7.08e Mixed strategies: optimal strategy using equations or graphical method

88 questions

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AQA D2 2010 January Q3
12 marks Easy -1.3
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\(\mathbf { B } _ { \mathbf { 1 } }\)\(\mathbf { B } _ { \mathbf { 2 } }\)\(\mathbf { B } _ { \mathbf { 3 } }\)
    \(\mathbf { A } _ { \mathbf { 1 } }\)-10-2
    \(\mathbf { A } _ { \mathbf { 2 } }\)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\(\mathbf { R } _ { \mathbf { 1 } }\)- 47- 3
    \cline { 2 - 5 }\(\mathbf { R } _ { \mathbf { 2 } }\)2- 11
    1. Find the optimal mixed strategy for Russ.
    2. Find the value of the game.
AQA D2 2011 January Q3
13 marks Easy -1.8
3 Two people, Rhona and Colleen, play a zero-sum game. The game is represented by the following pay-off matrix for Rhona.
\cline { 2 - 5 }Colleen
\cline { 2 - 5 } Strategy\(\mathbf { C } _ { \mathbf { 1 } }\)\(\mathbf { C } _ { \mathbf { 2 } }\)\(\mathbf { C } _ { \mathbf { 3 } }\)
\cline { 2 - 5 } Rhona\(\mathbf { R } _ { \mathbf { 1 } }\)264
\cline { 2 - 5 }\(\mathbf { R } _ { \mathbf { 2 } }\)3- 3- 1
\cline { 2 - 5 }\(\mathbf { R } _ { \mathbf { 3 } }\)\(x\)\(x + 3\)3
\cline { 2 - 5 }
\cline { 2 - 5 }
It is given that \(x < 2\).
    1. Write down the three row minima.
    2. Show that there is no stable solution.
  2. Explain why Rhona should never play strategy \(R _ { 3 }\).
    1. Find the optimal mixed strategy for Rhona.
    2. Find the value of the game.
AQA D2 2012 January Q3
13 marks Easy -2.5
3 Two people, Roz and Colum, play a zero-sum game. The game is represented by the following pay-off matrix for Roz.
Colum
\cline { 2 - 5 }Strategy\(\mathbf { C } _ { \mathbf { 1 } }\)\(\mathbf { C } _ { \mathbf { 2 } }\)\(\mathbf { C } _ { \mathbf { 3 } }\)
\multirow{3}{*}{\(\operatorname { Roz }\)}\(\mathbf { R } _ { \mathbf { 1 } }\)- 2- 6- 1
\cline { 2 - 5 }\(\mathbf { R } _ { \mathbf { 2 } }\)- 52- 6
\cline { 2 - 5 }\(\mathbf { R } _ { \mathbf { 3 } }\)- 33- 4
  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.
    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.
AQA D2 2010 June Q4
13 marks Moderate -0.5
4 Two people, Roger and Corrie, play a zero-sum game.
The game is represented by the following pay-off matrix for Roger.
Corrie
\cline { 2 - 5 }Strategy\(\mathbf { C } _ { \mathbf { 1 } }\)\(\mathbf { C } _ { \mathbf { 2 } }\)\(\mathbf { C } _ { \mathbf { 3 } }\)
\cline { 2 - 5 } Roger\(\mathbf { R } _ { \mathbf { 1 } }\)73- 5
\cline { 2 - 5 }\(\mathbf { R } _ { \mathbf { 2 } }\)- 2- 14
\cline { 2 - 5 }
\cline { 2 - 5 }
    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}
AQA D2 2011 June Q3
15 marks Easy -1.8
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 }StrategyABC
    TomI- 45- 3
    \cline { 2 - 5 }II- 3- 28
    \cline { 2 - 5 }III- 76- 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\(\mathbf { C } _ { \mathbf { 1 } }\)\(\mathbf { C } _ { \mathbf { 2 } }\)\(\mathbf { C } _ { \mathbf { 3 } }\)
    \(\mathbf { R } _ { \mathbf { 1 } }\)35- 1
    \(\mathbf { R } _ { \mathbf { 2 } }\)1- 24
    \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}
AQA D2 2013 June Q5
15 marks Easy -2.5
5 Romeo and Juliet play a zero-sum game. The game is represented by the following pay-off matrix for Romeo.
Juliet
\cline { 2 - 5 }StrategyDEF
A4- 40
\cline { 2 - 5 } RomeoB- 2- 53
\cline { 2 - 5 }C21- 2
\cline { 2 - 5 }
\cline { 2 - 5 }
  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.
