7.08d Nash equilibrium: identification and interpretation

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.

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)

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.

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.

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.

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

1. Stage	State	Action	Working	Minimax
   \multirow{3}{*}{1}	0	0	1
   	1	0	3
   	2	0	2
   \multirow{6}{*}{2}	\multirow{2}{*}{0}	0	(4,	\multirow{2}{*}{}
   		1	(2,
   	\multirow{2}{*}{1}	1	(3,	\multirow{2}{*}{}
   		2	(5,
   	\multirow{2}{*}{2}	0	(2,	\multirow{2}{*}{}
   		2	(4,
   \multirow{3}{*}{3}	\multirow{3}{*}{0}	0	(5,	\multirow{3}{*}{}
   		1	(3,
   		2	(1,

2. Minimax value = \(\_\_\_\_\) Minimax route = \(\_\_\_\_\)
3. \includegraphics[max width=\textwidth, alt={}, center]{95fbb09b-0301-4fc1-b694-838b8d0b64a6-10_958_1527_1539_351}

5 The local rugby club has challenged the local cricket club to a chess match to raise money for charity. Each of the top three chess players from the rugby club has played 10 chess games against each of the top three chess players from the cricket club. There were no drawn games. The table shows, for each pairing, the number of games won by the player from the rugby club minus the number of games won by the player from the cricket club. This will be called the score; the scores make a zero-sum game.

1. How many of the 10 games between Sanjeev and Doug did Sanjeev win? How many of the 10 games between Sanjeev and Euan did Euan win? Each club must choose one person to play. They want to choose the person who will optimise the score.
2. Find the play-safe choice for each club, showing your working. Explain how you know that the game is not stable.
3. Which person should the cricket club choose if they know that the rugby club will play-safe and which person should the rugby club choose if they know that the cricket club will play-safe?
4. Explain why the rugby club should not choose Tom. Which player should the cricket club not choose, and why? The rugby club chooses its player by using random numbers to choose between Sanjeev and Ursula, where the probability of choosing Sanjeev is \(p\) and the probability of choosing Ursula is \(1 - p\).
5. Write down an expression for the expected score for the rugby club for each of the two remaining choices that can be made by the cricket club. Calculate the optimal value for \(p\). Doug is studying AS Mathematics. He removes the row representing Tom and then models the cricket club's problem as the following LP. $$\begin{array} { l l } \operatorname { maximise } & M = m - 4 \\ \text { subject to } & m \leqslant 4 x \quad + 6 z \\ & m \leqslant 2 x + 10 y + 4 z \\ & x + y + z \leqslant 1 \\ \text { and } & m \geqslant 0 , x \geqslant 0 , y \geqslant 0 , z \geqslant 0 \end{array}$$
6. Show how Doug used the values in the table to get the constraints \(m \leqslant 4 x + 6 z\) and \(m \leqslant 2 x + 10 y + 4 z\). Doug uses the Simplex algorithm to solve the LP problem. His solution has \(x = 0\) and \(y = \frac { 1 } { 6 }\).
7. Calculate the optimal value of \(M\).

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.

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 \(M\).

5 A card game between two players consists of several rounds. In each round the players both choose a card from those in their hand; they then show these cards to each other and exchange tokens. The number of tokens that the second player gives to the first player depends on the colour of the first player's card and the design on the second player's card. The table shows the number of tokens that the first player receives for each combination of colour and design. A negative entry means that the first player gives tokens to the second, zero means that no tokens are exchanged. Let the stages be \(0,1,2,3,4,5\). Stage 0 represents arriving at the sanctuary entrance. Stage 1 represents visiting the first bird, stage 2 the second bird, and so on, with stage 5 representing leaving the sanctuary. Let the states be \(0,1,2,3,4\) representing the entrance/exit, kite, lark, moorhen and nightjar respectively.

