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Homework 9a: 20 points) In a union-management negotiation, the following are the percentages of annual wage increase gained by the Union for various combinations of union and management strategies: Management M1 M2 M3 U1 1 5 5 U2 2 2 2 Union U3 2 4 4 U4 3 1 2 Determine the best strategies and values of the game for the union and management respectively. 9b. For the Rock-Scissors-Paper two-person, zero-sum game, let X1, X2, and X3 be respectively the strategies representing Rock, Scissors, and Paper for the Row Player X and Y1, Y2, and Y3 be respectively the strategies representing Rock. Scissors, and Paper for the Column Player Y. Then the payoff matrix is given below: Column Player Y Y 1(Rock) Y2(Scissors) Y3(Paper) X1(Rock) 0 +1 -1 Row Player X X2(Scissors) - 1 0 +1 X3(Paper) +1 -1 0 To find the mixed strategy solution for this game based on Nash Equilibrium, the solution procedure is to assign probabilities p1, p2, and p3 respectively to X1, X2, and X3, then set up 3 linear equations for the 3 probabilities by equalizing the expected payoffs to X for different strategies of Y as well as the fact that the sum of all probabilities adds to 1 as follows: The expected payoff to X if Y plays Y1 equals the expected payoff to X if Y plays Y2: Op1-1p2+1p3= 1p1+Op2-p3 (equation 1) The expected payoff to Xif Y plays Y1 equals the expected payoff to X if Y plays Y3: Op 1-1p2+1p3--p1+1p2+Op3 (equation 2) All probabilities add to 1: 1p1 +1p2+1p3= 1 (equation 3) (10 points) Use the 3 equations to solve the probabilities and thus the mixed strategy solution and the expected payoff for X Homework 9a: 20 points) In a union-management negotiation, the following are the percentages of annual wage increase gained by the Union for various combinations of union and management strategies: Management M1 M2 M3 U1 1 5 5 U2 2 2 2 Union U3 2 4 4 U4 3 1 2 Determine the best strategies and values of the game for the union and management respectively. 9b. For the Rock-Scissors-Paper two-person, zero-sum game, let X1, X2, and X3 be respectively the strategies representing Rock, Scissors, and Paper for the Row Player X and Y1, Y2, and Y3 be respectively the strategies representing Rock. Scissors, and Paper for the Column Player Y. Then the payoff matrix is given below: Column Player Y Y 1(Rock) Y2(Scissors) Y3(Paper) X1(Rock) 0 +1 -1 Row Player X X2(Scissors) - 1 0 +1 X3(Paper) +1 -1 0 To find the mixed strategy solution for this game based on Nash Equilibrium, the solution procedure is to assign probabilities p1, p2, and p3 respectively to X1, X2, and X3, then set up 3 linear equations for the 3 probabilities by equalizing the expected payoffs to X for different strategies of Y as well as the fact that the sum of all probabilities adds to 1 as follows: The expected payoff to X if Y plays Y1 equals the expected payoff to X if Y plays Y2: Op1-1p2+1p3= 1p1+Op2-p3 (equation 1) The expected payoff to Xif Y plays Y1 equals the expected payoff to X if Y plays Y3: Op 1-1p2+1p3--p1+1p2+Op3 (equation 2) All probabilities add to 1: 1p1 +1p2+1p3= 1 (equation 3) (10 points) Use the 3 equations to solve the probabilities and thus the mixed strategy solution and the expected payoff for X