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

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 \(q\).

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 \(M\).

4 Ross and Collwen are playing a game in which each secretly chooses a magic spell. They then reveal their choices, and work out their scores using the tables below. Ross and Collwen are both trying to get as large a score as possible.

1. Explain how this can be rewritten as the following zero-sum game.

   		Collwen's choice
   		Fire	Ice	Gale
   \cline { 2 - 5 }	Fire	- 3	3	- 2
   \cline { 2 - 5 } Ross's choice	Ice	2	- 2	0
   \cline { 2 - 5 }	Gale	1	- 3	- 1
   \cline { 2 - 5 }

2. If Ross chooses Ice what is Collwen's best choice?
3. Find the play-safe strategy for Ross and the play-safe strategy for Collwen, showing your working. Explain how you know that the game is unstable.
4. Show that none of Collwen's strategies dominates any other.
5. Explain why Ross would never choose Gale, hence reduce the game to a \(2 \times 3\) zero-sum game, showing the pay-offs for Ross. Suppose that Ross uses random numbers to choose between Fire and Ice, choosing Fire with probability \(p\) and Ice with probability \(1 - p\).
6. Use a graphical method to find the optimal value of \(p\) for Ross.

6 At the final battle in a war game, the two opposing armies, led by General Rose, \(R\), and Colonel Cole, \(C\), are facing each other across a wide river. Each army consists of four divisions. The commander of each army needs to send some of his troops North and send the rest South. Each commander has to decide how many divisions (1,2 or 3) to send North. Intelligence information is available on the number of thousands of soldiers that each army can expect to have remaining with each combination of strategies. Thousands of soldiers remaining in \(R\) 's army \(C\) 's choice \(R\) 's choice Thousands of soldiers remaining in \(C\) 's army \(C\) 's choice \(R\) 's choice

1. Construct a table to show the number of thousands of soldiers remaining in \(R\) 's army minus the number of thousands of soldiers remaining in \(C\) 's army (the excess for \(R\) 's army) for each combination of strategies. The commander whose army has the greatest positive excess of soldiers remaining at the end of the game will be declared the winner.
2. Explain the meaning of the value in the top left cell of your table from part (i) (where each commander chooses strategy 1). Hence explain why this table may be regarded as representing a zero-sum game.
3. Find the play-safe strategy for \(R\) and the play-safe strategy for \(C\). If \(C\) knows that \(R\) will choose his play-safe strategy, which strategy should \(C\) choose? One of the strategies is redundant for one of the commanders, because of dominance.
4. Draw a table for the reduced game, once the redundant strategy has been removed. Label the rows and columns to show how many divisions have been sent North. A mixed strategy is to be employed on the resulting reduced game. This leads to the following LP problem:
   Maximise \(\quad M = m - 25\) Subject to \(\quad m \leqslant 15 x + 25 y + 35 z\) \(m \leqslant 45 x + 20 y\) \(x + y + z \leqslant 1\) and
5. Interpret what \(x , y\) and \(z\) represent and show how \(m \leqslant 15 x + 25 y + 35 z\) was formed. A computer runs the Simplex algorithm to solve this problem. It gives \(x = 0.5385 , y = 0\) and \(z = 0.4615\).
6. Describe how this solution should be interpreted, in terms of how General Rose chooses where to send his troops. Calculate the optimal value for \(M\) and explain its meaning. Elizabeth does not have access to a computer. She says that at the solution to the LP problem \(15 x + 25 y + 35 z\) must equal \(45 x + 20 y\) and \(x + y + z\) must equal 1 . This simplifies to give \(y + 7 z = 6 x\) and \(x + y + z = 1\).
7. Explain why there can be no valid solution of \(y + 7 z = 6 x\) and \(x + y + z = 1\) with \(x = 0\). Elizabeth tries \(z = 0\) and gets the solution \(x = \frac { 1 } { 7 }\) and \(y = \frac { 6 } { 7 }\).
8. Explain why this is not a solution to the LP problem.