1. Calculate how many minutes it takes to travel the route $$( 0 ; 0 ) - ( 1 ; 1 ) - ( 2 ; 2 ) - ( 3 ; 3 ) - ( 4 ; 4 ) - ( 5 ; 0 ) .$$ The friends then realise that if they try to find the quickest route using dynamic programming with this (stage; state) formulation, they will get the route \(( 0 ; 0 ) - ( 1 ; 1 ) - ( 2 ; 2 ) - ( 3 ; 3 ) - ( 4 ; 1 ) - ( 5 ; 0 )\), or this in reverse, taking 27 minutes.
2. Explain why the route \(( 0 ; 0 ) - ( 1 ; 1 ) - ( 2 ; 2 ) - ( 3 ; 3 ) - ( 4 ; 1 ) - ( 5 ; 0 )\) is not a solution to the friends' problem. Instead, the friends set up a dynamic programming tabulation with stages and states as described above, except that now the states also show, in brackets, any birds that have already been visited. So, for example, state \(1 ( 234 )\) means that they are currently visiting the kite and have already visited the other three birds in some order. The partially completed dynamic programming tabulation is shown opposite.
3. For the last completed row, i.e. stage 2, state 1(3), action 4(13), explain where the value 18 and the value 6 in the working column come from.
4. Complete the table in the insert and hence find the order in which the birds should be visited to give a quickest route and find the corresponding minimum journey time.

   Stage	State	Action	Working	Suboptimal minimum
   \multirow{4}{*}{4}	1(234)	0	10	10
   	2(134)	0	14	14
   	3(124)	0	12	12
   	4(123)	0	17	17
   \multirow{12}{*}{3}	1(23)	4(123)	\(17 + 6 = 23\)	23
   	1(24)	3(124)	\(12 + 2 = 14\)	14
   	1(34)	2(134)	\(14 + 3 = 17\)	17
   	2(13)	4(123)	\(17 + 4 = 21\)	21
   	2(14)	3(124)	\(12 + 2 = 14\)	14
   	2(34)	1(234)	\(10 + 3 = 13\)	13
   	3(12)	4(123)	\(17 + 3 = 20\)	20
   	3(14)	2(134)	\(14 + 2 = 16\)	16
   	3(24)	1(234)	\(10 + 2 = 12\)	12
   	4(12)	3(124)	\(12 + 3 = 15\)	15
   	4(13)	2(134)	\(14 + 4 = 18\)	18
   	4(23)	1(234)	\(10 + 6 = 16\)	16
   \multirow{12}{*}{2}	1(2)	3(12) 4(12)	\(20 + 2 = 22\)	21
   	1(3)	2(13) 4(13)	\(21 + 3 = 24 18 + 6 = 24\)	24
   	1(4)
   	2(1)
   	2(3)
   	2(4)
   	3(1)
   	3(2)
   	3(4)
   	4(1)
   	4(2)
   	4(3)
   \multirow{4}{*}{1}	1
   	2
   	3
   	4
   0	0	1 2 3 4

6 Rowena and Colin play a game in which each chooses a letter. The table shows how many points Rowena wins for each combination of letters. In each case the number of points that Colin wins is the negative of the entry in the table. Both Rowena and Colin are trying to win as many points as possible. Colin's letter \begin{table}[h] \end{table}

1. Write down Colin's play-safe strategy, showing your working. What is the maximum number of points that Colin can win if he plays safe?
2. Explain why Rowena would never choose the letter \(W\). Rowena uses random numbers to choose between her three remaining options, so that she chooses \(X , Y\) and \(Z\) with probabilities \(x , y\) and \(z\), respectively. Rowena then models the problem of which letter she should choose as the following LP. $$\begin{array} { c l } \text { Maximise } & M = m - 1 \\ \text { subject to } & m \leqslant 2 x + 6 y + z , \\ & m \leqslant 4 x + 2 y , \\ & m \leqslant 3 y + 2 z , \\ & m \leqslant 2 x + 2 z , \\ & x + y + z \leqslant 1 \\ \text { and } & m \geqslant 0 , x \geqslant 0 , y \geqslant 0 , z \geqslant 0 \end{array}$$
3. Show how the expression \(2 x + 6 y + z\) was formed. The Simplex algorithm is used to solve the LP problem. The solution has \(x = 0.3 , y = 0.2\) and \(z = 0.5\).
4. Show that the optimal value of \(M\) is 0.6 . Colin then models the problem of which letter he should choose in a similar way. When Rowena plays using her optimal solution, from above, Colin finds that he should never choose the letter \(N\). Letting \(p , q\) and \(t\) denote the probabilities that he chooses \(P , Q\) and \(T\), respectively, Colin obtains the following equations. $$- 3 p + q - t = - 0.6 \quad - p - 2 q + t = - 0.6 \quad p - q - t = - 0.6 \quad p + q + t = 1$$
5. Explain how the equation \(- 3 p + q - t = - 0.6\) is obtained.
6. Use the third and fourth equations to find the value of \(p\). Hence find the values of \(q\) and \(t\).

