DF5-DF6 Optimal mixed strategies and linear programming Quiz: AQA Further Maths, Unit 5

The 20 questions

1. What defines a mixed strategy in game theory?

   • Probability distribution
   • Deterministic choice
   • Pay-off matrix
   • Saddle point

2. What does an optimal mixed strategy maximise for the row player?

   • Guaranteed expected payoff
   • Total matrix sum
   • Maximum possible payoff
   • Minimum opponent score

3. For a $2 \times n$ game, which point on the lower envelope gives the row player's optimal mix?

   • Y-intercept
   • Highest point
   • Lowest point
   • Origin

4. A zero-sum game has pay-off matrix (gains to the row player A) with rows A1: 1 3; A2: 4 2, and columns B1, B2. What is the value of this game?

   • 3
   • 2.5
   • 2
   • 3.5

5. A zero-sum game has pay-off matrix (gains to the row player A) with rows A1: 1 3; A2: 4 2, and columns B1, B2. What probability should the row player assign to A1 in an optimal mixed strategy?

   • 0.4
   • 0.25
   • 0.5
   • 0.75

6. A zero-sum game has pay-off matrix (gains to the row player A) with rows A1: 4 0 3; A2: 1 3 2, and columns B1, B2, B3. What is the value of this game?

   • 1.5
   • 2.25
   • 2
   • 3

7. In a game with rows (4, 0, 3) and (1, 3, 2), what probability should row 1 have?

   • 1/2
   • 2/3
   • 1/4
   • 1/3

8. In a game with rows (4, 0, 3) and (1, 3, 2), which columns form the optimal mix?

   • B2 and B3
   • B1 and B3
   • All three columns
   • B1 and B2

9. A zero-sum game has pay-off matrix (gains to the row player A) with rows A1: 4 0 3; A2: 1 3 2, and columns B1, B2, B3. In an optimal mixed strategy for the column player, what probability is assigned to column B1?

   • 1/3
   • 0.5
   • 2/3
   • 0.25

10. In a linear programme for a zero-sum game, what does the row player maximise?

    • Sum of probabilities
    • Column player loss
    • Maximum entry value
    • Guaranteed payoff v

11. What must the sum of probabilities in any mixed strategy equal?

    • Number of rows
    • 1
    • 0
    • Payoff value v

12. How many payoff constraints does a $3 \times 3$ zero-sum game linear programme have for the row player?

    • 1
    • 2
    • 3
    • 9

13. If a zero-sum game contains a saddle point, what type are its optimal strategies?

    • Pure strategies
    • Dominant columns
    • Mixed strategies
    • Infinite strategies

14. A zero-sum game has pay-off matrix (gains to the row player A) with rows A1: 3 -1; A2: -2 4, and columns B1, B2. In an optimal mixed strategy for the column player, what probability does B1 receive?

    • 0.4
    • 0.5
    • 0.3
    • 0.6

15. Which theorem states that maximin and minimax values are equal in mixed strategy games?

    • Minimax theorem
    • Fundamental theorem
    • Bayes' theorem
    • Central limit theorem

16. A calculated strategy probability is $p = 1.2$. Why is this result invalid?

    • Negative probability
    • Equals zero
    • Exceeds maximum 1
    • Must be integer

17. In the graphical method for a 2 x n game, where is the optimal mix found?

    • Lowest upper envelope
    • Maximum gradient point
    • Highest lower envelope
    • Intersection of all lines

18. Why is a row assigned a probability of zero in an optimal mixed strategy?

    • Highest payoff row
    • Sum exceeds one
    • Equals total value
    • Dominated or unneeded

19. A zero-sum game has pay-off matrix (gains to the row player A) with rows A1: 1 3; A2: 4 2, and columns B1, B2. If the row player uses A1 with probability 0.25 and A2 with probability 0.75, what is the expected payoff against column B1?

    • 2.75
    • 2.5
    • 3.75
    • 3.25

20. In an optimal mixed strategy, what property do the opponent's used pure strategies share?

    • Equal expected payoff
    • Equal probability
    • Zero payoff sum
    • Equal maximum payoff

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