4D.5.4 Mixed strategies by graphical methods Quiz: Pearson Edexcel Further Maths, Unit 44

The 20 questions

1. In a 2 by n game, what is plotted on the y-axis against probability p?

   • Expected payoff
   • Minimax value
   • Game value
   • Strategy choice

2. What game sizes can be solved using the graphical method in A Level Further Maths?

   • 4 by 4
   • 3 by 3
   • 2 by n
   • n by n

3. What is a mixed strategy in game theory?

   • A single action
   • A dominant move
   • A probability distribution
   • A fixed payoff

4. For a 2 by n game, what point on the lower envelope gives the game value?

   • The minimum point
   • The x-intercept
   • The maximum point
   • The y-intercept

5. In a 2 by n game graph, what does the lower envelope represent?

   • Maximum expected payoff
   • Mean expected payoff
   • Total expected payoff
   • Minimum expected payoff

6. Where does the optimal mixed strategy occur on a 2 by n graphical envelope?

   • At graph midpoints
   • At line intersections
   • At axis origins
   • At y-intercepts

7. How is optimal probability p found algebraically for a 2 by 2 game without saddle points?

   • Maximising column sums
   • Minimising row sums
   • Equalising expected payoffs
   • Calculating the determinant

8. For the matrix [[2, -1], [-1, 3]], if the row player plays row 1 with probability p, what is the expected payoff against column 1?

   • 2p - 1
   • 3p - 1
   • p - 1
   • 3p + 1

9. For the matrix [[2, -1], [-1, 3]], what is the expected payoff against column 2 when row 1 is played with probability p?

   • 4p - 3
   • 3 - 4p
   • 3p - 1
   • 3 - p

10. For the matrix [[2, -1], [-1, 3]], what is the optimal probability p of playing row 1?

    • 4/7
    • 5/7
    • 1/2
    • 3/7

11. For the matrix [[2, -1], [-1, 3]], what is the value of the game?

    • 4/7
    • 5/7
    • 2/7
    • 1

12. For the matrix [[1, -2], [-3, 4]], what is the probability of playing row 1 in the optimal strategy?

    • 7/10
    • 4/5
    • 3/10
    • 1/2

13. What is the value of the game with pay-off matrix [[1, -2], [-3, 4]]?

    • -1/5
    • -2
    • 7/10
    • 1/5

14. For the matrix [[3, 0], [1, 2]], what is the value of the game?

    • 3/2
    • 5/4
    • 1
    • 2

15. For the matrix [[3, 0], [1, 2]], what is the probability that the row player plays row 2 in the optimal strategy?

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

16. For a 2 by 2 game matrix [[2, -1], [-1, 3]], how is optimal probability p found?

    • Equating column payoffs
    • Subtracting diagonal values
    • Calculating matrix trace
    • Summing row entries

17. If the row player plays p = 1/2 in the matrix [[2, -1], [-1, 3]], what is the worst-case payoff?

    • 1/2
    • 1
    • 5/7
    • 0

18. For the matrix [[3, 0], [1, 2]], what is the optimal probability that the column player plays column 1?

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

19. For the matrix [[2, -1], [-1, 3]], what is the optimal probability that the column player plays column 1?

    • 5/7
    • 3/7
    • 1/2
    • 4/7

20. What is the value of the zero-sum game with payoff matrix [[2, -1], [-1, 3]]?

    • 1/2
    • 4/7
    • 2/7
    • 5/7

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