i comma X star minus i is less than equal to U i of X i comma X star minus i for all this and also for all i in N. Now, only thing left to argue is that these X star i's are actually in S hat i and that is that is given to us by assumption we have been given this given this right here that X star i is in S hat i. So, in other words the the point X star satisfies the conditions of a Nash equilibrium for for the game G hat. So, it means X star is a Nash equilibrium of G hat. So, this is all I have used here is that see basically player in X star being a Nash equilibrium of the bigger game G he has no incentive to deviate to any other strategy in G and since S hat is a smaller set is a subset of of the strategies in G. So, therefore, he has no incentive to deviate to any other strategy in G hat also. And only thing for me to guarantee that I that is left for me is to is that for me to prove is that X star actually is an is available in G it is not gotten eliminated out when you created G hat. So, X star is actually available in G hat and it has not been eliminated out all right. So, this is this is one property. So, what are we learning here is seeing that if you if you eliminate strategies and if the original equilibrium is available in the in the in the smaller game then it is also an equilibrium of the smaller game almost obvious sort of statement. Okay. So, now let us let us now talk about elimination of weakly dominated and strictly dominated strategy. Now, suppose G hat let G hat be obtained from G by elimination of a weakly dominated strategy. Now, here is a statement in this you can say roughly in the opposite direction. So, G hat is now obtained from G by eliminating a weakly dominated strategy X hat J of a player J. So, there is a player J who is who has a weakly dominated strategy X hat J and that strategy has been eliminated and the resulting game is called G hat. Okay. Now, the claim is this then every Nash equilibrium of G hat is also a Nash equilibrium of G. Okay. Now, what does this mean? So, the previous theorem what did it tell you if you eliminated some strategies and the original Nash equilibrium is still available after elimination then it is a Nash equilibrium of the new one. Okay. This theorem what it is saying is that every Nash equilibrium of the smaller set of the smaller game is necessarily a Nash equilibrium of the original game which means that elimination of strategies cannot create new Nash equilibrium. It can add the most eliminate Nash equilibrium but cannot create new Nash equilibrium. Okay. All right. So, what do we what is S hat now? Now, S hat has the following structure S hat for this player J it has particular player J it has it is obtained from eliminating some dominated weakly dominated strategy whereas for others it is the same as before. So, this is equal to so S hat i is equal to S i for i not equal to J and it is equal to S i minus this sorry S j minus this specific strategy X hat J for i equal to J. So, for i equal to J you have removed the strategy X hat J. Okay. So, for all other players i S hat i is equal to S i and for player J it is equal to S j minus this weakly dominated strategy that has been removed. Okay. All right. So, now let X star be a Nash equilibrium of G hat. So, which means what that means U i of X i star comma X minus i star is less than equal to U i of X i comma X minus i star for all X i in S i and for all sorry S hat S hat i and for all i in. Okay. Now, this note this is for all i in X i in S hat i but S hat i is equal to S i for i not equal to J. So, that means for i not equal to J I can write this as basically this condition for into two places. So, i not equal to J this is actually the same as saying X i in S i and for i equal to J this is all S i except for all S j except for X i and J. Okay. All right. Now, what do we know about so essentially now if I want to show that X star is a Nash equilibrium of I have started assuming that X star is a Nash equilibrium of G hat. I want to show that it is a Nash equilibrium of G which means that I would have to show this for this inequality for all i so for all players i and for all X i in S i. Now, for i not equal to J S hat i is equal to S i so there is no issue here. Okay. So, this takes care of all players i not equal to J. But for player i for player j I can only do take care of it partially because I can take care of it except for this strategy X hat j. So, I do not know if strategy X hat j could be better than basically X star J here. Right. So, but then let us see what X hat j actually was right. So, if you see what was X hat j, X hat j was weakly dominated. Okay. Which means what if it is since it is weakly dominated it means that there is at least one other strategy which is better than this one. Right. Means there exists and it is weakly dominated where it was weakly dominated in G. Right. G hat has been obtained after removing X hat j. So, it was weakly dominated in G. So, that means there exists a strategy let us call it say T j in S j. Okay. Such that playing T j is better than playing X hat j regardless of what the other guys play. Is this clear? So, regardless of what the others play it is better for player j to play T j as opposed to X hat j. Okay. And I have a weak inequality here because this is weakly