5 Rose and Colin are playing a game in which they each have four cards. Each player chooses a card from those in their hand, and simultaneously they show each other the cards they have chosen. The table below shows how many points Rose wins for each combination of cards. In each case the number of points that Colin wins is the negative of the entry in the table. Both Rose and Colin are trying to win as many points as possible.

1. What is the greatest number of points that Colin can win when Rose chooses and which card does Colin need to choose to achieve this?
2. Explain why Rose should never choose and find the card that Colin should never choose. 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 and show whether or not the game is stable. Rose makes a random choice between her cards, choosing with probability \(x\) with probability \(y\), and with probability \(z\). She formulates the following LP problem to be solved using the Simplex algorithm:
   maximise \(\quad M = m - 6\),
   subject to \(\quad m \leqslant 4 x + 10 y\), \(n \leqslant 9 x + 3 y + 11 z\), \(n \leqslant 2 x + 10 y + z\), \(x + y + z \leqslant 1\),
   and \(x \geqslant 0 , y \geqslant 0 , z \geqslant 0 , m \geqslant 0\).
   (You are not required to solve this problem.)
4. Explain how \(9 x + 3 y + 11 z\) was obtained. The Simplex algorithm is used to solve the LP problem. The solution has \(x = \frac { 7 } { 48 } , y = \frac { 27 } { 48 } , z = \frac { 14 } { 48 }\).
5. Calculate the optimal value of \(M\).

2 The table gives the pay-off matrix for a zero-sum game between two players, A my and Bea. The values in the table show the pay-offs for A my. A my makes a random choice between strategies \(\mathbf { P }\) and \(Q\), choosing strategy \(P\) with probability \(p\) and strategy Q with probability \(1 - \mathrm { p }\).

1. Write down and simplify an expression for the expected pay-off for Amy when Bea chooses strategy X . Write down similar expressions for the cases when B ea chooses strategy Y and when she chooses strategy \(Z\).
2. Using graph paper, draw a graph to show A my's expected pay-off against p for each of Bea's choices of strategy. Using your graph, find the optimal value of pfor A my. A my and Bea play the game many times. A my chooses randomly between her strategies using the optimal value for p.
3. Showing your working, calculate A my's minimum expected pay-off per game. W hy might A my gain more points than this, on average?
4. W hat is B ea's minimum expected loss per game? How should B ea play to minimise her expected loss?

3 The 'Rovers' and the 'Collies' are two teams of dog owners who compete in weekly dog shows. The top three dogs owned by members of the Rovers are Prince, Queenie and Rex. The top four dogs owned by the Collies are Woof, Xena, Yappie and Zulu. In a show the Rovers choose one of their dogs to compete against one of the dogs owned by the Collies. There are 10 points available in total. Each of the 10 points is awarded either to the dog owned by the Rovers or to the dog owned by the Collies. There are no tied points. At the end of the competition, 5 points are subtracted from the number of points won by each dog to give the score for that dog. The table shows the score for the dog owned by the Rovers for each combination of dogs.

