4D.5.1-4D.5.2 Zero-sum games and stable solutions Quiz: Pearson Edexcel Further Maths, Unit 44

The 20 questions

1. In a two-person zero-sum game, what is true of the players' gains and losses?

   • The sum of the losses for one player equals the sum of the gains for the other player.
   • Both players have zero payoff in every outcome.
   • The value of the game is always twice the largest entry.
   • The number of strategies is the same for both players.

2. From whose point of view is the pay-off matrix written in the specification?

   • The row player's point of view, unless directed otherwise.
   • The point of view of an impartial referee.
   • The column player's point of view.
   • Whichever player moves first.

3. In a zero-sum game, when is there a stable solution?

   • If and only if the row maximin equals the row minimax.
   • If and only if every entry is positive.
   • If and only if the row maximin equals the column minimax.
   • If and only if the matrix is square.

4. What is a saddle point in a pay-off matrix for the row player?

   • An entry that is the smallest in its row and the largest in its column.
   • An entry that equals zero.
   • An entry that is the largest in its row and the smallest in its column.
   • Any entry in the middle of the matrix.

5. What does the row player's maximin strategy mean?

   • Choosing the largest entry overall.
   • Choosing the column with the smallest maximum.
   • Choosing the row whose maximum payoff is smallest.
   • Choosing the row whose minimum payoff is largest.

6. What does the column player's minimax strategy mean?

   • Choosing the row whose maximum entry is smallest.
   • Choosing the column whose maximum entry is smallest.
   • Choosing the largest entry overall.
   • Choosing the column whose minimum entry is largest.

7. When a zero-sum game has a stable solution, what is its value?

   • The sum of all the entries in the matrix.
   • The common value of the row maximin and the column minimax.
   • Always zero.
   • The average of the row maximin and the column minimax.

8. What is the maximin of the pay-off matrix [[3, -1, 2], [1, 0, 4]]?

   • 0
   • 2
   • 1
   • -1

9. What is the minimax of the pay-off matrix [[3, -1, 2], [1, 0, 4]]?

   • 2
   • 0
   • 1
   • 4

10. Why is a two-person zero-sum game called zero-sum?

    • Each player's gain is exactly the other's loss, so the total of the payoffs is always zero.
    • Both players always finish with a score of zero.
    • The game always ends in a draw.
    • The matrix always contains a zero entry.

11. Where is the saddle point of the pay-off matrix [[3, -1, 2], [1, 0, 4]]?

    • Row 2, column 3
    • Row 1, column 1
    • Row 2, column 2
    • Row 1, column 2

12. Does the matrix [[2, -1], [-3, 4]] have a stable solution?

    • Yes, because the maximin and minimax are both 2.
    • No, because the maximin is -1 but the minimax is 2.
    • Yes, because both maxima equal 4.
    • Yes, because the maximin and minimax are both -1.

13. For the matrix [[2, -1], [-3, 4]], what is the row maximin?

    • -3
    • 2
    • 4
    • -1

14. For the matrix [[2, -1], [-3, 4]], what is the column minimax?

    • -1
    • -3
    • 2
    • 4

15. What is the value of the game with pay-off matrix [[4, 6], [2, 1]]?

    • 2
    • 1
    • 6
    • 4

16. In a game with a stable solution, is the row player's optimal strategy pure?

    • Yes, the row player uses the saddle row with certainty.
    • No, the row player must mix two columns.
    • No, the row player must mix two rows with probabilities one half each.
    • Only if the matrix is 2 by 2.

17. A pay-off matrix has the saddle point 0 at (row 2, column 2). What does this tell the row player about row 2?

    • Row 2 guarantees at least 0, and the column player cannot force a loss above 0 by changing column.
    • Row 1 guarantees 3, so row 1 is always better.
    • Both players should switch to mixed strategies.
    • Column 1 guarantees zero loss only, so the row player should switch.

18. Why is the maximin of a matrix never greater than its minimax?

    • The minimax is always the average of the maximin.
    • There are fewer rows than columns.
    • Every row minimum is at most every column maximum, so the largest row minimum cannot exceed the smallest column maximum.
    • All payoffs are non-negative.

19. What is the value of the game with pay-off matrix [[1, 3, 2], [4, 3, 5], [2, 1, 0]]?

    • 5
    • 3
    • 2
    • 4

20. For a zero-sum game with row maximin 2 and column minimax 5, what can be concluded?

    • The game value is 3.5.
    • The row maximin must be recalculated.
    • The game is a draw.
    • There is no stable solution, so mixed strategies are needed.

Related quizzes

• Reducing pay-off matrices by dominance Quiz · 4D.5.3 · 20 questions
• Proof by mathematical induction Quiz · 1.1 · 20 questions
• Quadratic equations and complex arithmetic Quiz · 2.1-2.2 · 20 questions
• Matrix arithmetic and inverses Quiz · 3.1-3.2 · 20 questions
• Expectation of discrete random variables Quiz · 3B.1.1 · 20 questions
• The Poisson distribution Quiz · 3B.2.1 · 20 questions
• Geometric and negative binomial models Quiz · 3B.3.1 · 20 questions
• Hypothesis tests for the Poisson distribution Quiz · 3B.4.1 · 20 questions
• Applying the Central Limit Theorem Quiz · 3B.5.1 · 20 questions
• Goodness of fit tests for discrete distributions Quiz · 3B.6.1 · 20 questions