    1. Explain why the following is a suitable pay-off matrix for Juliet.
      45- 1
      0- 32
    2. Hence find the optimal strategy for Juliet.
    3. Find the value of the game for Juliet.
Edexcel D2 2006 January Q5
13 marks Moderate -0.5
5. A two-person zero-sum game is represented by the following pay-off matrix for player A.
B plays 1B plays 2B plays 3B plays 4
A plays 1- 213- 1
A plays 2- 1321
A plays 3- 420- 1
A plays 41- 2- 13
  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
    - 213
    - 132
    1- 2- 1
    constraints as inequalities. Define your variables clearly.
Edexcel D2 2002 June Q4
8 marks Moderate -0.5
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)
Edexcel D2 2003 June Q4
14 marks Moderate -0.5
4. A two person zero-sum game is represented by the following pay-off matrix for player \(A\).
\cline { 2 - 4 } \multicolumn{1}{c|}{}\(B\) plays I\(B\) plays II\(B\) plays III
\(A\) plays I2- 13
\(A\) plays II130
\(A\) plays III01- 3
  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 I2- 13
    \(A\) plays II130
  4. Find the best strategy for player \(A\), and the value of the game.
Edexcel D2 2005 June Q7
17 marks Standard +0.3
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 \(A\).
IIIIII
I523
II354
(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)
Edexcel D2 2007 June Q2
13 marks Moderate -0.5
2. Denis (D) and Hilary (H) play a two-person zero-sum game represented by the following pay-off matrix for Denis.
H plays 1H plays 2H plays 3
D plays 12- 13
D plays 2- 34- 4
  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)
Edexcel D2 2008 June Q5
16 marks Moderate -0.8
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.
Mark plays 1Mark plays 2Mark plays 3
Liz plays 1532
Liz plays 2456
Liz plays 3643
(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)
Edexcel D2 2009 June Q3
13 marks Standard +0.3
3. A two-person zero-sum game is represented by the following pay-off matrix for player A.
B plays 1B plays 2B plays 3
A plays 1- 56- 3
A plays 21- 413
A plays 3- 23- 1
  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.
Edexcel D2 2011 June Q4
9 marks Easy -1.8
4. Laura and Sam play a zero-sum game. This game is represented by the following pay-off matrix for Laura.
S plays 1S plays 2S plays 3
L plays 1- 4- 11
L plays 23- 1- 2
L plays 3- 302
Find the best strategy for Laura and the value of the game to her.
Edexcel D2 2012 June Q5
9 marks Moderate -0.3
5. Agent Goodie is planning to break into Evil Doctor Fiendish's secret base. He uses game theory to determine whether to approach the base from air, sea or land.
Evil Doctor Fiendish decides each day which of three possible plans he should use to protect his base. Agent Goodie evaluates the situation. He assigns numbers, negative indicating he fails in his mission, positive indicating success, to create a pay-off matrix. The numbers range from - 3 (he fails in his mission and is captured) to 5 (he successfully achieves his mission and escapes uninjured) and the pay-off matrix is shown below.
Fiendish uses plan 1Fiendish uses plan 2Fiendish uses plan 3
Air045
Sea2-31
Land-23-2
  1. Reduce the game so that Agent Goodie has only two choices, explaining your reasoning.
  2. Use game theory to determine Agent Goodie's best strategy.
  3. Find the value of the game to Agent Goodie.
Edexcel D2 2013 June Q4
9 marks Standard +0.8
4. Robin (R) and Steve (S) play a two-person zero-sum game which is represented by the following pay-off matrix for Robin.
S plays 1S plays 2S plays 3
R plays 1213
R plays 21- 12
R plays 3- 13- 3
Find the best strategy for Robin and the value of the game to him.
