4D.5.5 Mixed strategies by linear programming Quiz: Pearson Edexcel Further Maths, Unit 44

The 20 questions

1. Why is a constant added to a payoff matrix before linear programming?

   • Make matrix symmetric
   • Reduce variable count
   • Eliminate matrix fractions
   • Ensure positive value

2. Adding constant k to every matrix entry alters the game value v by how much?

   • Unchanged
   • Decreases by k
   • Multiplied by k
   • Increases by k

3. In game linear programming, what substitution defines x_i using probability p_i and value v?

   • x_i = p_i + v
   • x_i = v / p_i
   • x_i = p_i * v
   • x_i = p_i / v

4. What does the Simplex tableau use to determine the entering column variable in game linear programming?

   • Most negative indicator
   • Constant column
   • Most positive indicator
   • Zero indicator

5. What is the inequality direction for column constraints in the row player's linear programme?

   • Exactly 1
   • At most 1
   • At least 0
   • At least 1

6. In linear programming for games, what objective function does the column player maximise?

   • Game value v
   • Matrix determinant
   • Sum of y_j
   • Sum of p_i

7. Which algorithm is standard for solving converted game theory linear programs?

   • Dijkstra's algorithm
   • Simplex algorithm
   • Kruskal's algorithm
   • Hungarian algorithm

8. For the shifted matrix [[4, 1], [1, 5]] obtained by adding 2 to [[2, -1], [-1, 3]], what is the value of the game?

   • 19/7
   • 12/7
   • 5/7
   • 3

9. For the matrix [[4, 1], [1, 5]], what is x_1 from the equations 4x_1 + x_2 = 1 and x_1 + 5x_2 = 1?

   • 3/19
   • 7/19
   • 1/4
   • 4/19

10. For the matrix [[4, 1], [1, 5]], what is x_2 at the optimum?

    • 1/5
    • 7/19
    • 3/19
    • 4/19

11. For the matrix [[4, 1], [1, 5]], what is the minimum of x_1 + x_2?

    • 19/7
    • 12/19
    • 1/7
    • 7/19

12. For the matrix [[2, -1], [-1, 3]], what is the optimal probability p_1 of row 1, given the shifted solution?

    • 5/7
    • 4/19
    • 4/7
    • 3/7

13. For the matrix [[2, -1], [-1, 3]], what is the optimal probability p_2 of row 2?

    • 2/7
    • 3/7
    • 4/7
    • 1/7

14. For the matrix [[3, 1], [1, 3]], what is the value of the game?

    • 3
    • 2
    • 4/3
    • 1

15. For the matrix [[3, 1], [1, 3]], what is the minimum of x_1 + x_2 in the linear program with constraints 3x_1 + x_2 ≥ 1 and x_1 + 3x_2 ≥ 1?

    • 1
    • 1/4
    • 1/2
    • 2

16. For the matrix [[3, 1], [1, 3]], what is the optimal probability of row 1?

    • 1/2
    • 3/4
    • 1/4
    • 2/3

17. For the matrix [[3, 1], [1, 3]] with the optimal row strategy (1/2, 1/2), what is the expected payoff against column 1?

    • 1
    • 2
    • 3
    • 4

18. Why must decision variables x_i in game linear programming be non-negative?

    • Game values zero
    • Matrix dimensions fixed
    • Payoffs are negative
    • Probabilities are non-negative

19. What does the sum of the variables x equal in the row player's linear programme?

    • Equal to one
    • The game value
    • Equal to zero
    • One over v

20. For the matrix [[1, -2], [-3, 4]], what is the value after adding 2 to every entry?

    • 1/5
    • 7/5
    • 11/5
    • 9/5

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