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 the sum of both players' payoffs?

   • Zero
   • Positive one
   • Undefined
   • Negative one

2. From whose perspective is a standard game theory payoff matrix written?

   • Row player
   • Referee
   • Column player
   • Observer

3. When does a zero-sum game possess a stable solution in pure strategies?

   • Payoffs are zero
   • Matrix is square
   • Maximin equals minimax
   • Rows equal columns

4. What is a saddle point in a pay-off matrix?

   • Maximin equals minimax
   • Matrix is symmetrical
   • Row sum equals zero
   • All entries are positive

5. For the row player, what does a maximin strategy involve?

   • Minimising row totals
   • Maximising average gains
   • Minimising maximum losses
   • Maximising minimum payoffs

6. For the column player, what does a minimax strategy involve?

   • Minimising average gains
   • Minimising maximum payoffs
   • Maximising column totals
   • Maximising minimum losses

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

   • Zero always
   • Common maximin value
   • Sum of all payoffs
   • Maximum matrix entry

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 game termed zero-sum?

    • Payoffs cancel out
    • No player can win
    • Total risk is zero
    • All entries are zero

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. What prevents a pay-off matrix from having a stable solution?

    • Matrix size is odd
    • Maximin unequal to minimax
    • Row sums are non-zero
    • All entries are negative

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. What type of optimal strategy does a row player use when a stable solution exists?

    • Pure strategy
    • Randomised strategy
    • Mixed strategy
    • Null strategy

17. In a game with a saddle point value of 0, what is guaranteed?

    • Minimum payoff of zero
    • Guaranteed total win
    • Equal player strategies
    • No possible losses

18. What inequality always holds between the maximin and minimax values in a zero-sum game?

    • Maximin ≥ minimax
    • Maximin = minimax
    • Maximin ≤ minimax
    • Maximin > minimax

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. If maximin is 2 and minimax is 5, what type of strategy must be used?

    • Pure strategy
    • Dominant strategy
    • Stable strategy
    • Mixed strategy

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