7.08c Pure strategies: play-safe strategies and stable solutions

4

1. Two people, Adam and Bill, play a zero-sum game. The game is represented by the following pay-off matrix for Adam. 4
2. Roza plays a different zero-sum game against a computer. The game is represented by the following pay-off matrix for Roza.

2 Alex and Roberto play a zero-sum game. The game is represented by the following pay-off matrix for Alex. \begin{table}[h] \end{table}

1. Show that this game has a stable solution and state the play-safe strategy for each player.
2. List any saddle points.

5 Mark and Owen play a zero-sum game. The game is represented by the following pay-off matrix for Mark.

1. Explain why Mark should never play strategy B.
2. It is given that the value of the game is 0.6 . Find the optimal strategy for Owen.
   (You are not required to find the optimal mixed strategy for Mark.)
   [0pt] [7 marks]

2 Stan and Christine play a zero-sum game. The game is represented by the following pay-off matrix for Stan. \begin{table}[h] \end{table}

1. Find the play-safe strategy for each player.
2. Show that there is no stable solution.
3. Explain why a suitable pay-off matrix for Christine is given by

4 Monica and Vladimir play a zero-sum game. The game is represented by the following pay-off matrix for Monica.

3. A two-person zero-sum game is represented by the following pay-off matrix for player A.

1. Identify the play safe strategies for each player.
2. State, giving a reason, whether there is a stable solution to this game.
3. Find the best strategy for player A.
4. Find the value of the game to player B.

2. A two-person zero-sum game is represented by the following pay-off matrix for player A.

1. Identify the play safe strategies for each player.
2. State, giving a reason, whether there is a stable solution to this game.
3. Explain why the game above can be reduced to the following \(3 \times 3\) game.

   - 3	2	5
   - 2	5	4
   2	- 3	- 1

4. Formulate the \(3 \times 3\) game as a linear programming problem for player A, defining your variables clearly and writing the constraints as inequalities.

4. Eugene and Stephen play a zero-sum game. The pay-off matrix shows the number of points that Eugene scores for each combination of strategies.

1. Find the play-safe strategies for each of Eugene and Stephen, and hence show that this zero-sum game does not have a stable solution.
2. Suppose that Eugene knows that Stephen will use his play-safe strategy. Explain why Eugene should change from his play-safe strategy. You should state as part of your answer which strategy Eugene should now play.
3. Formulate the game as a linear programming problem for Stephen. Define your variables clearly. Write the constraints as equations.

3 Rose and Colin repeatedly play a zero-sum game. The pay-off matrix shows the number of points won by Rose for each combination of strategies.

1. What is the greatest number of points that Colin can win when Rose plays strategy \(A\) and which strategy does Colin need to play to achieve this?
2. Show that strategy \(B\) dominates strategy \(C\) and also that strategy \(Y\) dominates strategy \(Z\). Hence reduce the game to a \(3 \times 3\) pay-off matrix.
3. Find the play-safe strategy for each player on the reduced game. Is the game stable? Rose makes a random choice between the strategies, choosing strategy \(A\) with probability \(p _ { 1 }\), strategy \(B\) with probability \(p _ { 2 }\) and strategy \(D\) with probability \(p _ { 3 }\). She formulates the following LP problem to be solved using the Simplex algorithm: $$\begin{array} { l l } \text { maximise } & M = m - 5 , \\ \text { subject to } & m \leqslant 4 p _ { 1 } + 10 p _ { 2 } , \\ & m \leqslant 9 p _ { 1 } + 3 p _ { 2 } + 11 p _ { 3 } , \\ & m \leqslant 2 p _ { 1 } + 10 p _ { 2 } + p _ { 3 } , \\ & p _ { 1 } + p _ { 2 } + p _ { 3 } \leqslant 1 , \\ \text { and } & p _ { 1 } \geqslant 0 , p _ { 2 } \geqslant 0 , p _ { 3 } \geqslant 0 , m \geqslant 0 . \end{array}$$ (You are not required to solve this problem.)
4. Explain how \(9 p _ { 1 } + 3 p _ { 2 } + 11 p _ { 3 }\) was obtained. A computer gives the solution to the LP problem as \(p _ { 1 } = \frac { 7 } { 48 } , p _ { 2 } = \frac { 27 } { 48 } , p _ { 3 } = \frac { 14 } { 48 }\).
5. Calculate the value of \(M\) at this solution.

