 While discussing Nash Equilibrium, we had made a very crucial assumption that all the players are rational and intelligent. What rationality means is that they are always trying to maximize their own utilities and intelligence means that they understand the rules of the game and they are always going to pick the most optimal thing like the game theorist. So for instance in a game where Nash Equilibrium exists, a rational and intelligent player will always find that and it has the computational ability to find it and it will play according to that Nash Equilibrium. Now we are going to address a different kind of a situation where one player might be rational and intelligent but it is not sure whether the other player is also rational and intelligent or not. So there is a certain amount of risk while playing the Nash Equilibrium strategy when the other player might not be as intelligent as the current player. So let us look at one example for that. So suppose in this game on the right hand side player one has three strategies, player two has two strategies and it is not very difficult to figure out which one is the Nash Equilibrium. Essentially B comma R is the Nash Equilibrium for this game. So this is giving this ability. Now if you assume that the player two it is not capable of taking the most intelligent decision then if it changes from R to L its strategy from R to L then for player one playing B is very risky because it can certainly jump because of that wrong choice of player two from 3 to minus 100. How can we handle that? So how can an agent who is unsure about the intelligence of the other player still play a strategy and not be super worse off and that is exactly what we are going to discuss in this module. It is called the Max Ming strategy. So it is something like a worst case optimal choice. So if you look at the other strategies, so in this particular strategy for player one that is strategy B the minimum value of that player can be minus 100. While if it plays M the minimum value reduces a little bit, the negative value reduces a little bit goes to minus 10. So if the other player is unsure and you are not certain whether that player will play according to the most intelligent outcome then you can imagine that the worst thing is going to happen to you and therefore you are counting how much worse you can get in each of these strategies. So in M you can get as bad as minus 10 but if you look at T you can actually get a much better outcome which is one. So which is not as good as the Nash equilibrium but it is not too bad either for player one. So that is in some sense averting the risk. If this player decides that I should play T instead of playing B because I am not sure whether the other player is also playing according to the Nash equilibrium then I can at least guarantee this much amount of outcome which is this much amount of utility which is one and that will be my risk averse choice of. So that is exactly what we mean by a pessimistic estimate the agent one is actually doing a pessimistic estimate of the other players. So this is what we call the max mean strategy. You are assuming as if the other players or all the other players together is trying to minimize your utility as much as possible and you are trying to pick the action or the strategy that maximizes it and therefore you can first look at this this expression this argument max mean from inside out. So first look at the utility here a specific utility of player i is given by the strategy that it picks and the strategy that all the other players pick. First you look at the worst possible situation what can happen if you pick a specific strategy si then the worst thing that can happen to you is the minimum of s minus i and then you look at the strategy which maximizes that minimum value and that we are going to call the max max mean strategy for player i. Now we also can define equivalently the the max mean value which is essentially nothing but the the max mean the utility that you get at this max mean value. So you just look at the max over all the strategies of that player and minima over all the strategies of the other players and the utility that agent i gets that is going to be the max mean value which we are going to denote by v lower bar i. So just to make sure that this is not v minus i it is v v i lower bar and it is not very difficult to see that the utility in this context that player i will get when it plays this max mean strategy it is always going to be at least as much as the max mean value. So you can always guarantee as player i no matter what is the strategy that has been chosen by the other players you can always guarantee this max mean value as your utility. So in this example that max mean value for player 1 happens to be 1. Similarly you can find the max mean value for player 2 as well. Now once we have defined a different strategy we'll also have to somehow connect for our own understanding how it compares with the dominant strategies or the pure strategy Nash equilibrium etc and that is what we are going to do. So let's look at the max mean and the dominant strategies what are what is the relationship and here is the theorem it says that if si star is a dominant strategy for player i then it is also max mean strategy for player i. So in some sense it is saying that if you are playing a dominant strategy then you can be safe you can be safe that even the other player is not rational and intelligent you can still guarantee the max mean value that you are going to get. So that's that's a good news let us try to prove it in a little formal way. Let me first give you some sort of an intuition of this proof. So essentially you are looking at all the dominant strategies and you are going to claim that this particular strategy is going to be also max mean strategy. So what happens in a dominant strategy? So let's look at this step by step. So here I am going to give the arguments for strictly dominant strategies but the same argument will even follow for quickly dominant strategies just the inequalities will become a little weaker. Now if si star is a strictly dominant strategy for player i then by the definition of strictly dominant strategy we know that this inequality should hold the strict inequality should hold for all the strategies of the other players that is the definition of a strictly dominant strategy that is if the other players are picking some s minus i then for every possible s minus i si star is going to be giving player i a higher utility than any other strategy that that the agent might pick. So