2016 Syst465 HW 5 Solutions

P a g e | 1 SYST 465 001 ECON 496 002 Math 493 002 Homework 5 – Solutions Spring 2016 Problem 1 For the following problem use complementarity conditions to check the optimality of the proposed solution: a) (6 points) (i) Check the feasibility of the proposed primal solution. and The solution is primal feasible. (ii) Which primal constraints are binding? The third and fourth primal constraints are binding. (iii) Which primal constraints are not binding? What do we know about the dual variables? The first and second primal constraints are not binding. Therefore, (iv) Which primal variables are > 0? What do we know about dual constraints? The first and sixth dual constraints are binding. b) (8 points) Reduce the primal LP by elimination all columns that refer to a variable at 0 and find the dual LP. Use the information from above to solve the reduced dual LP (any way you want).

P a g e | 2 c) ( 2 points ) Calculate the primal and dual objective function values. d) (3 points) Check the feasibility of the non-binding dual constraints. The second, fourth , fifth and sixth dual constraints are not binding. One of the dual constraints is not feasible. e) ( 1 points ) Final Conclusion: Is the proposed solution an optimal solution to the LP? The proposed solution is not optimal. Problem 2 (20 poin ts) Consider the following two-person zero-sum game: 8 12 20 18 10 4 a) Is there a solution in pure strategies? Explain. row player : column player : These two values are not the same. There is no solution in pure strategies. b) Find the optimal strategies for the row player graphically. 8 12 20 18 10 4 Row Player:

P a g e | 4 column player : There is no equilibrium in pure strategies. Problem 4 (20 points) Constant Sum Game Two competing firm must simultaneously determine how much of a product to produce. The total profit earned by the two firms is always $9,000. If firm 1’s production level is low and firm 2’s is also low, then firm 1 earns a profit of $3000; if firm 1’s level is low and 2’s is high, then firm 1’s profit is $8000. If firm 1’s production level is high and so is firm 2’s, then firm 1’s profit is $2400; but if firm 1’s level is high while firm 2’s level is low, then firm 1’s profit is only $5400. a) Find the matrix and the optimal strategies for this constant-sum game. Assume that company 1 is the row player. Company 2 Company 1 There is no solution in pure strategies. Assume the row player uses mixed strategy (x ₁ ,x ₂ ). His expected reward if the column player plays column 1: 3000x ₁ +5400x ₂ = 3000x ₁ + 5400(1-x ₁ ) = - 2400x ₁ + 5400 His expected reward if the column player plays column 2: 8000x ₁ + 2400x ₂ = 8000x ₁ + 2400(1-x ₁ ) = 5600x ₁ + 2400 -2400x ₁ +5400 = 5600x ₁ + 2400; 8000x ₁ = 3000; x ₁ =3/8; x ₂ =5/8 The value of the game : -$2400 (3/8) + $5400 = $4,500 . Assume the column player uses mixed strategy (y ₁ ,y ₂ ). The row player’s expected reward if the row player plays row 1: low high low 3000 8000 high 5400 2400

P a g e | 5 3000y ₁ + 8000y ₂ = 3000y ₁ + 8000 (1-y ₁ ) = - 5000y ₁ + 8000 The row player’s expected reward if the row player plays row 2: 5400y ₁ + 2400y ₂ = 5400y ₁ + 2400 (1-y ₁ ) = 3000y ₁ + 2400 -5000y ₁ + 8000 = 3000y ₁ + 2400; -8000y ₁ = -5600; y ₁ =56/80=7/10; y ₂ =3/10 The reward for company 2 : $9000 - $4.500 = $4,500 . b) Determine the value of the game and the reward for company 2. The value of the game : -$2400 (15/16) + $5400 = $3,250. The reward for the column player: $9,000 - $3,250=$5,750. Problem 5 (15 poin ts) Consider the following two-person zero-sum game: Player B Player A There is a claim that the optimal strategy for player A is (0.8,0,0.2) and for player B is (0,0.2,0.8). a) Determine the value of the game. b) Use the LP formulations and the complementary slackness conditions to verify that these stratgies are optimal for both players. > 0, therefore the first and third dual constraints (y-player) are binding. are > 0, therefore the second and third primal constraints (x-player) are binding. Formulation 1 row player: The solution is primal feasible. The first primal constraint (x-player) is not binding, therefore 5 0 1 2 3 -2 -1 4 0

P a g e | 7 Problem 6 (20 points) Two companies, R and C, that manufacture the same product are bidding on a contract that has been put up for bids. Each company has three bids that it can submit: a high bid, a modest bid, and a small bid. The two sets of possible bids are not the same numerically. Therefore, one company always wins the contract. The following matrix represents the relative values to the two companies in certain monetary units. The matrix shows the reward for the row player. R \ C High Normal Small High 4 -3 -2 Normal -2 5 -1 Small -1 4 3 a) Show that this matrix does not possess an equlibrium in pure strategies. There is no equilibrium in pure straetgies . b) Formulate one LP for each player solving which yields his optimal strategy. Is scaling required? Solve in step c). Formulation 1 No scaling required. A transpose A Formulation 2 Scaling required. A transpose A c) What are the optimal strategies for Company R (row player) and Company C (column player)? Follow the example from class (Ex. 3 at end of file LP Formulations) and use simplex tableaux. Use the method from our class! I choose to pivot on y3 first. It. 0: Y1 Y2 Y3 T1 T2 T3 rhs

P a g e | 8 -1 -1 -1 0 0 0 0 ratio 4 -3 -2 1 0 0 1 -2 5 -1 0 1 0 1 -1 4 3 0 0 1 1 1/3 It. 1: y3 enters the basis, t1 leaves the basis. Y1 Y2 T3 T1 T2 Y3 rhs -4 1 1 0 0 0 1 10 -1 2 1 0 0 5 /3 -7 19 1 0 1 0 4 -1 4 1 0 0 1 1 Y1 Y2 T3 T1 T2 Y3 rhs -4/3 1/3 1/3 0 0 0 1/3 10/3 -1/3 2/3 1 0 0 5/3 -7/3 19/3 1/3 0 1 0 4/3 -1/3 4/3 1/3 0 0 1 1/3 It. 2 y1 enters the basis and T1 leaves the basis. T1 Y2 T3 Y1 T2 Y3 rhs ratio 4/3 6/9 18/9 0 0 0 30/9 1 -1/3 2/3 1 0 0 5/3 *(3/10) 7/3 183/9 24/9 0 1 0 75/9 1/3 39/9 12/9 0 0 1 15/9 T1 Y2 T3 Y1 T2 Y3 rhs ratio 4/10 2/10 6/10 0 0 0 10/10 3/10 -1/10 2/10 1 0 0 5/10 7/10 61/10 8/10 0 1 0 25/10 1/10 13/10 4/10 0 0 1 5/10 This is optimal. (Normally scaling is necessary for the formulation used above: Take the inverse of the objective function value as the value of the game. Divide every strategy by the value of the game to normalize.) Here is a special case as the objection function value is 1. No normalization necessary. d) Determine the value of the game: 1