Edexcel D2 2013 June Q4
11 marks Moderate -0.5
4. A two-person zero-sum game is represented by the following pay-off matrix for player A.
B plays 1B plays 2B plays 3
A plays 154- 6
A plays 2- 1- 23
A plays 31- 12
  1. Reduce the game so that player B has only two possible actions.
  2. Write down the reduced pay-off matrix for player B.
  3. Find the best strategy for player B and the value of the game to him.
Edexcel D2 2014 June Q3
10 marks Moderate -0.5
3. A two-person zero-sum game is represented by the following pay-off matrix for player A.
B plays 1B plays 2B plays 3
A plays 1- 22- 3
A plays 211- 1
A plays 32- 11
  1. Starting by reducing player B's options, find the best strategy for player B.
  2. State the value of the game to player B.
Edexcel D2 2014 June Q4
11 marks Challenging +1.2
4. A two-person zero-sum game is represented by the following pay-off matrix for player A.
B plays 1B plays 2B plays 3B plays 4
A plays 12- 11- 3
A plays 2- 32- 21
  1. Verify that there is no stable solution to this game.
  2. Find the best strategy for player A.
Edexcel D2 2015 June Q2
16 marks Easy -1.8
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.
Greg plays 1Greg plays 2Greg plays 3
Rani plays 1- 312
Rani plays 2021
Rani plays 324- 5
  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.
Edexcel D2 Q3
7 marks Moderate -0.5
3. A two-person zero-sum game is represented by the following pay-off matrix for player A.
B plays 1B plays 2B plays 3
A plays 1- 243
A plays 24- 12
Find the best strategy for player A and the value of the game.
(Total 7 marks)
Edexcel D2 Specimen Q8
11 marks Standard +0.8
8. A two person zero-sum game is represented by the following pay-off matrix for player \(A\).
IIIIII
I523
II354
  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)
OCR D2 2006 January Q6
15 marks Moderate -1.0
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.
\cline { 2 - 5 }\(X\)\(Y\)\(Z\)
\(A\)2- 34
\cline { 2 - 5 } Lucy's\(B\)- 351
\cline { 2 - 5 } strategyy\(C\)42- 3
  1. Show that strategy \(A\) does not dominate strategy \(B\) and also that strategy \(B\) does not dominate strategy \(A\).
  2. Show that Maria will not choose strategy \(Y\) if she plays safe.
  3. Give a reason why Lucy might choose to play strategy \(B\). Lucy decides to play strategy \(A\) with probability \(p _ { 1 }\), strategy \(B\) with probability \(p _ { 2 }\) and strategy \(C\) with probability \(p _ { 3 }\). She formulates the following LP problem to be solved using the Simplex algorithm: $$\begin{array} { l l } \text { maximise } & M = m - 3 , \\ \text { subject to } & m \leqslant 5 p _ { 1 } + 7 p _ { 3 } , \\ & m \leqslant 8 p _ { 2 } + 5 p _ { 3 } , \\ & m \leqslant 7 p _ { 1 } + 4 p _ { 2 } , \\ & p _ { 1 } + p _ { 2 } + p _ { 3 } \leqslant 1 , \\ \text { and } & p _ { 1 } \geqslant 0 , p _ { 2 } \geqslant 0 , p _ { 3 } \geqslant 0 , m \geqslant 0 . \end{array}$$ [You are not required to solve this problem.]
  4. Explain why 3 has to be subtracted from \(m\) in the objective row.
  5. Explain how \(5 p _ { 1 } + 7 p _ { 3 } , 8 p _ { 2 } + 5 p _ { 3 }\) and \(7 p _ { 1 } + 4 p _ { 2 }\) were obtained.
  6. Explain why \(m\) has to be less than or equal to each of the expressions in part (v). Lucy discovers that Maria does not intend ever to choose strategy \(Y\). Because of this she decides that she will never choose strategy \(B\). This means that \(p _ { 2 } = 0\).
  7. Show that the expected number of points won by Lucy when Maria chooses strategy \(X\) is \(4 - 2 p _ { 1 }\) and find a similar expression for the number of points won by Lucy when Maria chooses strategy \(Z\).