for any other si prime it is going to be strictly better. So if this inequality holds then we can define the following thing. So we can define a function which takes as input so imagine your player i and you have fixed a specific strategy si prime. Now what is the that strategy profile of all the other players that is giving you the minimum utility for that particular choice of si prime. So if you go back to this example suppose your si prime was m then what is that strategy profile of the other player in this case there is only one player what is that s minus i that gives the minimum utility to player one and for m if your si prime is m then that s minus i minimum that value is actually r while if your si prime is b then it is it is going to be l and so on. So that is what we are going to capture so this this particular term that we have written on the left s minus i mean si prime is that minimum strategy profile minimizing strategy profile of all the other players when player i is playing si prime that minimizes the strategy utility of player i right fair enough so this is this is the thing that we have just explained. Now I know that this because this si star is a strictly dominant strategy then it should hold this inequality should hold for every s minus i so even in particular if you look at this si prime and plug that particular value so m or b or t whatever it is it is always going to be following this strict inequality and this should hold not only for si prime if you change si prime what we change is the the value of this particular thing but for that new s minus i mean si prime also this inequality will hold all right so that is that is a good thing now from this particular part from this particular part we can actually say before going to the last line we can say that this value so ui if I look at the the value si prime and s minus i mean si prime then among all these things because si prime is strictly greater than this value so si star is going to be the arg max arg max over all si primes right for all si prime that does not belong I mean belongs to this set without this si star so it is going to be the the maximizer in fact here you can actually make it even larger so it is the arg max over all si because now you can also put in si star and you know that that value is going to be maximum so this is just an implication of this inequality here now we can expand that this inner part the inner part of this ui si s minus i mean which is nothing but the minimum right so this this term is what is this term by the definition of s minus i star s minus i mean si prime it is just the minimum so this particular term here is just nothing but this utility right so this yeah so this utility here is essentially can be replaced by this minimum so I know that si star is the arg max over this si and also the minimum over this s minus i and that is essentially proving that the strategy si star which was a strictly dominant strategy for player i is also a max mean strategy and you can just do the same exercise for weekly dominant strategies as well okay so let us now go move into the the pure strategy Nash equilibrium and see what relationship does it have with the the max mean value so now we are not looking at the max mean strategy we are looking at the max mean value and this claim says that if you have a pure strategy Nash equilibrium on a normal form game then it must satisfy the following inequality if you look at the utility of every player at that Nash equilibrium strategy profile then it must be at least as much as the max mean value so by playing a pure strategy Nash equilibrium you are still safe you you will always have the the max mean value that the max mean value that you could have achieved by being pessimistic so what it means that if the if all the players so it does not take away that risk factor if all the other players are also rational then they will perhaps play a pure strategy Nash equilibrium and in that pure strategy Nash equilibrium utility of all the players is going to be guaranteed to be at least as much as the their max mean value so how should we go about proving this so we do it in two steps the first step is just an observation that if you look at this this particular term utility of agent i when player i plays si and the other players are playing s minus i star which is the the pure pure strategy Nash equilibrium profile for all the other players then by definition you can by definition of minima you can say that this is this is at least as much as the minimum over all the s minus i's while player i is sticking to this si and this must be true for all si now if you look at the the other thing the second point which is the the the utility of player i at the Nash equilibrium profile then you also know that this inequality should hold and this is by the definition of the pure strategy Nash equilibrium if all the other players are still playing the Nash equilibrium profile then player i will get at least as much as as much utility by playing any other strategy in its Nash equilibrium strategy okay so this in this two inequalities hold now what we can show is that you start with this left hand side so you start with the utility on on the left and this by definition we have we have already shown this you can also use the best response strategy in order to best response definition of this of this Nash equilibrium it says that if you are maximizing with respect to the strategy when the other players are playing the Nash equilibrium strategy then your utility will be maximized at the Nash equilibrium so this equality is obvious from the definition of Nash equilibrium now what we can what we know is that this is going to be so this inequality that I am going to maximize with respect to si and the and we are minimizing with respect to s minus i because a max was already here and this this inequality is coming from this first inequality here this is just by the definition of of minima we can just replace the second part this this utility part with the minima here and that's it now now that we have replaced this max and mean in this sequence this max mean value is so max mean where you are taking the max with respect to that player's strategy and you are picking the minima with respect to the strategies of all the other players that essentially gives you the max mean value for that player so this is the v lower bar i so that is exactly what we have we have shown that the utility at the Nash equilibrium is at least because there is any inequality here at least as much as the max mean value