 Welcome back to NPTEL course on game theory. In this session, we start understanding the Lemke-Hausen algorithm. So, this requires some preliminary results. So, we will start working with those preliminary results. So, first to recap the notation, so let us say that we have this bi-matrix game AB, there of order m by n. That means the player 1 has m pure strategies. So, let me write them as 1 to m and player 2 has n pure strategies. We will write them as m plus 1, m plus 2, m plus n. So, this is for a notational convenience. And recall the mixed strategies delta 1 of player 1 is nothing but set of all x in Rm. Of course, they are all non-negative such that summation xi is 1. Similarly, the mixed strategies of delta player 2 is set of all y in Rn equals to 0 such that summation yj is 1. So, the condition that sigma xi is 1, I will also start writing it as x transpose 1 is equals to 1. So, here 1 is basically 1, 1, the vector of 1s. And here this I will write it as y transpose 1 is 1. So, this is again vector of 1s. Of course, this is n by 1, this is m by 1. So, this is the notation. And then we also have this payoff function, the player 1's mixed payoff function is u1xy this is nothing but x transpose Ay. We also have used the inner product notation. Similarly, the player 2's payoff is x transpose By which is inner product of x with By. So, this is the notation that we have already introduced in the previous sessions and we will look at this one. So, the another definition that I need to introduce now here is that the support of a mixed strategy support of a mixed strategy. So, let us take x is a mixed strategy in delta 1. So, look at the set i that belongs to 1, 2, m such that summation xi, xi greater than 0. Look at all those i's for which the xi is greater than 0, this is nothing but the support of x. Similarly, if you take a mixed strategy of delta 2 that is a player 2 mixed strategy from delta 2 then support of y is nothing but j in m plus 1 to m plus n such that yj is greater than 0. So, the support of this mixed strategy of player 1, support of mixed strategy of player 2 both are this thing. Now, so let me mention an interesting result now which is known as a best response condition which is in some way we have seen it already in previous results even if we do not mention it formally it is in some sense it is done. So, what it says is the following thing that x and y be mixed strategies of player 1 and 2 respectively then x is best response to y if and only if for all i in s 1 xi greater than 0 implies ay, ay is a vector and look at its ith component that should be u which is nothing but the maximum of ay kth element such that x i is in s 1. So, what this says? This is not really a hard fact to see it. So, xi greater than 0 x is best response to y what does that mean? This is the you can say this is a theorem. So, what it is really saying here is that x is a best response that means what? So, x belongs to r max of let us say x prime comma ay that x prime is inside s 1. So, now this particular thing can be written as summation x prime i ay i and i is 1 to m. Now so, if x is a best response so that means look at this particular term this is a convex combination of ay i's. Now if x is the best response that means x is going to put the largest probability on the i's where this ay i is maximized that means wherever it is putting ay i has to be maximum. This immediately says that x i has to be positive if and only if ay i is the maximum of ay j's. So, that is exactly translates into this particular condition. So, in fact this argument we have seen it earlier. So, I will not go more formally here but this is a very important fact which we use at many places. This is basically the convexity argument. So, once we see this one. So, our next thing is to say that this actually helps something. So, for this we need to introduce what is called non-degenerate game. So, what this says is the following thing. A two player game is called non-degenerate no-mixed strategy of support size K has more than K pure best responses. So, what this condition is saying is that if there is a mixed strategy with support size K then it will have best responses. Let us look at the pure best responses. The size of the pure best responses cannot be more than K. Now, this immediately gives the following proposition in any non-degenerate game every Nash equilibrium x star, y star, x star and y star have supports of equal size. So, this is a quickly follows from the definition of non-degenerate game because in the non-degenerate game what we said is that every mixed strategy of support size K cannot have more than K pure best responses. So, therefore, if x star has support K, y star cannot have more than K. The support of y star cannot be more than K because if the support of y star means what they are nothing but the pure best responses of x star because that is the condition of the Nash equilibrium. X star, y star is a Nash equilibrium. Therefore, every j where y star has a positive value, y star j has a positive value then that j is going to be pure best response to x star. So, therefore, the support of y star cannot be more than support of x star. A symmetric argument says that x star and y star have, they have to have same support size. Their supports must have equal size. So, that is basically follows from the definition. Now, in fact, I should also mention that if a game is not non-degenerate, we call that as degenerate game. But we concentrate mostly on the non-degenerate games. The reason is if you have a degenerate game by suitably changing certain things, we can actually convert this into non-degenerate game. But we will not get into this details now. We will see it later on. But right now we concentrate only on non-degenerate games. The reason for non-degeneracy is essentially this proposition which says that in any equilibrium, both the strategies, mixed strategies have same supports of equal size. So, this in fact can be used as an algorithm. So, this is known as equilibrium by support enumeration. So, what is the input for this algorithm? Non-degenerate game. We input a non-degenerate game and then the method is the output of course is going to be the Nash equilibrium. So, the method is the following thing. For each k that is basically going to be anything from 1 to up to minimum of m comma n. So, and each pair i comma j of k sized subsets of S1 and S2 respectively solve the equations. Summation i in i x i b i j is equals to v for j in j. Similarly, summation of course I should also add here summation x i should be 1 summation a