4D.5.3 Reducing pay-off matrices by dominance Quiz: Pearson Edexcel Further Maths, Unit 44

The 20 questions

1. For the row player, when is row A dominated by row B?

   • Row A contains the largest entry in the matrix.
   • Every entry in row A is at most the corresponding entry in row B, so the row player never prefers row A.
   • Row A has a higher minimum than row B.
   • Row A has a larger total than row B.

2. Why can a dominated row be removed from a pay-off matrix?

   • It contains a zero entry.
   • The column player always chooses that row.
   • It has the lowest row minimum.
   • The row player would never choose it, since another row always does at least as well.

3. For the column player, who wants low losses, when is column j dominated by column k?

   • Every entry in column j is at least the corresponding entry in column k.
   • Column j contains a zero entry.
   • Every entry in column j is at most the entry in column k.
   • Column j has a larger maximum than the minimum of column k.

4. What does a reduced pay-off matrix keep after dominated rows and columns are removed?

   • A matrix with no saddle point.
   • A smaller pay-off matrix with the same value as the original game.
   • A matrix that is always 1 by 1.
   • A matrix with a larger value, since dominated options are removed.

5. Can removing dominated strategies change the value of a game?

   • No, removing dominated strategies leaves the value unchanged.
   • Yes, the value always falls by the removed entry.
   • Only for mixed strategies.
   • Yes, the value always rises by one.

6. When a row is dominated by another row, what is the reduced form of the matrix?

   • The dominated row is kept but its entries are doubled.
   • The dominating row is deleted instead.
   • Both rows are merged into their average.
   • The dominated row is deleted and the remaining rows are kept.

7. Why is it useful to reduce a pay-off matrix before solving it?

   • It makes the value of the game larger.
   • It removes all mixed strategies.
   • It gives a smaller matrix, so graphical or linear programming methods are easier, without changing the value.
   • It guarantees a stable solution.

8. In the matrix [[2, 3, 1], [1, 2, 0]] for the row player, which row is dominated?

   • Column 2
   • Row 2
   • Neither row
   • Row 1

9. After removing row 2 from [[2, 3, 1], [1, 2, 0]], which column is dominated for the column player?

   • Column 1
   • Column 2
   • No column is dominated.
   • Column 3

10. After removing column 2 from [[2, 3, 1]], what is the reduced matrix?

    • [2, 1, 3]
    • [2, 1]
    • [2, 3]
    • [1, 3]

11. What is the value of the reduced 1 by 2 game [2, 1]?

    • 1
    • 3
    • 0
    • 2

12. For the matrix E = [[4, 1], [3, 2]], which column is dominated?

    • Neither column
    • Row 1
    • Column 2
    • Column 1

13. After removing the dominated column from E = [[4, 1], [3, 2]], what is the reduced matrix?

    • [[1, 2]]
    • [[2], [1]]
    • [[4], [3]]
    • [[1], [2]]

14. What is the value of the game E = [[4, 1], [3, 2]]?

    • 3
    • 1
    • 4
    • 2

15. In the matrix [[5, 4, 6], [5, 3, 6]], which column can the column player discard?

    • No column is dominated.
    • Column 2, since its entries are the smallest.
    • Column 1, since each of its entries is at least the matching entry in column 3.
    • Column 3, since each of its entries is at least the matching entry in column 1.

16. What is the value of the game [[5, 4, 6], [5, 3, 6]]?

    • 6
    • 3
    • 5
    • 4

17. Could a strictly dominated strategy have positive probability in an optimal mixed strategy?

    • Yes, it is always used with probability one half.
    • No, a strictly dominated strategy has zero probability in every optimal mixed strategy.
    • Yes, it is used whenever the game has a stable solution.
    • Only if the matrix is 2 by 2.

18. For the matrix G = [[1, 2, 3], [0, 1, 2], [2, 3, 4]], what is the value of the game?

    • 3
    • 1
    • 2
    • 4

19. For G = [[1, 2, 3], [0, 1, 2], [2, 3, 4]], which rows are removed by dominance?

    • Row 1 only.
    • Rows 1 and 2.
    • Rows 2 and 3.
    • Row 3 only.

20. Does removing dominated strategies change the optimal strategies of the remaining game?

    • It leaves the game unchanged but reverses the players.
    • It makes the value of the reduced game larger.
    • It always doubles the probabilities of the remaining strategies.
    • It removes only the dominated strategies, and every optimal strategy of the original game puts zero probability on them.

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