4 Rowan and Colin are playing a game of 'scissors-paper-rock'. In each round of this game, each player chooses one of scissors ( \(\$$ ), paper ( \)\square\( ) or rock ( \)\bullet$ ). The players reveal their choices simultaneously, using coded hand signals. Rowan and Colin will play a large number of rounds. At the end of the game the player with the greater total score is the winner. The rules of the game are that scissors wins over paper, paper wins over rock and rock wins over scissors. In this version of the game, if a player chooses scissors they will score \(+ 1,0\) or - 1 points, according to whether they win, draw or lose, but if they choose paper or rock they will score \(+ 2,0\) or - 2 points. This gives the following pay-off tables. \includegraphics[max width=\textwidth, alt={}, center]{490ff276-6639-40a1-bffb-dc6967f3ab21-5_476_773_667_239} \includegraphics[max width=\textwidth, alt={}, center]{490ff276-6639-40a1-bffb-dc6967f3ab21-5_478_780_667_1071}

1. Use an example to show that this is not a zero-sum game.
2. Write down the minimum number of points that Rowan can win using each strategy. Hence find the strategy that maximises the minimum number of points that Rowan can win. Rowan decides to use random numbers to choose between the three strategies, choosing scissors with probability \(p\), paper with probability \(q\) and rock with probability \(( 1 - p - q )\).
3. Find and simplify, in terms of \(p\) and \(q\), expressions for the expected number of points won by Rowan for each of Colin's possible choices. Rowan wants his expected winnings to be the same for all three of Colin's possible choices.
4. Calculate the probability with which Rowan should play each strategy.

6 Rose is playing a game against a computer. Rose aims a laser beam along a row, \(A , B\) or \(C\), and, at the same time, the computer aims a laser beam down a column, \(X , Y\) or \(Z\). The number of points won by Rose is determined by where the two laser beams cross. These values are given in the table. The computer loses whatever Rose wins.

1. Find Rose's play-safe strategy and show that the computer's play-safe strategy is \(Y\). How do you know that the game does not have a stable solution?
2. Explain why Rose should never choose row \(C\) and hence reduce the game to a \(2 \times 3\) pay-off matrix.
3. Rose intends to play the game a large number of times. She decides to use a standard six-sided die to choose between row \(A\) and row \(B\), so that row \(A\) is chosen with probability \(a\) and row \(B\) is chosen with probability \(1 - a\). Show that the expected pay-off for Rose when the computer chooses column \(X\) is \(4 - 3 a\), and find the corresponding expressions for when the computer chooses column \(Y\) and when it chooses column \(Z\). Sketch a graph showing the expected pay-offs against \(a\), and hence decide on Rose's optimal choice for \(a\). Describe how Rose could use the die to decide whether to play \(A\) or \(B\). The computer is to choose \(X , Y\) and \(Z\) with probabilities \(x , y\) and \(z\) respectively, where \(x + y + z = 1\). Graham is an AS student studying the D1 module. He wants to find the optimal choices for \(x , y\) and \(z\) and starts off by producing a pay-off matrix for the computer.
4. Graham produces the following pay-off matrix.

   3	1	0
   0	1	2

   Write down the pay-off matrix for the computer and explain what Graham did to its entries to get the values in his pay-off matrix.
5. Graham then sets up the linear programming problem: $$\begin{array} { l l } \text { maximise } & P = p - 4 , \\ \text { subject to } & p - 3 x - y \leqslant 0 , \\ & p - y - 2 z \leqslant 0 , \\ & x + y + z \leqslant 1 , \\ \text { and } & p \geqslant 0 , x \geqslant 0 , y \geqslant 0 , z \geqslant 0 . \end{array}$$ The Simplex algorithm is applied to the problem and gives \(x = 0.4\) and \(y = 0\). Find the values of \(z , p\) and \(P\) and interpret the solution in the context of the game. {}

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

Find the optimal strategy for each player and the value of the game.

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

1. Using a graphical method, find the optimal strategy for player \(X\).
2. Find the optimal strategy for player \(Y\).
3. Find the value of the game.

3. 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.

6. 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 \(B\).
3. Find the value of the game.

4. 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.

4 Kareem and Sam play a game in which each holds a hand of three cards.

• Kareem's cards are numbered 1, 2 and 5.
• Sam's cards are numbered 3, 4 and 6 .

In each round Kareem and Sam simultaneously choose a card from their hand, they show their chosen card to the other player and then return the card to their own hand.

