 All of this is well and good provided we know that we are per bar is equal to be lower bar. And that condition the if there is irritating because it tells you all of this nice story works provided we are per bar is equal to be lower bar and we already have seen an example where it is where the two are not equal. So question then becomes what can how can you resolve this? This is where we basically need a next new idea, you need a new idea you need a new way to think about the strategic situation. All this reasoning is so far is good, but we have we have sort of limited ourselves to a particular way of viewing the options at hand, which is that a player has to play a particular row or a particular column. If you think of essentially the strategies that players are exposed to in a game the strategies that they have in a game in many cases these strategies can be thought of as let us say you know suppose a general wants to decide you know how much which of his he has to decide whether he wants to take this route or that route that is thought of one kind of way of visualizing the strategy. Instead of saying that I want to should I go to this route or should I go to that route instead of having that sort of dilemma a general might also ask if I have 1000 troops how many should I send to this route and how many should I send to that route. So many times the kind of strategies that we have with us can be interpreted as something on as a set of options on such that we do not have to pick only one of those options but rather we can decide that how much of our resources are to be allocated to each option. Say for example, so in the case of the general he has to decide I have 1000 troops maybe I send 300 here and 700 here or if you are a fund manager you have 1 lakh rupees maybe I put 25000 in this project and 25000 in that project. So this way once you start thinking about it this way then the options that you have that a player has are not just merely to commit to a particular row or a column but rather to decide the extent of commitment to each row or column. Now this extent of commitment can be thought of in many different ways one is that as I said is a resource allocation so you can think how much resources am I devoting to each option. The other way is to say well what I am going to do is I am going to pick a row or a column at random and I am going to pick it at random but I am going to choose very wisely the probability with which I am going to be choosing each of my choices. This now expands the strategy space. So now you are not just picking a particular row or a particular column as a player what you are actually doing is picking a every row essentially with some probability and what you have to decide is the probabilities that you are going to assign to each row or each column. So this is what is called a mixed strategy. A mixed strategy is basically a randomized choice of a pure strategy. So this is one way of interpreting it but this is mathematically this is good enough so we will just use this description is a randomized choice of a pure strategy. What is a pure strategy whatever we have these strategies we were talking of so far they are what are called we will call them pure strategies. Pure strategies are definite choices of rows or column mixed strategies are randomized choices on top of pure strategies. So these pure strategies are what you know rows and columns and a randomized choice of the same is what is called a mixed strategy. So formally a mixed strategy is a probability distribution on the space of pure strategies or equivalently the space of pure strategies currently a random variable takes values in the set of so mixed strategies just like pure strategies are for each player the mixed strategy is also for each player. Each player now has to do make a randomized choice over his set of pure strategies. So the probability with which you play with which you the player plays is the choice that he has to nominate. So it is then many people use this word randomized in a kind of loose way many people use the word randomized in a kind of loose way where they when they say pick something at random they actually means of pick it uniformly at random ok everything with equal probability but that is not what we mean here by random we mean with the probability distribution and the probability distribution has to be chosen with in a strategic way is this clear. So then in this case now can you tell me mathematically what is the what would be the probability distribution for the row player. So the row player the pure strategies for the row player for the row player the pure strategies are the rows are the rows ok and suppose there are m rows m rows suppose for then what are his what what is a mixed strategy for the row player a mixed strategy for the row player is now a probability distribution on this set of m rows right so how do I describe that I need to tell you a probability with which I am going to choose every row ok. So let us let us write it like this let y i is the probability of choosing row i probability of choosing row i ok this is the probability of choosing row i now what must these y i satisfy how many y i's are there there are m of them ok how and what should these y i's satisfy yeah so this because this is a probability distribution it has to satisfy that so firstly so let me just write it like this let y be this y 1 to y m ok a composite vector just I put them together y 1 to y m there are m of them and these are all greater than equal to 0 right and they also satisfy that the sum of these y i's is equal to 1 so this here this let us call this capital Y this capital Y is the set of mixed strategies for the row player so every mixed strategy for the row player can be written in is basically x can be expressed in this form it basically is a vector of length m now the components are greater than equal to 0 and the sum to 1 ok and you interpret the ith component as the probability with which player that the row player is going to play row i ok alright what is similarly now for the column player let us denote it by z suppose there are n columns then the what is a mixed strategy for the