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

3

1. Two people, Ann and Bill, play a zero-sum game. The game is represented by the following pay-off matrix for Ann.

   \multirow{5}{*}{Ann}	Bill
   	Strategy	\(\mathbf { B } _ { \mathbf { 1 } }\)	\(\mathbf { B } _ { \mathbf { 2 } }\)	\(\mathbf { B } _ { \mathbf { 3 } }\)
   	\(\mathbf { A } _ { \mathbf { 1 } }\)	-1	0	-2
   	\(\mathbf { A } _ { \mathbf { 2 } }\)	4	-2	-3
   	\(\mathbf { A } _ { \mathbf { 3 } }\)	-4	-5	-3

   Show that this game has a stable solution and state the play-safe strategies for Ann and Bill.
2. Russ and Carlos play a different zero-sum game, which does not have a stable solution. The game is represented by the following pay-off matrix for Russ.

   	Carlos
   \cline { 2 - 5 }	Strategy	\(\mathbf { C } _ { \mathbf { 1 } }\)	\(\mathbf { C } _ { \mathbf { 2 } }\)	\(\mathbf { C } _ { \mathbf { 3 } }\)
   \cline { 2 - 5 } Russ	\(\mathbf { R } _ { \mathbf { 1 } }\)	- 4	7	- 3
   \cline { 2 - 5 }	\(\mathbf { R } _ { \mathbf { 2 } }\)	2	- 1	1

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

3 Two people, Rhona and Colleen, play a zero-sum game. The game is represented by the following pay-off matrix for Rhona. It is given that \(x < 2\).

1. 1. Write down the three row minima.
   2. Show that there is no stable solution.
2. Explain why Rhona should never play strategy \(R _ { 3 }\).
3. 1. Find the optimal mixed strategy for Rhona.
   2. Find the value of the game.

3 Two people, Roz and Colum, play a zero-sum game. The game is represented by the following pay-off matrix for Roz.

1. Explain what is meant by the term 'zero-sum game'.
2. Determine the play-safe strategy for Colum, giving a reason for your answer.
3. 1. Show that the matrix can be reduced to a 2 by 3 matrix, giving the reason for deleting one of the rows.
   2. Hence find the optimal mixed strategy for Roz.

2 Harry and Will play a zero-sum game. The game is represented by the following pay-off matrix for Harry.

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

6 Kate and Pippa play a zero-sum game. The game is represented by the following pay-off matrix for Kate. \includegraphics[max width=\textwidth, alt={}, center]{3ba973a1-6a45-4381-b634-e9c4673ef1fb-18_2482_1707_223_155}

4 Two people, Roger and Corrie, play a zero-sum game.
The game is represented by the following pay-off matrix for Roger.

1. 1. Find the optimal mixed strategy for Roger.
   2. Show that the value of the game is \(\frac { 7 } { 13 }\).
2. Given that the value of the game is \(\frac { 7 } { 13 }\), find the optimal mixed strategy for Corrie.
   \includegraphics[max width=\textwidth, alt={}]{c4dc61a7-47ee-4d5c-bf6d-30a5da2ee8dd-09_2484_1709_223_153}

3

1. Two people, Tom and Jerry, play a zero-sum game. The game is represented by the following pay-off matrix for Tom.

   	Jerry
   \cline { 2 - 5 }	Strategy	A	B	C
   Tom	I	- 4	5	- 3
   \cline { 2 - 5 }	II	- 3	- 2	8
   \cline { 2 - 5 }	III	- 7	6	- 2

   Show that this game has a stable solution and state the play-safe strategy for each player.
2. Rohan and Carla play a different zero-sum game for which there is no stable solution. The game is represented by the following pay-off matrix for Rohan. \begin{table}[h]
   \captionsetup{labelformat=empty} \caption{Carla} Rohan

   Strategy	\(\mathbf { C } _ { \mathbf { 1 } }\)	\(\mathbf { C } _ { \mathbf { 2 } }\)	\(\mathbf { C } _ { \mathbf { 3 } }\)
   \(\mathbf { R } _ { \mathbf { 1 } }\)	3	5	- 1
   \(\mathbf { R } _ { \mathbf { 2 } }\)	1	- 2	4

   \end{table}
   1. Find the optimal mixed strategy for Rohan and show that the value of the game is \(\frac { 3 } { 2 }\).
   2. Carla plays strategy \(\mathrm { C } _ { 1 }\) with probability \(p\), and strategy \(\mathrm { C } _ { 2 }\) with probability \(q\). Find the values of \(p\) and \(q\) and hence find the optimal mixed strategy for Carla.
      (4 marks)
      
      \includegraphics[max width=\textwidth, alt={}]{1aca4e91-d1b3-4a78-8880-e42a4fbf3ddb-10_2486_1714_221_153}
      \includegraphics[max width=\textwidth, alt={}]{1aca4e91-d1b3-4a78-8880-e42a4fbf3ddb-11_2486_1714_221_153}

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

1. Find the play-safe strategy for each player.
2. Show that there is no stable solution.
3. Explain why Juliet should never play strategy D.
4. 1. Explain why the following is a suitable pay-off matrix for Juliet.