i j y j is equals to u and this j is in j and of course this is for i in i then summation y j should be 1 and also x greater than equals to 0 y greater than equals to 0. What we are really saying here is that so we are trying to solve this system of linear equations here with these constraints if we solve this system of linear equations if we can find some subsets i and j such that this happens then this correspond the output whatever it gives x and y is going to be a equilibrium. In fact, this comes basically from the fact that if we go back this best respond condition theorem that we what we have done here this theorem actually proves this fact. So, I will leave this for you to verify this one. So, we will now go on to the later parts. So, before I proceed further and like to introduce some notation which is the notion of polytope. So, this requires some notation. So, let me introduce first the Fn combination. So, what is Fn combination? So, take some points z 1, z 2, z k in some Euclidean space. There what is an Fn combination means it is simply sum of lambda i z i where i runs from 1 to k where lambda i's are in R and summation lambda i is 1. So, this is what is otherwise this is simply a linear combination. So, sigma summation lambda i is 1 that is there and it is a convex combination if these lambda i's are all greater than or equals to 0. So, and remember a convex set means all convex combinations are in the set. So, we have introduced the convex set earlier. So, I would not go more into the definition of convex set. What convex set says that you take any of its points convex combination and they all lie inside this set then you call that as a convex set. So, that is convex set we have seen it earlier. Next we need to define what is called Fn independent. So, what this Fn independent means is that take z 1, z 2, z k in some Euclidean space they are Fn independent if no z i is Fn combination of other points. So, none of the z i's will be a Fn combination of other points. If that happens you will call that as a Fn independent this point set of points z 1, z 2, k they are called Fn independent if no z i is a Fn combination of others. Now, a convex set has dimension let us say some D prime if and only if it has D prime plus 1 Fn independent points and no more. It has a convex set look at a subset of this pick some points which are Fn independent and no Fn independent set can have more than D prime plus 1 points and you can always find D prime plus 1 points which are Fn independent then you say that the convex set has a dimension D prime. So, this is basically in fact an exercise for you is take this in R2 take this simple like here what is its Fn dimension what is its dimension you can calculate and just check it just for a this is a good exercise in fact we can even look at our mixed strategies what is its Fn what is its dimension. So, next I would I will introduce polyhedron. So, a polyhedron P is a subset of Rd let us take is basically is a set of this it should be something like this set such that C z is less than equals to q for some matrix C and vector q. So, basically what we are saying is that this is C z less than equals to q is essentially a certain number of linear inequalities. So, q is a vector C is a matrix so C z is again a vector. So, each component of the C z is less than equals to corresponding component of the vector q and look at all those points z for which this is happening and such sets are called polyhedron. So, essentially they are the solution set of certain linear inequalities. So, and now we call we say that it has say that it has full dimension it has dimension t that means this polyhedron should have d plus 1 affinely independent points then you are we are going to say that this has a full dimension. So, remember that the by the very definition of this set the z in Rd this thing you can easily verify that this is a convex and everything. So, therefore this polyhedron by very definition this is a convex set and that is why we are talking about it dimension affinely independent and everything here. So, if this is bounded then we call this as so polytope is essentially a bounded polyhedron. So, because this inequality does not say that this set is going to be bounded or not if the set z in Rd such that C z less than equals to q is bounded set then we call this as a polytope instead of using the word polyhedron. Polyhedron is a more general word and polytope is for bounded polyhedrons. So, the next thing what is a phase of P let us take P is a polyhedron. So, it will be basically z in P such that C transpose z is given by some q naught for some C in Rd and q naught in R. So, basically the phase is essentially a this signifies a line C transpose z is q naught and then you are looking at the intersection. This is a plane sorry because we are in the higher dimensions it is a hyper plane this is a hyper plane this is a hyper plane and basically we are looking at a polyhedron intersection with some hyper plane and one more condition we need to say here is that for all z for all z C transpose z must be less than equals to q naught. If you take any other point in P then the C transpose z must be less than equals to q naught that means you have a you have a hyper plane here like this and then the the polyhedron lies this side of it and then whatever intersects with this one that is basically the phase. So, the poly this particular hyper plane is separating this polyhedron. So, the polyhedron is going to be on only on one side of it. So, this is known as a phase. So, then vertex. So, a vertex of P is unique element of a zero dimensional phase of P. So, what it means is that so when you take a hyper plane if the hyper plane actually intersects this polyhedron exactly at one unique point and the polyhedron lies only on one side. So, that unique point is known as a vertex because there are more algebraic way of defining it that you will be able to learn it in any convex optimization course. So, this vertex is a very important in fact in convex analysis there is a very important theorem which is known as a Crain-Wilman theorem which actually says that any bounded convex set has a extreme point which is actually in the sense vertex here in this case. So, an edge is basically one dimensional phase of P. If you have a one dimensional phase that is called edge a phase set is basically D minus it is a phase of dimension D minus 1 it is one dimension less. So, these are some notations that we require and we will continue the rest in the next session and we will stop this session at this place. Thank you.