Given a RxC Matrix, A, i.e. R rows and C columns we define a Saddle-Point as
Saddle_Pt (A(i,j))
≡ A(i,j) is the minimum of Row i and the maximum of Col j.
e.g.
1 2 3
4 5 6
7 8 9
-- 7 is Saddle_Pt.
at position (3,1)
There may be more than one Saddle-Pt,
In game theory, we look at the minimum value of all the rows which is this case is
(1,4, and 7,) We then take the max of those which is 7. This is the maxmin.
Then we look at the max of all the columns which are (7,8, and 9) and we take the min of those which is 7. So position (3,1) is the number 7 which would be the saddle point. The idea is if, the number in C means that person pays R that amount, then we want the be number in the matrix fro both R and C.
For example if R picks first and he pick row 1, then to minimize his payout, C picks 1. However, row three always has a better payout for R so he would never pick row 1 or 2. We can eliminate them from the matrix. We are left with
(7,8,9) But if you are C, want to pay the least and would always pick 7 if R picked that row. The point that is the greatest of the mimima and the least of the maxima is called a saddle point.
Sometimes there is no saddle point. We then pick the "best" policy and doing so involves some randomized strategies.
One more example is
2 -2 -3
1, 0, 2
-1 -1 3
In this example -2 means C pays -2 to R in other words R pays C 2 units.
The saddle point is 0 because it is the max(min of row) and it equals the
min ( max of columns)
There will be no saddle point; no point in the table will be an optimum.
when entity is minimum of a row & respective column is maximum i.e. pay off matrix
An inflection point is not a saddle point, but a saddle point is an inflection point. To be precise, a saddle point is both a stationary point and an inflection point. An inflection point is a point at which the curvature changes sign, so it is not necessary to be a stationary point.
A saddle point is a point in the range of a smooth function every neighbourhood of which contains points on each side of its tangent plane.
cyl
A saddle.
The pommell
The low point of a mountain ridge is called a col or a saddle. It is the point where two slopes converge and is often a pass or a corridor between two peaks.
Yes, it is possible for a function to have a negative semidefinite Hessian matrix at a critical point.
importjava.io.*; classsaddle { publicstaticvoidmain(Stringargs[])throwsIOException { inti,k,j,c1=0,t=0,c2=0,z=0; BufferedReader br=newBufferedReader(newInputStreamReader(System.in)); System.out.println("enter the number of rows in the matrix:"); intn=Integer.parseInt(br.readLine()); System.out.println("enter the number of columns in the matrix:"); intm=Integer.parseInt(br.readLine()); inta[][]=newint[n][m]; System.out.println("enter the elements of the matrix:"); for(i=0;i<n;i++)<span=""> { for(j=0;j<m;j++)<span=""> a[i][j]=Integer.parseInt(br.readLine()); } System.out.println("matrix is:"); for(i=0;i<n;i++)<span=""> { for(j=0;j<m;j++)<span=""> System.out.print(a[i][j]+" "); System.out.println(); } for(i=0;i<n;i++)<span=""> { for(j=0;j<m;j++)<span=""> { z=a[i][j]; c1=0; c2=0; for(k=0;k<m;k++)<span=""> { if(z<=a[i][k]) c1++; } for(k=0;k<n;k++)<span=""> { if(z>=a[k][j]) c2++; } if(c1==m&&c2==n) { t=1; System.out.println("saddle pt="+z); } } } if(t!=1) System.out.println("there is no saddle point"); } }
A saddle in geography refers to a low point or dip between two higher points on a ridge or mountain. It has a characteristic saddle shape, resembling a horse saddle, hence the name. Saddle points are important features for hiking and mountaineering as they often provide a passageway or route to cross over a mountain ridge.
When looking at a horse saddle, the saddle point (i.e. the centre) is the lowest point on the structure in the longitudinal plane whilst also being the highest point on the structure in the lateral plane. Therefore it can be considered geometrically to simultaneously be a point of maximum and minumum height in space. This describes well the equilibrium situation in game theory where the agent tries tominimize the maximum possible loss. Alternatively, it can be thought of as maximizing the minimum gain (maximin).