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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).
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2 boys were playing a game and they came up with the name of the game
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John bowled 136 on his first game and 148 on his second game Jerry had a total score of 483 for three games What would John have to bowl on his third game to beat Jerry?
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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)
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