Group properties and structure

1

1. A group \(G\) of order 6 has the combination table shown below.

   	\(e\)	\(a\)	\(b\)	\(p\)	\(q\)	\(r\)
   \(e\)	\(e\)	\(a\)	\(b\)	\(p\)	\(q\)	\(r\)
   \(a\)	\(a\)	\(b\)	\(e\)	\(r\)	\(p\)	\(q\)
   \(b\)	\(b\)	\(e\)	\(a\)	\(q\)	\(r\)	\(p\)
   \(p\)	\(p\)	\(q\)	\(r\)	\(e\)	\(a\)	\(b\)
   \(q\)	\(q\)	\(r\)	\(p\)	\(b\)	\(e\)	\(a\)
   \(r\)	\(r\)	\(p\)	\(q\)	\(a\)	\(b\)	\(e\)

   1. State, with a reason, whether or not \(G\) is commutative.
   2. State the number of subgroups of \(G\) which are of order 2 .
   3. List the elements of the subgroup of \(G\) which is of order 3 .
2. A multiplicative group \(H\) of order 6 has elements \(e , c , c ^ { 2 } , c ^ { 3 } , c ^ { 4 } , c ^ { 5 }\), where \(e\) is the identity. Write down the order of each of the elements \(c ^ { 3 } , c ^ { 4 }\) and \(c ^ { 5 }\).

1

1. For the infinite group of non-zero complex numbers under multiplication, state the identity element and the inverse of \(1 + 2 \mathrm { i }\), giving your answers in the form \(a + \mathrm { i } b\).
2. For the group of matrices of the form \(\left( \begin{array} { l l } a & 0 \\ 0 & 0 \end{array} \right)\) under matrix addition, where \(a \in \mathbb { R }\), state the identity element and the inverse of \(\left( \begin{array} { l l } 3 & 0 \\ 0 & 0 \end{array} \right)\).

2 The set \(S = \{ a , b , c , d \}\) under the binary operation * forms a group \(G\) of order 4 with the following operation table.

1. Find the order of each element of \(G\).
2. Write down a proper subgroup of \(G\).
3. Is the group \(G\) cyclic? Give a reason for your answer.
4. State suitable values for each of \(a , b , c\) and \(d\) in the case where the operation \(*\) is multiplication of complex numbers.

5 A local hockey league has three divisions. Each team in the league plays in a division for a year. In the following year a team might play in the same division again, or it might move up or down one division. This question is about the progress of one particular team in the league. In 2007 this team will be playing in either Division 1 or Division 2. Because of its present position, the probability that it will be playing in Division 1 is 0.6 , and the probability that it will be playing in Division 2 is 0.4 . The following transition probabilities apply to this team from 2007 onwards.

• When the team is playing in Division 1, the probability that it will play in Division 2 in the following year is 0.2 .
• When the team is playing in Division 2, the probability that it will play in Division 1 in the following year is 0.1 , and the probability that it will play in Division 3 in the following year is 0.3 .
• When the team is playing in Division 3, the probability that it will play in Division 2 in the following year is 0.15 .

This process is modelled as a Markov chain with three states corresponding to the three divisions.

1. Write down the transition matrix.
2. Determine in which division the team is most likely to be playing in 2014.
3. Find the equilibrium probabilities for each division for this team. In 2015 the rules of the league are changed. A team playing in Division 3 might now be dropped from the league in the following year. Once dropped, a team does not play in the league again.
   -The transition probabilities from Divisions 1 and 2 remain the same as before.
   • When the team is playing in Division 3, the probability that it will play in Division 2 in the following year is 0.15 , and the probability that it will be dropped from the league is 0.1 .
   The team plays in Division 2 in 2015.
   The new situation is modelled as a Markov chain with four states: 'Division1', 'Division 2', 'Division 3' and 'Out of league'.
4. Write down the transition matrix which applies from 2015.
5. Find the probability that the team is still playing in the league in 2020.
6. Find the first year for which the probability that the team is out of the league is greater than 0.5 .

