 We have defined the mixed strategy in the previous module which means that there is a probability distribution, mixed strategy is a probability distribution over all the pure strategies that a player has. So in some sense probabilistically you are mixing all the strategies and that is why the name mixed strategy. Now once we have developed the notion of mixed strategy it is also very natural to ask about the equilibrium notion and the equilibrium that we are going to discuss in this case is called the mixed strategy Nash equilibrium or MSNE in short. So what is a mixed strategy Nash equilibrium? It is very similar to the pure strategy Nash equilibrium. The difference being that we are now using the profile, the strategy profile where both the players, all the players are choosing mixed strategies instead of a pure strategy and the definition remains almost equivalent. So if all the other players, players except I, so if you are focusing on player I, all the players except I are already playing the mixed strategy profile sigma minus I star and player I is considering to deviate from its own strategy sigma star to sigma I prime where sigma I prime is some other strategy. So any other strategy in the set of strategies, in the set of mixed strategies for player I then that utility, so this player I is not going to get better off and that is the definition of mixed strategy Nash equilibrium. This inequality should hold for every agent I in N and if that happens then we call this strategy profile sigma I star sigma minus I star to be a mixed strategy Nash equilibrium. Now it is very natural to draw the relationship between pure strategy Nash equilibrium and mixed strategy Nash equilibrium. You can very easily see that pure strategy Nash equilibrium is a very special case of mixed strategy Nash equilibrium. The case where it puts degenerate probability distribution on the strategy set. So in pure strategies the whole probability mass is concentrated on one specific strategy and when all the players are doing that then the equilibrium that you get is a pure strategy Nash equilibrium. You can think of that equivalently in the mixed strategy world as the probability being exactly equal to one for that strategy. So therefore pure strategy Nash equilibrium is by definition a special case of the mixed strategy Nash equilibrium. So all the games that has pure strategy Nash equilibrium definitely has a mixed strategy Nash equilibrium but the converse might not be true. Alright so the next definition that we are going to give or next result that we are going to give is essentially establishing an alternative definition for mixed strategy Nash equilibrium. So oftentimes we will see that this definition of mixed strategy Nash equilibrium is more appropriate and easier to use for certain proof of a theorem. So let us first state that result formally. So a mixed strategy profile sigma star sigma minus i star is a mixed strategy Nash equilibrium if and only if so this is a necessary and sufficient condition. To write the same inequality what you replace is instead of the strategy of player i which was a mixed strategy in the original definition so in this original definition it was a mixed strategy. Now what we are changing is this is going to be a pure strategy and we are definitely overloading this notation ui so what it means is that when you write ui si sigma minus i this is nothing but the probability so we are taking the sum over the probability mass so I am using the shorthand notation the sigma minus i of s minus i that means this is just the product of the corresponding sigma j sigma k's and corresponding s j's case which are the probability of choosing that particular strategy. So essentially there will be n minus one summation so just for a shorthand I am using this shorthand notation s minus i is is living in the capital s minus i and then you are taking the utility of si so this is the same si that player i has picked and all the other players are picking this s minus i. So what it does is it is picking the utility when player i is playing si and the other players are choosing s minus i and multiplying with the probability of choosing that s minus i and taking its expected sum so the expectation expected utility of player i when it is choosing the pure strategy si and all the other players are choosing the mixed strategy sigma minus i is given by this number that is what we mean on the right hand side in this inequality alright so it it says that so how does this definition or this is a sufficient condition change anything it is just saying that instead of looking at all possible mixed strategies which is an uncountably large set you are you are sufficient it is sufficient to look at only a finite set you can only look at the strategies the strategies in your strategy set well which is finite the strategy set is finite we have assumed that so that verification will ensure that this is a mixed strategy Nash equilibrium and that is the that is the advantage of using this equivalent definition so why are we calling it a equivalent definition because it is a necessary and sufficient condition if it is if this if you use this as a definition then there is no change because MSNE is this and this also means MSNE okay so let us try to prove this first prove the only direction of the necessary direction that is quite obvious that si is a special case so we are in this direction what are we trying to prove we are trying to prove that if it is an MSNE by this original definition of MSNE then definitely this inequality should hold this will hold because si is a special case you can just think of si as a degenerate case of that probability distribution now the interesting direction is the other direction where we are given this inequality this inequality holds for every si in capital si and for every