In my problem is saddle point is defined as " A saddle point is an element which is less than all the other elements in that row and greater than all the other elements in that column".
On the internet, you may get some other definition for saddle point but stick to my definition.
I think you may have misunderstood what a saddle point is:
In mathematics, a saddle point or minimax point is a point [...] a function where the slopes (derivatives) in orthogonal directions are all zero (a critical point), but which is not a local extremum of the function.
One simple example with a saddle point at (0,0) is x.^3. One example with 2 saddle points is
x = -3:0.1:2;
y = 2*x.^5 + 5*x.^4 - 10*x.^3 - 20*x.^2 + 40*x + 30 % function
plot(x,y)
hold on
plot([-2,1],[-34,47],'o','MarkerSize',5)
with saddle points at (-2,-34) and (1,47).
"A necessary and sufficient condition for a saddle point to exist is the presence of a payoff matrix element which is both a minimum of its row and a maximum of its column. A game may have more than one saddle point, but all must have the same value."
source: Game Saddle Point
Therefore, a matrix may have more than one saddle point if all have the same value. However number of unique saddle points will at max be one.