  8. Set your two expressions from part (vii) equal to each other and solve for \(p _ { 1 }\). Calculate the expected number of points won by Lucy with this value of \(p _ { 1 }\) and also when \(p _ { 1 } = 0\) and when \(p _ { 1 } = 1\). Use these values to decide how Lucy should choose between strategies \(A\) and \(C\) to maximise the expected number of points that she wins.
OCR D2 2007 January Q4
10 marks Standard +0.3
4 The table gives the pay-off matrix for a zero-sum game between two players, Rowan and Colin. \begin{table}[h]
\captionsetup{labelformat=empty} \caption{Colin}
\cline { 2 - 5 }Strategy \(X\)Strategy \(Y\)Strategy \(Z\)
\cline { 2 - 5 } RowanStrategy \(P\)5- 3- 2
\cline { 2 - 5 }Strategy \(Q\)- 431
\cline { 2 - 5 }
\cline { 2 - 5 }
\end{table} Rowan makes a random choice between strategies \(P\) and \(Q\), choosing strategy \(P\) with probability \(p\) and strategy \(Q\) with probability \(1 - p\).
  1. Write down and simplify an expression for the expected pay-off for Rowan when Colin chooses strategy \(X\).
  2. Using graph paper, draw a graph to show Rowan's expected pay-off against \(p\) for each of Colin's choices of strategy.
  3. Using your graph, find the optimal value of \(p\) for Rowan.
  4. Rowan plays using the optimal value of \(p\). Explain why, in the long run, Colin cannot expect to win more than 0.25 per game.
OCR D2 2010 January Q5
16 marks Easy -1.8
5 Robbie received a new computer game for Christmas. He has already worked through several levels of the game but is now stuck at one of the levels in which he is playing against a character called Conan. Robbie has played this particular level several times. Each time Robbie encounters Conan he can choose to be helped by a dwarf, an elf or a fairy. Conan chooses between being helped by a goblin, a hag or an imp. The players make their choices simultaneously, without knowing what the other has chosen. Robbie starts the level with ten gold coins. The table shows the number of gold coins that Conan must give Robbie in each encounter for each combination of helpers (a negative entry means that Robbie gives gold coins to Conan). If Robbie's total reaches twenty gold coins then he completes the level, but if it reaches zero the game ends. This means that each attempt can be regarded as a zero-sum game.
Conan
\cline { 2 - 5 }GoblinHagImp
\cline { 2 - 5 }Dwarf- 1- 42
\cline { 2 - 5 } RobbieElf31- 4
\cline { 2 - 5 }Fairy1- 11
\cline { 2 - 5 }
\cline { 2 - 5 }
  1. Find the play-safe choice for each player, showing your working. Which helper should Robbie choose if he thinks that Conan will play-safe?
  2. How many gold coins can Robbie expect to win, with each choice of helper, if he thinks that Conan will choose randomly between his three choices (so that each has probability \(\frac { 1 } { 3 }\) )? Robbie decides to choose his helper by using random numbers to choose between the elf and the fairy, where the probability of choosing the elf is \(p\) and the probability of choosing the fairy is \(1 - p\).
  3. Write down an expression for the expected number of gold coins won at each encounter by Robbie for each of Conan's choices. Calculate the optimal value of \(p\). Robbie's girlfriend thinks that he should have included the possibility of choosing the dwarf. She denotes the probability with which Robbie should choose the dwarf, the elf and the fairy as \(x , y\) and \(z\) respectively. She then models the problem of choosing between the three helpers as the following LP. $$\begin{aligned} \text { Maximise } & M = m - 4 , \\ \text { subject to } & m \leqslant 3 x + 7 y + 5 z \\ & m \leqslant 5 y + 3 z \\ & m \leqslant 6 x + 5 z \\ & x + y + z \leqslant 1 , \\ \text { and } & m \geqslant 0 , x \geqslant 0 , y \geqslant 0 , z \geqslant 0 . \end{aligned}$$
  4. Explain how the expression \(3 x + 7 y + 5 z\) was formed. Robbie's girlfriend uses the Simplex algorithm to solve the LP problem. Her solution has \(x = 0\) and \(y = \frac { 2 } { 7 }\).
  5. Calculate the optimal value of \(M\).