column player the mixed strategy for the column player is a vector now z 1 to z n these z's are all greater than equal to 0 these z's are all greater than equal to 0 and they have the property that if you sum them sum these z's they sum to 1 let us let us call this as a set capital Z and this is the set of mixed strategies for the column player now if they play these strategies a strategy y if the row player plays a strategy y from capital Y and z and column player plays a strategy z from capital Z then is they are going to now get a payoff which is going to be which is going to be a random variable because it is going to randomly choose an actual column and an actual an actual row he just decides the probability with which they are being chosen using capital Y using these y and z ok so players decide the probabilities with which these are to be chosen but they have to actually act a column or a row ok so which means that now the row that is chosen so row i is going to be chosen randomly row j is going to be chosen randomly so the players would get a payoff a i j ok when row i is chosen and row j is chosen now what is the probability with which i is chosen and j is chosen y i z j y ok but why are they independent non cooperative the game is non cooperative players cannot communicate with each other they have to randomize independently right so because they randomize independently see at the end of the day if i want to define a random payoff i have to define for you the joint distribution of the row and column i have only defined for you the marginal distributions which is the probability of choosing a row and the probability of choosing a column separately right now from the marginals if i want to be able to regenerate the joint in general i would need some more more information but then that would also require the players to coordinate between themselves if they have to correlate them their choices right so since the players cannot communicate with each other this automatically means that player the choices are being made independently ok so the probability that you see row i and column j being chosen is the probability that you see row i being chosen times the probability that you see row j being chosen ok so the payoff now that the players are going to get is this random payoff they get this payoff with this probability ok and then so whenever there is a stochastic payoff involved so there is you can this can be shown very very formally but essentially remember that what you need what the player must do then is to look at the expected value so the expected value of this is what players have to have to know this is actually the payoff that there is so yeah so there is a there is a theory of utility from which you can show that under the set of axioms of rationality what a player must do is maximize the expected payoff or expected utility and in short so whenever there is a stochastic outcome what you need to look at is the expected value of it you could have looked at other things you could have looked at its variance you could have looked at its tail probability you could have looked at its max etc etc why should you look at its mean well there is a there is a there is a reason for that yeah yes yes yes yeah yeah so naturally because every you know I have to compute the expected value yeah okay so the in other words the the let us call this something let us call this j of y comma z that is equal to equal to this and in fact I can write this out in more compact form and this is just simply y transpose a z so I can think of y and z these as column vectors okay so I come these two capital y and capital z these two I will think of these as column vectors column vector if these are column vectors then this sum is essentially just can be written in this sort of form it is y transpose a z is clear so now what you have is a zero sum game but where the strategies are y and z earlier the strategies were i and j now the strategies are y and z where the row player is looking to pick a distribution on the rows to minimize to get the least expected value here which is the least value of j the column player is looking to pick a distribution on the columns to get the maximum value of j okay it is still a zero sum game but with these now as their strategies the distributions as their strategies so what the players have to do is strategically decide with what probabilities are they going to be playing each of their each of their particular rows or row or column so we can once again define something analogous to what we did earlier since this is just again a zero sum game but in a different strategy space I can once again talk of the same logic that I did so far each player is looking to make maximum damage to the other each player can think of the worst case damage that the other can do and try to minimize that worst case damage yeah correct yeah so the strategies now are the y's and z's yes which conditional probability yes so the players pick a distribution and that is and play with that that is the that is the assumption so it is the player the game is just played once just what they need to decide is the distribution okay so the randomized choice is a very is a very convenient thing because it can be applied in any situation that interpretation of a mixed strategy but the problem with that is also that you know and it is also you know gives you a proper interpretation as a probability measure and then you can compute expectations and so on the problem with that is you know there is the way to if you want to give it an operational interpretation like exactly how is the game played out many many people then try to think of the game being played multiple times okay so if the game so but remember that is not what is being done here we are not assuming that the game is played multiple times and this is not the frequency with which so why is not the frequency with which player i chooses rho i if the game was being played multiple times the players will also strategize in a different way knowing that you know this is the game is going to be played in such and such way and that leads to a different analysis altogether it is