agent i in n then we will have to show that this is a mixed strategy Nash equilibrium according to this original definition according to this definition so we will have to start with this this bunch of inequalities here and we will have to show this inequality so let us do try to do that so first let us look at the right hand side so we will somehow have to arrive at this expression here so let us write down what is the utility when player i is choosing a mixed strategy of sigma i and other players are choosing a mixed strategy profile of sigma minus i star so by the definition you can write it as sigma i of si I am just looking at player i's mixed strategy multiplying that with the utility when player i is picking that pure strategy and other players are picking a mixed strategy of sigma minus i star and taking the sum over all si's so this is the this is this equality so this is the same thing as the left hand side now what we have already seen or already know because we are given this particular expression is that this is going to be less than or equal to so this is actually coming from the given condition this is less than or equal to ui of sigma i star sigma minus i star now this sigma i star and sigma minus i star are very special strategies very special mixed strategies so they do not depend on on this si anymore so if you look at this particular expression ui sigma i star sigma minus i star that is completely independent of si so therefore in this summation I can take that quantity outside outside this summation okay so once we do that then this we are only left with this expression and we know it what this value is this is a probability distribution and essentially you are summing over all possible realizations of of that random variable so therefore this will definitely be equal to one so the right hand side becomes ui of sigma i star sigma minus i star so that essentially proves this fact that you are this is the sigma i so the sigma i star sigma minus i star is essentially a mixed strategy Nash equilibrium so we can use that definition the inequality as an equivalent definition of mixed strategy Nash equilibrium let us first look at some examples of mixed strategy Nash equilibrium in in some of the games that we have already seen so this game of penalty shootout we have already seen and let us try to see whether this mixed strategy profile where player 1 sigma 1 is two third and one third for these two strategies l and r and sigma 2 is this mixed strategy four fifth and one fifth for player 2 if we say that this is a mixed strategy Nash equilibrium we should be able to prove that if it is not then we should be able to provide a counter example so the first thing that we observe is that if you if you look at player 2 right so player 2 and look at the two different strategies it has l and r what is the expected utility of player 2 so you can write this as utility of player 2 when that player that second player is choosing l and the first player is choosing this sigma 1 so just i am using the same sigma 1 here which means that two third and one third and contrast that with the same utility of player 2 when player 1 is choosing sigma 1 and the other player is choosing r second player is choosing r so you can see that this is going to be you are going to multiply this utility so because we are looking at player 2 so this utility two third and minus one with one third that that is going to give you one third which is the expected utility so this utility is going to be one third while this utility you can calculate that is going to be minus one third so if you if you are a rational player then why should you even play r because in the in the world where player 1 is mixing these two strategies l and r in this way then player 1 should not be playing r at all because this is going to give them one third minus one third and this is going to give one third any probability mass any epsilon amount of probability mass if you can move from r to l that is going to give you some benefit because l gives you a larger expected utility so therefore there is no reason for player 2 to stick to this this mixed strategy of four fifth and one fifth it will essentially turn to the other one and so therefore this is definitely not an equilibrium point and for that very reason we can conclude that this strategy profile is not a mixed strategy next equilibrium so we will have to find something else essentially the the whole intuition that I am trying to give you here is that there needs to be some amount of balance so if these two numbers so here it is one third and minus one third if these two numbers were equal then there was no reason for any of this for player 2 to move its probability mass from one to another it does not gain anything so and that should also have happened for for player 1 then it would have been a sort of a equilibrium point so this example essentially gives us some sort of a intuition that this expected utility over this two different strategies should be somewhat equal otherwise there is no reason for for this player to play the strategy which is giving less expected utility so if you change this probability masses to half and half for both these players now this is a different sigma 1 so let's say sigma 1 tilde and sigma 2 tilde this has changed now you can see that the the previous condition that we have given so now you can look at the same expected utility of player 2 for when player 1 is choosing sigma 1 tilde and it is choosing L that is going to be equal to you to when player 1 is choosing sigma 1 and it is choosing R and this actually gives a kind of a foundational result for the characterization of mixed strategy Nash equilibrium if it is a mixed strategy Nash equilibrium then something like this has to happen and that is what we are going to see in the next lecture.