The lowest point V in the shaded region indicates the value of game. From the above figure, the value of the game is 3.
Likewise, we can draw a graph for player B. The point of optimal solution i. Comparing the above two equations, we have. Substituting the values of p 1 and p 2 in equation E1. Games where one player has only two courses of action while the other has more than two, are called 2 X n or n X 2 games. If these games do not have a saddle point or are reducible by the dominance method, then before solving these games we write all 2 X 2 sub-games and determine the value of each 2 X 2 sub-game. This method is illustrated by the following example.
Determine the solution of game for the pay-off matrix given below:. Obviously, there is no saddle point and also no course of action dominates the other. Therefore, we consider each 2 X 2 sub-game and obtain their values. The saddle point is 1. So the value of game, V1 is 1. This game has no saddle point, so we use the algebraic method. The 2 X 2 sub-game with the lowest value is c and hence the solution to this game provides the solution to the larger game. Using algebraic method :.
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A value for n-person games, Contrib Theory Games, 2, — R Two-person zero-sum games, Springer, Berlin. Game theory problem solving using linear programming method and examples. Model building in mathematical programming, 5th Edition. Quantitative Methods-Theory and Applications, Macmillan, - All rights reserved. Pay-off Matrix for Players A and B. Table 2. Table 6. Figure 1. Singh, A. Bhuiyan, B. Darkwah, K. Morrow, J. Shoham, Y. Hillier, F. If the column player chooses column three, then the row player is going to choose row one, yielding a payoff of 3 for the column player.
The column player is seeking to minimize the payoff, and she knows that the row player is seeking to maximize, so he's always going to choose column two. The row player is seeking to maximize the payoff, but since he knows that the column player is going to choose the column that yields 1 no matter which row he chooses, he is indifferent between playing row one or row two.
Sign up to join this community. The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group. Create a free Team What is Teams? Learn more. Asked 8 years ago. Active 8 years ago. Viewed 5k times. Please help to solve this problem. BlackAdder 3, 18 18 silver badges 30 30 bronze badges. Add a comment. The game can be solved without reducing the size also.
After reducing the above game with the help of dominance property we get the following game. Step 2: Let x be the probability of selection of alternative 1 by player A and 1 — x be the probability of selection of alternative 2 by player A.
Derive the expected gain function of player A with respect to each of the alternatives of player B. Similarly, the second alternative of player B is column number 2, so multiply 2 with x and -9 with 1 — x and add them. Similarly, the third alternative of player B is the column number 4, so multiply -6 with x and 4 with 1 — x and add them. Please refer the shown table. See the table below:.
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