      4	5	- 1
      0	- 3	2

   2. Hence find the optimal strategy for Juliet.
   3. Find the value of the game for Juliet.

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

1. Verify that there is no stable solution to this game.
2. Explain why the \(4 \times 4\) game above may be reduced to the following \(3 \times 3\) game.
3. Formulate the \(3 \times 3\) game as a linear programming problem for player A. Write the

   - 2	1	3
   - 1	3	2
   1	- 2	- 1

   constraints as inequalities. Define your variables clearly.

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

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

4. Andrew ( \(A\) ) and Barbara ( \(B\) ) play a zero-sum game. This game is represented by the following payoff matrix for Andrew. $$A \left( \begin{array} { c c c } & B & \\ 3 & 5 & 4 \\ 1 & 4 & 2 \\ 6 & 3 & 7 \end{array} \right)$$

1. Explain why this matrix may be reduced to $$\left( \begin{array} { l l } 3 & 5 \\ 6 & 3 \end{array} \right)$$
2. Hence find the best strategy for each player and the value of the game.
   (8)

4. 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. Verify that there is no stable solution to this game.
3. Explain why the pay-off matrix above may be reduced to

   \cline { 2 - 4 } \multicolumn{1}{c|}{}	\(B\) plays I	\(B\) plays II	\(B\) plays III
   \(A\) plays I	2	- 1	3
   \(A\) plays II	1	3	0

4. Find the best strategy for player \(A\), and the value of the game.

7. (a) Explain briefly what is meant by a zero-sum game. A two person zero-sum game is represented by the following pay-off matrix for player \(A\). (b) Verify that there is no stable solution to this game.
(c) Find the best strategy for player \(A\) and the value of the game to her.
(d) Formulate the game as a linear programming problem for player \(B\). Write the constraints as inequalities and define your variables clearly.
(Total 17 marks)

2. Denis (D) and Hilary (H) play a two-person zero-sum game represented by the following pay-off matrix for Denis.

1. Show that there is no stable solution to this game.
2. Find the best strategy for Denis and the value of the game to him.
   (10) (Total 13 marks)

5. (a) In game theory, explain the circumstances under which column \(( x )\) dominates column \(( y )\) in a two-person zero-sum game. Liz and Mark play a zero-sum game. This game is represented by the following pay-off matrix for Liz. (b) Verify that there is no stable solution to this game.
(c) Find the best strategy for Liz and the value of the game to her. The game now changes so that when Liz plays 1 and Mark plays 3 the pay-off to Liz changes from 2 to
4. All other pay-offs for this zero-sum game remain the same.
(d) Explain why a graphical approach is no longer possible and briefly describe the method Liz should use to determine her best strategy.
(2) (Total 16 marks)

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

1. Verify that there is no stable solution to this game.
2. Reduce the game so that player B has a choice of only two actions.
3. Write down the reduced pay-off matrix for player B.
4. Find the best strategy for player B and the value of the game to player B.

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

1. Verify that there is no stable solution to this game.
2. Find the best strategy for player A.

2. Rani and Greg play a zero-sum game. The pay-off matrix shows the number of points that Rani scores for each combination of strategies.

1. Explain what the term 'zero-sum game' means.
2. State the number of points that Greg scores if he plays his strategy 3 and Rani plays her strategy 3.
3. Verify that there is no stable solution to this game.
4. Reduce the game so that Greg has only two possible strategies. Write down the reduced pay-off matrix for Greg.
5. Find the best strategy for Greg and the value of the game to him.

3. A two-person zero-sum game is represented by the following pay-off matrix for player A. Find the best strategy for player A and the value of the game.
(Total 7 marks)

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

1. Verify that there is no stable solution to this game.
2. Find the best strategy for player \(A\) and the value of the game to her.
   (Total 11 marks)

6 Lucy and Maria repeatedly play a zero-sum game. The pay-off matrix shows the number of points won by Lucy, who is playing rows, for each combination of strategies.