called a repeated game that is not what is being done here it is not the frequency with which they are playing these strategies it is a random choice so you should be comfortable thinking of probability without thinking of frequency of use okay so it is a random choice of arising out of a sophisticated design of the probability distribution is this clear so a vector y star in capital y is a mixed security strategy for the row player if look at this his payoff look at the worst case damage that the column player could do to him okay and the this here is called the mixed security level denoted as Vm okay Vm upper bar okay and likewise a vector that star nz is a mixed security strategy column player if okay so this this is how we can define now the corresponding security strategies so these are security strategies in the sense of security strategy district so they are in the sense of distributions so in terms of mixed strategies what are the security strategies they should be playing so what kind of distribution gives them a certain guarantees the players a certain amount of certain payoff that is what is being defined here okay now because now so how many mixed strategies does a player have how many mixed strategies does the player have infinitely many right necessarily because every distribution probability distribution is a mixed on pure strategies is a mixed strategy right so if he has at least two pure strategies he will have infinitely many mixed strategy so the set of mixed set of strategies now has become infinite okay it is the when you go to mixed strategies automatically the set of strategies now becomes becomes infinite so it is actually not that straightforward to now say even that a by a security strategy actually exists showing that there exists such a y star that satisfies this particular property is itself not itself requires a proof you need to prove that there exists a y star which satisfies it earlier all you did was you know you could just check over columns then rows and so on and you could immediately derive that now why does that even exist a distribution like this has to be shown distribution that is better than all other distribution okay so that is that is one this thing and so once you show once the question of existence is along along with the question of existence you also have therefore a question of whether mixed security levels can be thought of as well defined alright so so this this needs some amount of work so this needs some analysis to you need to show that there exists actually a mixed strategy for each layer but assuming there exists one we now we we will we have the same properties that we had earlier we have that we always have vm upper bar greater than equal to vm lower bar this was this can be shown in just the same logic that we did we had earlier okay so we always have this particular thing now we move to this particular regime of where where now players are allowed to randomize but we need to still justify what is it that is being how exactly are players gaining from randomization okay so to see how players gain from randomization let us let us do one let us observe one simple property can you tell me what is the relation between vm upper bar and v upper bar how are these related vm upper bar is smaller why you are taking okay so one that is one of the arguments that that is so so vm upper bar is the claim is vm upper bar is smaller than v upper bar and the reason that is the case is because mixed strategies are more general than pure strategy right pure strategy can be thought of as a mixed strategy with where only particular pure strategy is given probability 1 and all other strategies are given probability 0 but is that enough because if you see how vm has been defined remember okay yes the mixed strategies are more general for than the pure strategies but you are also maximizing over now earlier you are doing max over j now you are doing max over z so the maximization is also being done over a larger set okay so so that is so that is where the subtlety is basically and the answer is correct so it is true that vm is smaller so in fact I will write out the more general thing we of course have this thing that I have written out here but we also have v upper bar of a is greater than equal to vm upper bar of a is greater than equal to vm lower bar of a which is in turn greater than equal to v lower bar of a okay and so let us let us actually prove one of these inequalities so let us prove this one for example okay so I am going to assume that a mixed security strategy exists okay for the moment so what is a mixed security strategy it is a y star such that y star transpose az max over this this is less than equal to max over z y transpose az so actually we can only focus on the left hand side and just look at this more carefully this y star transpose a is a row vector multiplying with a column vector z having components z1 to zn right now what are these these z1 to zn z1 to zn are simply these these this is together a probability distribution right so these are all essentially some kind of weights they add up to 1 they are all greater than equal to so when you want to maximize this over z okay we have we wanted to maximize this thing over z okay we want to maximize this quantity over z we have to basically decide what weights should these components z1 to zn get how what should be the weights precisely so I look at the column that has the largest value okay here I look at the column that has the largest value and pick that one and give that z probability 1 and all the others probability 0 weight equal to 1 and all the others weight 0 so so this actually therefore is equal to so the left hand side here this is actually equal to the max over j of or actually let me just write this for the right hand side that will make things easier for me so I will write this for y transpose right so you pick the largest column largest value from this row vector y transpose a give that particular guy to weight 1 and everything else 0 that has to be the maximum value over all over all possible weighted combinations like this okay so now therefore now you have basically the same