4 Euan and Wai Mai play a zero-sum game. Each is trying to maximise the total number of points that they score in many repeats of the game. The table shows the number of points that Euan scores for each combination of strategies.

1. Explain what the term 'zero-sum game' means.
2. How many points does Wai Mai score if she chooses \(X\) and Euan chooses \(A\) ?
3. Why should Wai Mai never choose strategy \(Z\) ?
4. Delete the column for \(Z\) and find the play-safe strategy for Euan and the play-safe strategy for Wai Mai on the table that remains. Is the resulting game stable or not? State how you know. The value 3 in the cell corresponding to Euan choosing \(D\) and Wai Mai choosing \(X\) is changed to - 5 ; otherwise the table is unchanged. Wai Mai decides that she will choose her strategy by making a random choice between \(X\) and \(Y\), choosing \(X\) with probability \(p\) and \(Y\) with probability \(1 - p\).
5. Write down and simplify an expression for the expected score for Wai Mai when Euan chooses each of his four strategies.
6. Using graph paper, draw a graph showing Wai Mai's expected score against \(p\) for each of Euan's four strategies and hence calculate the optimum value of \(p\).

1. The payoff matrix for player \(A\) in a two-person zero-sum game with value \(V\) is shown below.

Formulate this information as a linear programming problem, the solution to which will give the optimal strategy for player \(B\).

1. Rewrite the matrix as necessary and state the new value of the game, \(v\), in terms of \(V\).
2. Define your decision variables.
3. Write down the objective function in terms of your decision variables.
4. Write down the constraints.
   

2. The payoff matrix for player \(A\) in a two-person zero-sum game is shown below.

1. Applying the dominance rule, explain, with justification, which strategy can be ignored by
   1. player \(A\),
   2. player \(B\).
2. For the reduced table, find the optimal strategy for
   1. player \(A\),
   2. player \(B\).

6. The payoff matrix for player \(A\) in a two-person zero-sum game is shown below.

1. Explain why the game does not have a saddle point.
2. Using a graphical method, find the optimal strategy for player \(B\).
3. Find the optimal strategy for player \(A\).
4. Find the value of the game.

2. A two-person zero-sum game is represented by the payoff matrix for player \(A\) shown below.

1. Represent the expected payoffs to \(A\) against \(B\) 's strategies graphically and hence determine which strategy is not worth considering for player \(B\).
2. Find the best strategy for player \(A\) and the value of the game.

3 Alex and Sam are playing a zero-sum game. The game is represented by the pay-off matrix for Alex. 3

1. Explain why the value of the game is 2
   3
2. Identify the play-safe strategy for each player.
   Each pipe is labelled with its upper capacity in \(\mathrm { cm } ^ { 3 } \mathrm {~s} ^ { - 1 }\) \includegraphics[max width=\textwidth, alt={}, center]{5a826f8b-4751-4589-ad0a-109fc5c821f2-04_620_940_450_550}

3 Summer and Haf play a zero-sum game. The pay-off matrix for the game is shown below. Haf 3

1. Show that the game has a stable solution.
   3
2. (i) State the value of the game for Summer. 3 (b) (ii) State the play-safe strategy for each player.

6 Xander and Yvonne are playing a zero-sum game. The game is represented by the pay-off matrix for Xander. \begin{table}[h] \end{table} 6

1. Show that the game has a stable solution.
   6
2. State the play-safe strategy for each player. Play-safe strategy for Xander is \(\_\_\_\_\) Play-safe strategy for Yvonne is \(\_\_\_\_\) 6
3. The game that Xander and Yvonne are playing is part of a marbles challenge. The pay-off matrix values represent the number of marbles gained by Xander in each game. In the challenge, the game is repeated until one player has 24 marbles more than the other player. Explain why Xander and Yvonne must play at least 3 games to complete the challenge.

6 Victoria and Albert play a zero-sum game. The game is represented by the following pay-off matrix for Victoria. 6

1. Find the play-safe strategies for each player.
   6
2. State, with a reason, the strategy that Albert should never play.
   6
3. (i) Determine an optimal mixed strategy for Victoria.
   [0pt] [5 marks]
   6 (c) (ii) Find the value of the game for Victoria.
   6 (c) (iii) State an assumption that must made in order that your answer for part (c)(ii) is the maximum expected pay-off that Victoria can achieve.

