Lesson DF1-DF4

DF1-DF4 Pay-off matrices, play-safe strategies and dominance Quiz: AQA Further Maths, Unit 5

20 questions

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Lesson DF1-DF4, Pay-off matrices, play-safe strategies and dominance: 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 pay-off matrix in game theory?

    • A table showing the order in which the two players choose their strategies
    • A table giving the cost of each strategy for both players, with equal totals
    • A table giving the gain to the row player for each pair of strategies, with the column player's gain being the negative of this
    • A table of probabilities for each outcome, with each row summing to one

  2. What is a zero-sum game?

    • A game in which one player's gain equals the other player's loss, so the total payoff is zero
    • A game in which both players receive zero payoff whenever they choose different strategies
    • A game that has no pay-off matrix
    • A game in which the row player can never win

  3. What is a play-safe strategy for the row player?

    • Choosing the column whose maximum entry is smallest
    • Choosing the row whose sum of entries is smallest
    • Choosing the row whose minimum entry is largest, guaranteeing the best worst-case payoff
    • Choosing the row whose maximum entry is largest, so that the best outcome is obtained

  4. What is the value of a two-person zero-sum game?

    • The payoff to the row player that is guaranteed when both players play optimally
    • The largest entry in the pay-off matrix
    • The average of all entries in the pay-off matrix
    • The payoff from a single play of the game chosen at random

  5. A zero-sum game has pay-off matrix (gains to the row player A) with rows A1: 2 4 1; A2: 3 5 4; A3: 1 3 0, and columns B1, B2, B3. What is the value of this game?

    • 4
    • 2
    • 5
    • 3

  6. A zero-sum game has pay-off matrix (gains to the row player A) with rows A1: 2 4 1; A2: 3 5 4; A3: 1 3 0, and columns B1, B2, B3. What is the stable solution (saddle point) of this game?

    • Row A2 and column B2, with entry 5
    • Row A1 and column B3, with entry 1
    • Row A3 and column B3, with entry 0
    • Row A2 and column B1, with entry 3

  7. A zero-sum game has pay-off matrix (gains to the row player A) with rows A1: 2 4 1; A2: 3 5 4; A3: 1 3 0, and columns B1, B2, B3. Which play-safe row should the row player choose?

    • A1, since its largest entry 4 is the largest of all entries
    • A2 with column B2, since the entry 5 is the largest in the matrix
    • A3, since it contains the smallest entry 0 in the matrix
    • A2, since its row minimum of 3 is the largest of the row minima

  8. 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. What is the play-safe value for the row player, using the maximin method?

    • 4
    • 1
    • 3
    • -1

  9. 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. Does this game have a stable solution?

    • Yes, at (A2, B2) with entry 4
    • Yes, at (A1, B1) with entry 3
    • No, since the maximin value -1 and the minimax value 3 are different, so no saddle point exists
    • Yes, since every entry is non-zero

  10. 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, what probability does the row player assign to A1?

    • 0.4
    • 0.6
    • 0.3
    • 0.5

  11. 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. What is the value of this game when both players use optimal mixed strategies?

    • 2
    • 0
    • -1
    • 1

  12. The row player's gains in a game are A1: 2 3 1 and A2: 1 2 0. Which statement is correct?

    • A1 dominates A2, so A2 can be removed from the analysis
    • Column B3 dominates both rows, so both rows can be removed
    • A2 dominates A1, so A1 can be removed from the analysis
    • Neither row dominates the other, since their entries differ

  13. After removing row A2 from the game with rows A1: 2 3 1 and A2: 1 2 0, which column should the column player choose?

    • B1, since its payoff equals that of A1
    • B2, since it gives the row player the largest payoff
    • Any column, since only one row remains and all are equal
    • B3, since it gives the row player the smallest payoff

  14. What is a stable solution in a zero-sum game?

    • A pair of strategies whose entry is zero
    • A pair of strategies whose entry is the largest in its row and column
    • A pair in which both players choose the middle strategy
    • A pair of strategies whose entry is the smallest in its row and the largest in its column

  15. When is a row of a pay-off matrix dominated?

    • When it has the largest entry in the whole matrix
    • When its minimum entry is zero
    • When it contains the smallest entry in the whole matrix
    • When another row is at least as good for the row player in every column, and strictly better in at least one

  16. A zero-sum game has pay-off matrix (gains to the row player A) with rows A1: 2 4 1; A2: 3 5 4; A3: 1 3 0, and columns B1, B2, B3. Which rows can be eliminated by dominance?

    • Only A1, dominated by A3
    • No row can be eliminated
    • A1 and A3, both dominated by A2
    • Only A3, dominated by A1

  17. For any pay-off matrix in a two-person zero-sum game, which inequality always holds between the maximin and minimax values?

    • Maximin ≥ minimax
    • Maximin + minimax = 0
    • Maximin = minimax always
    • Maximin ≤ minimax

  18. A zero-sum game has pay-off matrix (gains to the row player A) with rows A1: 2 4 1; A2: 3 5 4; A3: 1 3 0, and columns B1, B2, B3. After the dominated rows A1 and A3 are removed, which column should the column player choose?

    • B1, which gives the row player a payoff of 3
    • B1, which gives the row player a payoff of 2
    • B2, which gives the row player a payoff of 5
    • B3, which gives the row player a payoff of 4

  19. Why does a game with a stable solution have value equal to the saddle entry?

    • The saddle entry is always the largest entry in the matrix
    • Neither player can gain by changing strategy alone, so the saddle entry is the guaranteed payoff
    • The column player always chooses the largest entry in the matrix
    • The average of all entries equals the saddle entry

  20. Why might a player use a mixed strategy?

    • When a stable solution exists, mixing always increases the payoff
    • Because mixing removes the need for a pay-off matrix
    • When no stable solution exists, randomising stops the opponent exploiting a predictable choice
    • Because the column player always mixes with equal probabilities

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