# Game Theory 0-Sum Games

May 11th, 2015
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Solution: (a). Column 2 dominates column 1; then row 3 dominates row 4; then column 4 dominates column 3; then row 1 dominates row 2. The resulting submatrix consists of row 1 and 3 vs. columns 2 and 4. Solving this 2 by 2 game and moving back to the original game we find that value is 3/2, I’s optimal strategy is p (1 2, 0,1 2,0) and II’ (0,3 8, 0,5 8). (b). Column 2 dominates column 4; then (1/2)row 1+ (1/2)row 2 dominates row 3; then (1/2)col 1+(1/2)col 2 dominates col 3. The resulting 2 by 2 game is easily solved. Moving back to the original game we find that the value is 30/7, Is optimal strategy is q ’s optimal strategy is (2/7,5/7,0) and II’s optimal strategy is (3/7,4/7,0,0).

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Game Theory 0-Sum Games 1 2 4 3 1 2 2 1 0 1 1 0 4 3 (a) 5 4 MATH 4321 Spring 2013 Assignment Solution 0-Sum Games 2 1. Reduce by dominance to 2x2 games and solve. 6 3 3 5 2 6 4 7 (b) 10 0 7 1 Solution: (a). Column 2 dominates column 1; then row 3 dominates row 4; then column 4 dominates column 3; then row 1 dominates row 2. The resulting submatrix consists of row 1 and 3 vs. columns 2 and 4. Solving this 2 by 2 game and moving back to the original game we find that value is 3/2, Is optimal strategy is p (1 2, 0,1 2,0) and II (0,3 8, 0,5 8). (b). Column 2 dominates column 4; then (1/2)row 1+ (1/2)row 2 dominates row 3; then (1/2)col 1+(1/2)col 2 dominates col 3. The resulting 2 by 2 game is easily solved. Moving back to the original game we find that the value is 30/7, Is optimal strategy is q s optimal strategy is (2/7,5/7,0) and IIs optimal strategy is (3/7,4/7,0,0).2. Reduce by dominance to a 3x2 matrix game and solve: 4 3 12 . 8 4 6 0 8 5 4 * 12 . Then, we use the graphical method in the following. 8 4 * 0 8 *Solution: Note that 5/8xCol2 + 3/8xCol1 uniformly dominates Col3. Therefore, we can delete Col3 to get 4 3 12 0 8 4 6 8 /12 0 8 5 1/ 3 2 / 3 0 4 / 12 Answer: The optimal strategy for I is (4/12, 8/12, 0) The optimal strategy for II

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