• If the sum of the numbers on the cards shown is even then the number of points that Kareem scores is \(2 k\), where \(k\) is the number on Kareem's card.
• If the sum of the numbers on the cards shown is odd then the number of points that Kareem scores is \(4 - s\), where \(s\) is the number on Sam's card.
  1. Complete the pay-off matrix for this game, to show the points scored by Kareem.
  2. Write down which card Kareem should play to maximise the number of points that he scores for each of Sam's choices.
  3. Determine the play-safe strategy for Kareem.
  4. Explain why Kareem should never play the card numbered 1.

Sam chooses a card at random, so each of Sam's three cards is equally likely.

Calculate Kareem's expected score for each of his remaining choices.

2 In a game two players are each dealt five cards from a set of ten different cards.
Player 1 is dealt cards A, B, F, G and J.
Player 2 is dealt cards C, D, E, H and I. Each player chooses a card to play.
The players reveal their choices simultaneously. The pay-off matrix below shows the points scored by player 1 for each combination of cards. Pay-off for player 1 \begin{table}[h] \end{table}

1. Determine the play-safe strategy for player 1, ignoring any effect on player 2. The pay-off matrix below shows the points scored by player 2 for each combination of cards.
   Pay-off for player 2 Player 1 \begin{table}[h]
   \captionsetup{labelformat=empty} \caption{Player 2}

   		C	D	E	H
   \cline { 2 - 6 } A	2	2	0	1	I
   \cline { 2 - 6 } B	3	1	2	1	2
   \cline { 2 - 6 } F	3	2	2	1	0
   \cline { 2 - 6 } G	1	3	0	0	0
   \cline { 2 - 6 } J	2	1	0	3	1
   \cline { 2 - 6 }
   \cline { 2 - 6 }

   \end{table}
2. Use a dominance argument to delete two columns from the pay-off matrix. You must show all relevant comparisons.
3. Explain, with reference to specific combinations of cards, why the game cannot be converted to a zero-sum game.

5 The number of points won by player 1 in a zero-sum game is shown in the pay-off matrix below, where \(k\) is a constant. \begin{table}[h] \end{table}

1. In one game, player 2 chooses strategy H. Write down the greatest number of points that player 2 could win. You are given that strategy A is a play-safe strategy for player 1.
2. Determine the range of possible values for \(k\).
3. Determine the column minimax value.

6 The pay-off matrix for a game between two players, Sumi and Vlad, is shown below. If Sumi plays A and Vlad plays X then Sumi gets X points and Vlad gets 1 point. Sumi You are given that cell ( \(\mathrm { A } , \mathrm { X }\) ) is a Nash Equilibrium solution.

1. Find the range of possible values of X .
2. Explain what the statement 'cell (A, X) is a Nash Equilibrium solution' means for each player.
3. Find a cell where each player gets their maximin pay-off. Suppose, instead, that the game can be converted to a zero-sum game.
4. Determine the optimal strategy for Sumi for the zero-sum game.

5 In each turn of a game between two players they simultaneously each choose a strategy and then calculate the points won using the table below. They are each trying to maximise the number of points that they win. In each cell the first value is the number of points won by player 1 and the second value is the number of points won by player 2 .

1. Find the play-safe strategy for each player.
2. Explain why player 2 would never choose strategy Z .
3. Find the Nash equilibrium solution(s) or show that there is no Nash equilibrium solution. Player 2 chooses strategy X with probability \(p\) and strategy Y with probability \(1 - p\). You are given that when player 1 chooses strategy A, the expected number of points won by each player is the same.
4. 1. Calculate the value of \(p\).
   2. Determine which player expects to win the greater number of points when player 1 chooses strategy B.

3 Amir and Beth play a zero-sum game.
The table shows the pay-off for Amir for each combination of strategies, where these values are known. \begin{table}[h] \end{table} You are given that \(\mathrm { a } < 0 < \mathrm { b } < \mathrm { c }\).
Amir's play-safe strategy is R.

1. Determine the range of possible values of \(a\). Beth's play-safe strategy is Y.
2. Determine the range of possible values of \(b\).
3. Determine whether or not the game is stable.