4

1. The composition table for a group \(G\) of order 8 is given below.

   	\(a\)	\(b\)	\(c\)	\(d\)	\(e\)	\(f\)	\(g\)	\(h\)
   \(a\)	\(c\)	\(e\)	\(b\)	\(f\)	\(a\)	\(h\)	\(d\)	\(g\)
   \(b\)	\(e\)	\(c\)	\(a\)	\(g\)	\(b\)	\(d\)	h	\(f\)
   \(c\)	\(b\)	\(a\)	\(e\)	\(h\)	\(c\)	\(g\)	\(f\)	\(d\)
   \(d\)	\(f\)	\(g\)	\(h\)	\(a\)	\(d\)	\(c\)	\(e\)	\(b\)
   \(e\)	\(a\)	\(b\)	\(c\)	\(d\)	\(e\)	\(f\)	\(g\)	\(h\)
   \(f\)	\(h\)	\(d\)	\(g\)	\(c\)	\(f\)	\(b\)	\(a\)	\(e\)
   \(g\)	\(d\)	\(h\)	\(f\)	\(e\)	\(g\)	\(a\)	\(b\)	\(c\)
   \(h\)	\(g\)	\(f\)	\(d\)	\(b\)	\(h\)	\(e\)	\(c\)	\(a\)

   1. State which is the identity element, and give the inverse of each element of \(G\).
   2. Show that \(G\) is cyclic.
   3. Specify an isomorphism between \(G\) and the group \(H\) consisting of \(\{ 0,2,4,6,8,10,12,14 \}\) under addition modulo 16 .
   4. Show that \(G\) is not isomorphic to the group of symmetries of a square.
2. The set \(S\) consists of the functions \(\mathrm { f } _ { n } ( x ) = \frac { x } { 1 + n x }\), for all integers \(n\), and the binary operation is composition of functions.
   1. Show that \(\mathrm { f } _ { m } \mathrm { f } _ { n } = \mathrm { f } _ { m + n }\).
   2. Hence show that the binary operation is associative.
   3. Prove that \(S\) is a group.
   4. Describe one subgroup of \(S\) which contains more than one element, but which is not the whole of \(S\).

5 In this question, give probabilities correct to 4 decimal places.
A contestant in a game-show starts with one, two or three 'lives', and then performs a series of tasks. After each task, the number of lives either decreases by one, or remains the same, or increases by one. The game ends when the number of lives becomes either four or zero. If the number of lives is four, the contestant wins a prize; if the number of lives is zero, the contestant loses and leaves with nothing. At the start, the number of lives is decided at random, so that the contestant is equally likely to start with one, two or three lives. The tasks do not involve any skill, and after every task:

• the probability that the number of lives decreases by one is 0.5 ,
• the probability that the number of lives remains the same is 0.05 ,
• the probability that the number of lives increases by one is 0.45 .

This is modelled as a Markov chain with five states corresponding to the possible numbers of lives. The states corresponding to zero lives and four lives are absorbing states.

1. Write down the transition matrix \(\mathbf { P }\).
2. Show that, after 8 tasks, the probability that the contestant has three lives is 0.0207 , correct to 4 decimal places.
3. Find the probability that, after 10 tasks, the game has not yet ended.
4. Find the probability that the game ends after exactly 10 tasks.
5. Find the smallest value of \(N\) for which the probability that the game has not yet ended after \(N\) tasks is less than 0.01 .
6. Find the limit of \(\mathbf { P } ^ { n }\) as \(n\) tends to infinity.
7. Find the probability that the contestant wins a prize. The beginning of the game is now changed, so that the probabilities of starting with one, two or three lives can be adjusted.
8. State the maximum possible probability that the contestant wins a prize, and how this can be achieved.
9. Given that the probability of starting with one life is 0.1 , and the probability of winning a prize is 0.6 , find the probabilities of starting with two lives and starting with three lives. OCR is committed to seeking permission to reproduce all third-party content that it uses in its assessment materials. OCR has attempted to identify and contact all copyright holders whose work is used in this paper. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced in the OCR Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download from our public website (\href{http://www.ocr.org.uk}{www.ocr.org.uk}) after the live examination series.
   If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity.
   For queries or further information please contact the Copyright Team, First Floor, 9 Hills Road, Cambridge CB2 1GE.
   OCR is part of the Cambridge Assessment Group; Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge.

5 In this question, give probabilities correct to 4 decimal places.
The speeds of vehicles are measured on a busy stretch of road and are categorised as A (not more than 30 mph ), B (more than 30 mph but not more than 40 mph ) or C (more than 40 mph ).

• Following a vehicle in category A , the probabilities that the next vehicle is in categories \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) are \(0.9,0.07,0.03\) respectively.
• Following a vehicle in category B , the probabilities that the next vehicle is in categories \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) are \(0.3,0.6,0.1\) respectively.
• Following a vehicle in category C , the probabilities that the next vehicle is in categories \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) are \(0.1,0.7,0.2\) respectively.

This is modelled as a Markov chain with three states corresponding to the categories A, B, C. The speed of the first vehicle is measured as 28 mph .