dominated. Okay. So, I have a great, this is, oh sorry, you have written a greater or minus greater. It should be less than equal to. So, this is less than equal to. Yeah. So, T j is better than playing X hat j regardless of what the others play. Okay. Now, what do I do? Well, this has, this is true for all X minus j. Right. Regardless of what the others play. So, I can actually make the others play their star strategies from here. Okay. So, let us pick these star strategies. These star strategies are in G hat. Remember. So, they have not been eliminated. They are actually present in G also. Okay. So, I can make X minus j as just basically their respective star one. So, I can just put in place of this which means U i of T j comma X minus j star less than equal to X hat j X minus j star. Okay. So, I should come back here now. So, definition of weakly dominated. Okay. So, here actually I should, there is a small, there is a small error here in this definition. So, the weakly dominated was that I said that this inequality could be, you could have equality in one, actually in one. Right. Then it is when we say you could have equality and so, but it has to be strict for at least one. A strategy cannot weakly dominate itself. Okay. You cannot have equality for all. Okay. So, if the inequality above is weak, but holds with strict inequality for at least one X minus i, then X i is said to be weakly dominated. Okay. So, that means you have a, this here is replaced with a less than equal to, but there should be at least one case where there is a strict. Okay. The reason for the including this is basically you avoid degeneracy where you have equality throughout because then you know, you can effectively start repeating strategies one after just copy pasting and repeating strategies and that then they will all end up dominating each, weakly dominating each other and that creates a whole bunch of degeneracies. No, no, no. We will see that one second. All right. So, X j is weakly dominated. That means there exists a T j like this. So, tell me this T j is weakly dominating X j. Now, is T j present in S hat j or not? Why is it present in S hat j? Yeah. Okay. This is where I needed it. Okay. So, if T j could be equal to X hat j, then T j, then it would be a degenerate case where it is not the, it is dominating itself, weakly dominating it. Right, right. So, but this is that is, yeah, of course, that is that is a different thing. So, so this is essentially weakly dominated means that essentially it is not actually equal to, this actually implies that T j is is present is is distinct from X hat j and hence T j is in S hat j. Now, if T j is in S hat j, it what does this mean? If T j is in S hat j, then I have this this term here, this is for all X i in S hat i, right and for all i. So, in particular, I can take X j here as T j because T j is present in S hat j, right. So, therefore, I putting putting X j equal to T j gives me u i of X star is less than equal to, so u j of X star is less than equal to u j of u j comma X minus j star and T j comma X minus j star is in turn less than equal to, that in turn is less than equal to u j comma X hat j comma X minus j star. So, in other words, we have we have shown we got this for free S j minus X hat j and for X hat j also now we have shown the inequality. So, in other words, X star is a Nash equilibrium, all of this basically from here we conclude that this is a Nash equilibrium. No, no, no, no, no, no. Strict inequality was for at least one of the others, other strategies. See for at least one X minus i, this should be strict. That means at least one profile of strategies that the others would play, X i dash should be better than strictly better than X i. So, T j is the strategy which is the dominating one and dominated is X hat. So, from here, then we get we can conclude that is a Nash equilibrium. What this is saying is that once you again repeating that if you eliminate weakly dominated strategies, you cannot create new equilibrium. You could eliminate equilibrium, but you cannot create. So, it is just to avoid this degeneracy that a strategy cannot weakly dominate itself. Otherwise, you know, you can end up with a lot of vacuous statements, you know. I mean, what does it mean to eliminate a weakly dominated strategy? Every strategy will weakly dominate itself and you will keep eliminating every strategy. You can eliminate one of them. You can eliminate one of them if as a weakly dominated one. Sorry, what? Yeah, yeah, you cannot. If they are two, two are equivalent. Yeah, so that is that is actually a case more of a case of okay, yeah, I understand what you are saying. So, if you have two strategies that essentially are identical, right? It is just a you can say that is I think more of a case of poor form problem formulation. Actually, you have not, you have just unnecessarily created another strategy when there is, you know, concretely, there is no, they are in different strategies. No, but regardless, so the point is, so regardless of what the other is playing, these two strategies are identical, right? They are identical for every case. That is what yeah. So, then you cannot eliminate. No, then that is all the more reason why you cannot. No, no, no, this is, this is for the player I. For the other players, if