1. Explain why calculating the score by subtracting 5 from the number of points for each dog makes this a zero-sum game.
2. If the Rovers choose Prince and the Collies choose Woof, what score does Woof get, and how many points do Prince and Woof each get in the competition?
3. Show that column \(W\) is dominated by one of the other columns, and state which column this is.
4. Delete the column for \(W\) and find the play-safe strategy for the Rovers and the play-safe strategy for the Collies on the table that remains. Queenie is ill one week, so the Rovers make a random choice between Prince and Rex, choosing Prince with probability \(p\) and Rex with probability \(1 - p\).
5. Write down and simplify an expression for the expected score for the Rovers when the Collies choose Xena. Write down and simplify the corresponding expressions for when the Collies choose Yappie and for when they choose Zulu.
6. Using graph paper, draw a graph to show the expected score for the Rovers against \(p\) for each of the choices that the Collies can make. Using your graph, find the optimal value of \(p\) for the Rovers. If Queenie had not been ill, the Rovers would have made a random choice between Prince, Queenie and Rex, choosing Prince with probability \(p _ { 1 }\), Queenie with probability \(p _ { 2 }\) and Rex with probability \(p _ { 3 }\). The problem of choosing the optimal values of \(p _ { 1 } , p _ { 2 }\) and \(p _ { 3 }\) can be formulated as the following linear programming problem: $$\begin{array} { l l } \operatorname { maximise } & M = m - 4 \\ \text { subject to } & m \leqslant 6 p _ { 1 } + 5 p _ { 2 } , \\ & m \leqslant 3 p _ { 1 } + p _ { 2 } + 5 p _ { 3 } , \\ & m \leqslant 7 p _ { 1 } + 3 p _ { 2 } + 4 p _ { 3 } , \\ & p _ { 1 } + p _ { 2 } + p _ { 3 } \leqslant 1 \\ \text { and } & p _ { 1 } \geqslant 0 , p _ { 2 } \geqslant 0 , p _ { 3 } \geqslant 0 , m \geqslant 0 . \end{array}$$
7. Explain how the expressions \(6 p _ { 1 } + 5 p _ { 2 } , 3 p _ { 1 } + p _ { 2 } + 5 p _ { 3 }\) and \(7 p _ { 1 } + 3 p _ { 2 } + 4 p _ { 3 }\) were obtained. Also explain how the linear programming formulation tells you that \(M\) is a maximin solution. The Simplex algorithm is used to find the optimal values of the probabilities. The optimal value of \(p _ { 1 }\) is \(\frac { 5 } { 8 }\) and the optimal value of \(p _ { 2 }\) is 0 .
8. Calculate the optimal value of \(p _ { 3 }\) and the corresponding value of \(M\).

3 Basil runs a luxury hotel. He advertises summer breaks at the hotel in several different magazines. Last summer he won the opportunity to place a full-page colour advertisement in one of four magazines for the price of the usual smaller advertisement. The table shows the expected additional weekly income, in \(\pounds\), for each of the magazines for each possible type of weather. Basil wanted to maximise the additional income.

1. Explain carefully why no magazine choice can be rejected using a dominance argument.
2. Treating the choice of strategies as being a zero-sum game, find Basil's play-safe strategy and show that the game is unstable.
3. Calculate the expected additional weekly income for each magazine choice if the weather is rainy with probability 0.4 and sunny with probability 0.6 . Suppose that the weather is rainy with probability \(p\) and sunny with probability \(1 - p\).
4. Which magazine should Basil choose if the weather is certain to be sunny ( \(p = 0\) ), and which should he choose if the weather is certain to be rainy ( \(p = 1\) )?
5. Graph the expected additional weekly income against \(p\). Hence advise Basil on which magazine he should choose for the different possible ranges of values of \(p\).

4 A group of rowers have challenged some cyclists to see which team is fitter. There will be several rounds to the challenge. In each round, the rowers and the cyclists each choose a team member and these two compete in a series of gym exercises. The time by which the winner finishes ahead of the loser is converted into points. These points are added to the score for the winner's team and taken off the score for the loser's team. The table shows the expected number of points added to the score for the rowers for each combination of competitors. Rowers \begin{table}[h] \end{table}

1. Regarding this as a zero-sum game, find the play-safe strategy for the rowers and the play-safe strategy for the cyclists. Show that the game is stable. Unfortunately, Wendy and Kath are needed by their coaches and cannot compete.
2. Show that the resulting reduced game is unstable.
3. Suppose that the cyclists are equally likely to choose Chris or Jamie. Calculate the expected number of points added to the score for the rowers when they choose Andy and when they choose Zac. Suppose that the cyclists use random numbers to choose between Chris and Jamie, so that Chris is chosen with probability \(p\) and Jamie with probability \(1 - p\).
4. Showing all your working, calculate the optimum value of \(p\) for the cyclists.
5. The rowers use random numbers in a similar way to choose between Andy and Zac, so that Andy is chosen with probability \(q\) and Zac with probability \(1 - q\). Calculate the optimum value of \(q\).