4D.5.4 Mixed strategies by graphical methods Quiz: Pearson Edexcel Further Maths, Unit 44

The 20 questions

1. When a 2 by n game has no stable solution, what is plotted in the graphical method?

   • The expected payoff against each column, as a function of the probability p of the first row.
   • Each cell's payoff as a single point in the plane.
   • The row maxima against the column minima.
   • The objective function P against x and y.

2. Which games can the graphical method handle in this specification?

   • Any game with n greater than 10.
   • Only 2 by 2 games.
   • 2 by n or n by 2 games, where n = 1, 2, 3 or 4.
   • Only games that have a stable solution.

3. What is a mixed strategy?

   • A strategy that is always uniform over all options.
   • A strategy used only when the value of the game is zero.
   • A probability distribution over the pure strategies, chosen to make the opponent indifferent.
   • A strategy in which every entry is equal to the maximin.

4. How is the value of a 2 by n game read from its graph?

   • It is the height of the lowest point of the upper envelope.
   • It is the payoff in the top-left entry of the matrix.
   • It is the height of the highest point of the lower envelope of the column lines.
   • It is the average of the column payoffs at p = 1/2.

5. On the graph of a 2 by n game, what does the lower envelope show?

   • The maximum of the column lines at each p, which the row player wants to minimise.
   • The sum of all the column lines.
   • The single line through the origin.
   • The minimum of the column lines at each p, which the row player wants to maximise.

6. Why does the optimal probability occur where two lines of a graph cross?

   • Because probabilities must be whole numbers.
   • Because the best guaranteed payoff occurs where the minimum changes from one line to another.
   • Because intersections always give a payoff of zero.
   • Because the lines on the graph are parallel.

7. For a 2 by 2 game with no stable solution, how is the optimal probability found?

   • By setting the two row sums equal to one.
   • By setting the largest entry equal to zero.
   • By setting the two column payoffs equal and solving for p.
   • By adding the two columns together.

8. For the matrix [[2, -1], [-1, 3]], if the row player plays row 1 with probability p, what is the expected payoff against column 1?

   • 2p - 1
   • 3p - 1
   • p - 1
   • 3p + 1

9. For the matrix [[2, -1], [-1, 3]], what is the expected payoff against column 2 when row 1 is played with probability p?

   • 4p - 3
   • 3 - 4p
   • 3p - 1
   • 3 - p

10. For the matrix [[2, -1], [-1, 3]], what is the optimal probability p of playing row 1?

    • 4/7
    • 5/7
    • 1/2
    • 3/7

11. For the matrix [[2, -1], [-1, 3]], what is the value of the game?

    • 4/7
    • 5/7
    • 2/7
    • 1

12. For the matrix [[1, -2], [-3, 4]], what is the probability of playing row 1 in the optimal strategy?

    • 7/10
    • 4/5
    • 3/10
    • 1/2

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

    • -1/5
    • -2
    • 7/10
    • 1/5

14. For the matrix [[3, 0], [1, 2]], what is the value of the game?

    • 3/2
    • 5/4
    • 1
    • 2

15. For the matrix [[3, 0], [1, 2]], what is the probability that the row player plays row 2 in the optimal strategy?

    • 3/4
    • 1/2
    • 1/4
    • 2/3

16. For the matrix [[2, -1], [-1, 3]], why must the two column payoffs be equal at the optimum?

    • Otherwise the column player could shift probability to the cheaper column and lower the row player's guaranteed payoff.
    • Because probabilities must always sum to one.
    • Because the lines on the graph are parallel.
    • Otherwise the matrix would have a saddle point.

17. If the row player plays p = 1/2 in the matrix [[2, -1], [-1, 3]], what is the worst-case payoff?

    • 1/2
    • 1
    • 5/7
    • 0

18. For the matrix [[3, 0], [1, 2]], what is the optimal probability that the column player plays column 1?

    • 1/4
    • 1/2
    • 3/4
    • 2/3

19. For the matrix [[2, -1], [-1, 3]], what is the optimal probability that the column player plays column 1?

    • 5/7
    • 3/7
    • 1/2
    • 4/7

20. In the graph for [[2, -1], [-1, 3]] at p = 4/7, what happens to the column lines?

    • Only the column 2 line is active at height 3.
    • Both column lines are at height 4/7.
    • The lines are parallel at height 0.
    • Both column lines meet at height 5/7, which is the peak of the lower envelope.

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