they are not identical, that is all the more reason why you should not be eliminated, because you are changing the strategic considerations for the other player. So, you are basically saying that regardless of what you, what the other player does, I do not care. But that does not mean that they can be eliminated. You can drop one of them. The other guys care. I mean, so yes, yes. So, that means I can drop the, I can drop one of these. If I, if there is one that is strictly better for me, if I have a, if there is a third strategy that weakly dominates these two, I can eliminate these two and stick to the dominating one. But yes, if, then in that case, you cannot eliminate them. I think I have an example. So, this example, I mean, I do not know if it is actually a true story, but it has been referred to in multiple settings. I think it has appeared in some movies and all that as well. So, there are two parts. This is called the example of the Battle of the Bismarck Sea. So, there are two generals here. One is, this is the war between America and Japan. So, there are two generals who have to make decisions about which direction in which, about the direction in which they will be sending their troops. The generals are Kenny and Imamura and I will tell you the story. So, general Imamura has been ordered to send transport troops across this sea and general Kenny has to, has been ordered to bomb the troops. That is the situation. Now, he has, Imamura has two choices. There is, he can go north or he can go south. The north route is shorter, the south route is longer and Kenny has to decide where to send his planes. Now, if Kenny sends planes to the wrong route, he can recall them, but the number of days of bombing is reduced. That is the situation. So, the payoff matrix that you get is, you can think, you can write it like this that, so this is, so if Kenny goes along the north route, the north route is the shorter one. So, both players are looking for the maximum. So, the, so there is nothing that, so Kenny does not have a, does not have any dominated or dominant strategy because your two is greater than, you have two greater than one and two less than three. So, there is nothing that Kenny can eliminate, but Imamura can eliminate. So, he is, he is also looking for the largest value. So, now minus two is equal to minus two and minus one is greater than minus three. So, north is better than south for Imamura. Now, remember, but weakly, north weakly dominates south, not strictly. Now, because why, now so, so Kenny can reason in this way that, well, Imamura I know is rational. So, he will not go south, he will instead go north and when he is going north, what should I be doing? I should also be going north. So, what has happened is that, this is not exactly the stat, the thing that you guys were talking about, not the, it is not the completely degenerate type of case, but you see what has happened is Imamura has ended up eliminating a weakly dominated strategy and in the process effectively knowing that, then Kenny knows that north is weakly dominated. If Imamura eliminates that, then from that logic, I can then go, I can also eliminate south and then and bomb him in the north. So, effectively what has happened here? So, if you reason about it in this way, essentially the fact that north and south are indifferent for Imamura, has gotten basically completely eliminated by, because he has, he has chosen to eliminate weakly dominated strategy. So, if you, if you, you can try, when we go to mixed strategies and so on, we will see that there is a better way of solving this particular game than doing this, that this is not the, this is practically not the right way to approach this. And the, the fallacy is exactly in this, that you are eliminating something that is weakly dominated and so I have not got to that yet. So, when you eliminate something that is weakly dominated, for what you were talking about is an even more degenerate case, where there are two strategies that are same, identical, in terms of payoff. But even when you eliminate something that is weakly dominated, it is, you end up, there is a chance that you will end up eliminating equilibria. And you end up eliminating equilibria means what, there are possible strategic outcomes that you are all, you are basically letting go of. No, no, no, see that is not a good enough justification, that is the point. So, see, so we cannot, so what I have not completed my story yet. But basically weakly dominated strategies will in general lead to loss of equilibria. Strictly dominated do not lead to loss of equilibria. So, strictly you can freely eliminate, weakly you can eliminate at the risk of losing equilibria. So, at the risk of, no, but loss of equilibria means that it is a strategic outcome that could have been possible which you have now let go. It is, it is a Nash equilibrium, but there could have been another Nash equilibrium that you, that could have been better. No, but that is all, that is provided all you are looking for is this particular property. I mean, there is more to it than just that. See the point is, see weakly dominated strategy, the elimination of weakly dominated strategy is, has perils with it. And that is the, that is the point I will be making, I will be making soon.