Lesson DF5-DF6

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

20 questions

Written by Revision Ninja · Last reviewed 11 October 2026

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Lesson DF5-DF6, Optimal mixed strategies and linear programming: 20 multiple choice questions for the AQA Further Maths (7367), Unit 5: Optional application 3: discrete mathematics, written with Revision Ninja.

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The 20 questions

  1. What is a mixed strategy in a zero-sum game?

    • A strategy in which a player chooses each pure option with a set probability, randomising the choice over repeated play
    • A strategy in which a player always chooses the option with the largest entry
    • A strategy in which a player picks two options at once in a single play
    • A strategy in which a player always chooses the middle option

  2. What makes a mixed strategy optimal for a player?

    • It gives the best guaranteed expected payoff whatever the opponent does
    • It is a strategy that the opponent is forced to use
    • It gives equal probability to every option, so it is always fair
    • It gives the highest payoff against one specific opponent option only

  3. In the graphical method for a 2 × n zero-sum game, what is plotted and what is sought?

    • The payoff matrix is drawn as a bar chart and the tallest bar is read off
    • The expected payoff against each column is plotted against the probability of one row, and the highest point of the lower envelope is found
    • The payoff for each row is plotted against the column probability, and the lowest point of the upper envelope is found
    • A line through the origin is drawn for each strategy and where it meets the axis is read off

  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. 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 optimal mixed strategy should the row player use?

    • A1 with probability 2/3 and A2 with probability 1/3
    • A1 with probability 1/2 and A2 with probability 1/2
    • A1 with probability 1/4 and A2 with probability 3/4
    • A1 with probability 1/3 and A2 with probability 2/3

  8. 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. Which columns does the column player use in an optimal mixed strategy?

    • Columns B2 and B3 only, since their lines meet at the optimal point
    • Column B3 only, since it has the largest entries
    • All three columns with equal probability
    • Columns B1 and B2 only, since their lines meet at the optimal point

  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. To solve a zero-sum game as a linear programme, what does the row player's problem look like?

    • Minimise v subject to each row's expected payoff being at most v
    • Maximise v subject to each row's expected payoff being at most v
    • Maximise the sum of the probabilities subject to every entry being at most v
    • Maximise v subject to each column's expected payoff being at least v, with the probabilities summing to 1 and non-negative

  11. Why must the probabilities in a mixed strategy sum to 1?

    • Because the value of every game is always 1
    • Because a mixed strategy is a probability distribution over the pure strategies, so the probabilities must total 1
    • So that the value of the game equals the sum of all the entries
    • So that the column player's probabilities are then equal

  12. A 3 × 3 zero-sum game is solved as a linear programme for the row player. How many constraints, excluding non-negativity and the probability sum, are there?

    • 2, one fewer than the number of rows
    • 9, one for each entry of the matrix
    • 3, one for each column of the pay-off matrix
    • 1, the probability sum constraint only

  13. A 2 × 2 zero-sum game has a saddle point. What are its optimal strategies?

    • Pure strategies, with the saddle row and saddle column each played with probability 1
    • There are no optimal strategies for a game with a saddle point
    • Both rows played with probability 1/2
    • Each row played with a probability equal to its maximin value

  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. Why do the optimal mixed strategies of both players give the same value?

    • Because the value is the single payoff at which the maximin and minimax of the mixed game coincide
    • Because the row player's payoffs are always larger than the column player's
    • Because mixed strategies always assign equal probabilities
    • Because the value depends only on the number of rows and columns

  16. A calculation gives a probability p = 1.2 for row A1 in a 2 × 2 game. What does this indicate?

    • The value of the game must then be 1.2
    • The row player should play A1 more than all the time, which is allowed
    • The calculation is wrong, since probabilities must lie between 0 and 1
    • The row player should play A2 with probability -0.2 for a better result

  17. Which statement about the graphical solution of a 2 × n game is correct?

    • The optimum occurs at the lowest point of the upper envelope of the column lines
    • The graph gives the column player's probabilities directly without any calculation
    • The optimum occurs at the highest point of the lower envelope of the column lines, usually where two of them meet
    • The optimum is always at p = 0 or p = 1

  18. A game has an optimal mixed strategy that uses only two of its three rows. Why might the third row be excluded?

    • Because every row must be used in an optimal mixed strategy
    • Because the excluded row has the largest entry in the matrix
    • Because a probability of zero means the game has no value
    • Because it is not needed for the optimum, since the remaining rows already give the guaranteed value

  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 must be equal across the opponent's pure strategies that are actually used?

    • The expected payoff against each of them, which equals the value of the game
    • The probability assigned to each pure strategy
    • The difference between the row minima of the matrix
    • The largest entry in the pay-off matrix

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