Tag: Special

Suppose we have a game for two people as follows:

How can player X choose values? $ x in mathbb {R} ^ N $ To maximize $ x ^ Tcy $ and player Y chooses values $ y in mathbb {R} ^ M $ minimize $ x ^ Tcy $, given the matrix $ c in ( mathbb {R} ^ N x mathbb {R} ^ M) $. In addition there are restrictions:

$ x_i, y_j geq -10 $,

$ sum_i x_i = 1 $ Y

$ sum_j y_j = 1 $.

So he $ x_i $ Y $ y_j $ can be negative

According to https://en.wikipedia.org/wiki/Minimax_theorem, the minimax theorem is valid and the game has a value if:

x and y are compact, convex sets and $ f (x, y) = x ^ Tcy $ it is a continuous function that is concave-convex.

In my opinion, x and y with these constraints are compact and convex sets. The function $ f (x, y) = x ^ Tcy $ it is bilinear, therefore concave-convex. And I don't know why f is not a continuous function. So the minimax theorem must have a value of the game v. Can anyone confirm that?

Because my simulations show that there is no game value, if x and y can be negative.

If I calculate the optimal x and y with the "linprog" function in matlab:

%payoff matrix, e.g.:  
c=(2 -1 1;1 0 -1;-1 1 -2);

(N,M)=size(c);

%optimal x for Player X:      
x=linprog(-(1;zeros(N,1)),(ones(M,1) -c'),zeros(M,1),(0 ones(1,N)),1,(-inf;ones(N,1)*(-10)));  
x(1,:)=();

%optimal y for player Y:     
y=linprog((1;zeros(M,1)),(-ones(N,1) c),zeros(N,1),(0 ones(1,M)),1,(-inf;ones(M,1)*(-10)));  
y(1,:)=();

%the results are:  
% v1=0   
v1=min(c'*x)

% v2=-10.2   
v2=max(c*y)

% payoff p=0    
p=x'c*y

The result is for almost all matrices c: $ v1 neq v2 neq p $. So there is no game value. If the constraints are changed and the x and y are constrained so that they are not negative (as in a usual zero-sum game with mixed strategies), always $ v1 = v2 = p $ follow ..