thing that we had earlier because this is actually now so now minimizing this minimizing this over y in y minimizing the max over j of y transpose a j this is naturally more general than minimizing over every row because if I took up because I can take y to be a pure strategy and then I would get just aij there so if y is a pure strategy y transpose a j becomes equal to just the pure strategy i let us say and this becomes aij right so that means if y is a pure strategy means what it gives will probability 1 to rho i and all other rows get probability 0 then in that case this become so therefore this is therefore more general is than minimizing over rows itself so consequently the minimizing value has to be lower because you are minimizing over a larger set right so minimizing over uh minimizing the minimum value has to be lower has to be lower because the set of mixed strategies is a larger set so consequently we get this inequality here that v m upper bar has to be less than equal to v v upper bar same thing can be argued even on the other side and we get that v v m lower bar is greater than equal to v lower bar the first one yeah yeah yeah the same same sort of argument as before so I can just show this to you so so you just have you have j of y comma z as this right and take any any this this is always uh uh greater than equal to the min over min over y in y y transpose a z and it is always less than equal to the max over z in z of y transpose a z okay so and again I get if you want I can put a dash here if you are getting confused this is always the case and so now what I have on the left hand side is a function of y and I have right hand side is a function of z function of y alone and z alone and it is all the left hand side is always greater than equal to the right hand side so which means the least value of the left hand side is greater than equal to the largest value of the right hand side okay so that and then so the min over y y max over z in z of y transpose a z this is greater than equal to max over z min over y this is this is this here is your Vm lower bar and this is Vm upper bar so actually this this does not require you know this kind of form in fact you can do this for any function take a function of two variables the min max is always going to be greater than equal to the maximum that is that holds for any function okay all that it still has the same issues still there you know so okay so I do not want to take away the punch line but essentially we still have this here the inequality it is no different from the inequality we started with which was this is greater than equal to this okay so what will happen in the next class is that I will show that this is in fact equal V bar of a not equal to that is because there could be in general this is this is certainly more general than the than that okay all right so the reason for that is okay maybe I can just talk about that here yeah so why are these not equal so we we just showed that V upper bar is greater than equal to V Vm upper bar but then why are they not equal the reason they are not equal is if you see this this expression here I just said that if you take y to be a pure strategy take y to be a pure strategy i then the then this then this just reduces to aij now the reason this is this is not the same as minimizing over pure strategies is because this now the resulting expression that you have here is not linear in y anymore the this expression here was linear in z for a fixed y it was linear in z so therefore you you had to just find the weight at the coefficient with the larger the largest coefficient and give that the full weight whereas if you see this here this is not linear in y anymore this is the max of linear functions and the maximum of linear functions is actually is is you know the way it looks would be maximum of linear function take this function take this function what would be the maximum of these two it would be this region up to this point you know up to this point the second one is larger after that point the other one becomes larger and this is now a max of several such linear functions right n of these linear functions so it is not linear anymore and because it is not linear anymore it is not you cannot claim that this is that the the same way as before you know you cannot pick a component see because the component has now gone inside here it is hidden inside this maximization even if you did pick a component they would still be outside a max over j which will keep switching based on what what y you have chosen the maximizing j right for some wise for some wise it will be this j that is maximum some some otherwise it will be this j that is maximum yes so so okay so it that is I was going to show that next time but actually so what you are asking is this is this is this going to be strict in general right yes so there you can show many functions for which this is okay but but this is already telling us one justification for moving to mixed strategies the randomization has benefited both players each player has improved his security level by randomization right his pure strategies yeah no so that so that is where this is settled so so you will so we will again jumping ahead a little bit let us but the for the moment let us just just justify why we are why we have why it makes sense to allow players the option to randomize it because by randomization they are you know you cannot been once their security levels have improved you cannot basically deny them that option essentially they have they have the one's security level improves which means that it means that you know if they could they would randomize okay all right so what what I will show next time is is is basically the champion theorem here which is that this is equal okay so all the things that we were you know fantasizing about that would happen if we upper bar is equal to be lower bar would now happen for free because we upper bar and we will be m upper bar would be equal to be lower bar okay so there will always be a saddle point etc etc all those consequences will follow but in the space of mixed strategies not in the space of your strategies okay all that