1 Deanna and Will play a zero-sum game.
The game is represented by the following pay-off matrix for Deanna. Which strategy is Deanna's play-safe strategy?
Circle your answer.
A
B
C
D

7 Avon and Roj play a zero-sum game. The game is represented by the following pay-off matrix for Avon. 7 (c)

1. Find the optimal mixed strategy for Avon.
   7
2. Find the value of the game for Avon.
   

7 (d) Roj thinks that his best outcome from the game is to play strategy \(\mathbf { R } _ { \mathbf { 2 } }\) each time. Avon notices that Roj always plays strategy \(\mathbf { R } _ { \mathbf { 2 } }\) and Avon wants to use this knowledge to maximise his expected pay-off from the game. Explain how your answer to part (c)(i) should change and find Avon's maximum expected pay-off from the game. \includegraphics[max width=\textwidth, alt={}, center]{59347089-ea4a-4ee6-b40e-1ab78aa7cdc3-16_2490_1735_219_139}

7. A two person zero-sum game is represented by the pay-off matrix for player A shown above. It is given that \(k\) is an integer.

1. Show that Q is the play-safe option for player A regardless of the value of \(k\). Given that Z is the play-safe option for player B ,
2. determine the range of possible values of \(k\). You must make your working clear.
3. Explain why player B should never play option X. You must make your reasoning clear. Player A intends to make a random choice between options \(\mathrm { Q } , \mathrm { R }\) and S , choosing option Q with probability \(p _ { 1 }\), option R with probability \(p _ { 2 }\) and option S with probability \(p _ { 3 }\) Player A wants to find the optimal values of \(p _ { 1 } , p _ { 2 }\) and \(p _ { 3 }\) using the Simplex algorithm.
   Given that \(k > - 4\), player A formulates the following objective function for the corresponding linear program. $$\text { Maximise } P = V \text {, where } V = \text { the value of the original game } + 4$$
4. 1. Formulate the constraints of the linear programming problem for player A. You should write the constraints as equations.
   2. Write down an initial Simplex tableau, making your variables clear. The Simplex algorithm is used to solve the linear programming problem. It is given that in the final Simplex tableau the optimal value of \(p _ { 1 } = \frac { 7 } { 37 }\), the optimal value of \(p _ { 2 } = \frac { 17 } { 37 }\) and all the slack variables are zero.
5. Determine the value of \(k\), making your method clear.

A two-person zero-sum game is represented by the following pay-off matrix for player \(A\).

	I	II	III	IV
I	\(-4\)	\(-5\)	\(-2\)	4
II	\(-1\)	1	\(-1\)	2
III	0	5	\(-2\)	\(-4\)
IV	\(-1\)	3	\(-1\)	1

1. Determine the play-safe strategy for each player. [4]
2. Verify that there is a stable solution and determine the saddle points. [3]
3. State the value of the game to \(B\). [1]

Emma and Freddie play a zero-sum game. This game is represented by the following pay-off matrix for Emma. \(\begin{pmatrix} -4 & -1 & 3 \\ 2 & 1 & -2 \end{pmatrix}\)

1. Show that there is no stable solution. [3]
2. Find the best strategy for Emma and the value of the game to her. [8]
3. Write down the value of the game to Freddie and his pay-off matrix. [3]

(Total 14 marks)

The payoff matrix for player X in a two-person zero-sum game is shown below.

	Y
	\(Y_1\)	\(Y_2\)
\multirow{2}{*}{X}	\(X_1\)	\(-2\)	4
	\(X_2\)	6	\(-1\)

1. Explain why the game does not have a saddle point. [3 marks]
2. Find the optimal strategy for
   1. player X, [8 marks]
   2. player Y.
3. Find the value of the game. [2 marks]

The payoff matrix for player \(A\) in a two-person zero-sum game is shown below. \begin{array}{c|c|c|c|c} & & \multicolumn{3}{c}{B}
& & \text{I} & \text{II} & \text{III}
\hline \multirow{3}{*}{A} & \text{I} & -3 & 4 & 0
& \text{II} & 2 & 2 & 1
& \text{III} & 3 & -2 & -1
\end{array} Find the optimal strategy for each player and the value of the game. [4 marks]