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 every entry of the pay-off matrix before setting up the linear program?

   • So that all probabilities are zero.
   • So that the objective becomes a maximisation.
   • So that the simplex algorithm can handle negative slack variables.
   • So that the value of the game is positive and the substitution x_i = p_i / v gives a valid linear program.

2. If a constant k is added to every entry of a game, how does the value change, and does the optimal strategy change?

   • The value decreases by k.
   • The strategies are halved.
   • The value is unchanged but the strategies change.
   • The value increases by k, and the optimal strategies are unchanged.

3. In the linear programming method for the row player, what substitution is used with value v?

   • x_i = v - p_i
   • x_i = p_i + v
   • x_i = p_i × v
   • x_i = p_i / v, where p_i is the probability of row i.

4. In the linear program for the row player, what quantity is minimised?

   • The sum x_1 + x_2 + ... + x_m, which equals 1/v.
   • The largest entry of the matrix.
   • The sum of the probabilities p_i.
   • The value v directly.

5. In the linear program for the row player, what is the form of the constraint for each column j?

   • The sum over i of x_i is at least a_ij.
   • The sum over i of a_ij x_i is at most 1.
   • The sum over j of a_ij x_i equals 1.
   • The sum over i of a_ij x_i is at least 1.

6. What linear program does the column player solve, after the same scaling?

   • Minimise the sum of y_j subject to the sums at least 1 for each row.
   • Minimise the largest entry of the matrix.
   • Maximise the sum of y_j subject to the sum over j of a_ij y_j at most 1 for each row i.
   • Maximise the value v directly with no constraints.

7. Why can the simplex algorithm be used for these linear programs?

   • Because the objective is linear with no constraints.
   • Each player's problem can be written as a linear program that the simplex algorithm can solve.
   • Because all variables are integers.
   • Because the matrix is always 2 by 2.

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 do the x_i in these linear programs need to be non-negative?

    • Because every x_i must equal 1.
    • Because the objective requires all x_i to be integers.
    • Because the simplex method only accepts non-negative numbers in the Value column.
    • Because they are scaled probabilities, p_i divided by v, which cannot be negative.

19. What does the sum of the x_i equal in the row player's linear program?

    • v, because x_i = p_i × v.
    • The number of rows in the matrix.
    • 0 at the optimum.
    • 1/v, because the sum of the p_i is 1 and x_i = p_i / 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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