1. Write down the transition matrix \(\mathbf { P }\).
2. Find the probabilities that the 10th vehicle is in each of the three categories.
3. Find the probability that the 12th and 13th vehicles are in the same category.
4. Find the smallest value of \(n\) for which the probability that the \(n\)th and \(( n + 1 )\) th vehicles are in the same category is less than 0.8, and give the value of this probability.
5. Find the expected number of vehicles (including the first vehicle) in category A before a vehicle in a different category.
6. Find the limit of \(\mathbf { P } ^ { n }\) as \(n\) tends to infinity, and hence write down the equilibrium probabilities for the three categories.
7. Find the probability that, after many vehicles have passed by, the next three vehicles are all in category A. On a new stretch of road, the same categories are used but some of the transition probabilities are different.
   • Following a vehicle in category A , the probability that the next vehicle is in category B is equal to the probability that it is in category C .
   • Following a vehicle in category B , the probability that the next vehicle is in category A is equal to the probability that it is in category C .
   • Following a vehicle in category C , the probabilities that the next vehicle is in categories \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) are \(0.1,0.7,0.2\) respectively.
   In the long run, the proportions of vehicles in categories A, B, C are 50\%, 40\%, 10\% respectively.
8. Find the transition matrix for the new stretch of road.

5 Each level of a fantasy computer game is set in a single location, Alphaworld, Betaworld, Chiworld or Deltaworld. After completing a level, a player goes on to the next level, which could be set in the same location as the previous level, or in a different location. In the first version of the game, the initial and transition probabilities are as follows.
Level 1 is set in Alphaworld or Betaworld, with probabilities 0.6, 0.4 respectively.
After a level set in Alphaworld, the next level will be set in Betaworld, Chiworld or Deltaworld, with probabilities \(0.7,0.1,0.2\) respectively.
After a level set in Betaworld, the next level will be set in Alphaworld, Betaworld or Deltaworld, with probabilities \(0.1,0.8,0.1\) respectively.
After a level set in Chiworld, the next level will also be set in Chiworld.
After a level set in Deltaworld, the next level will be set in Alphaworld, Betaworld or Chiworld, with probabilities \(0.3,0.6,0.1\) respectively. The situation is modelled as a Markov chain with four states.

1. Write down the transition matrix.
2. Find the probabilities that level 14 is set in each location.
3. Find the probability that level 15 is set in the same location as level 14 .
4. Find the level at which the probability of being set in Chiworld first exceeds 0.5.
5. Following a level set in Betaworld, find the expected number of further levels which will be set in Betaworld before changing to a different location. In the second version of the game, the initial probabilities and the transition probabilities after Alphaworld, Betaworld and Deltaworld are all the same as in the first version; but after a level set in Chiworld, the next level will be set in Chiworld or Deltaworld, with probabilities \(0.9,0.1\) respectively.
6. By considering powers of the new transition matrix, or otherwise, find the equilibrium probabilities for the four locations. In the third version of the game, the initial probabilities and the transition probabilities after Alphaworld, Betaworld and Deltaworld are again all the same as in the first version; but the transition probabilities after Chiworld have changed again. The equilibrium probabilities for Alphaworld, Betaworld, Chiworld and Deltaworld are now 0.11, 0.75, 0.04, 0.1 respectively.
7. Find the new transition probabilities after a level set in Chiworld. OCR is committed to seeking permission to reproduce all third-party content that it uses in its assessment materials. OCR has attempted to identify and contact all copyright holders whose work is used in this paper. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced in the OCR Copyright Acknowledgements Booklet. This is produced for each series of examinations, is given to all schools that receive assessment material and is freely available to download from our public website (\href{http://www.ocr.org.uk}{www.ocr.org.uk}) after the live examination series.
   If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity. For queries or further information please contact the Copyright Team, First Floor, 9 Hills Road, Cambridge CB2 1PB.
   OCR is part of the Cambridge Assessment Group; Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge.

1 In this question \(G\) is a group of order \(n\), where \(3 \leqslant n < 8\).

1. In each case, write down the smallest possible value of \(n\) :
   (a) if \(G\) is cyclic,
   (b) if \(G\) has a proper subgroup of order 3,
   (c) if \(G\) has at least two elements of order 2 .
2. Another group has the same order as \(G\), but is not isomorphic to \(G\). Write down the possible value(s) of \(n\).

2 The elements of a group \(G\) are the complex numbers \(a + b \mathrm { i }\) where \(a , b \in \{ 0,1,2,3,4 \}\). These elements are combined under the operation of addition modulo 5 .

1. State the identity element and the order of \(G\).
2. Write down the inverse of \(2 + 4 \mathrm { i }\).
3. Show that every non-zero element of \(G\) has order 5 .

7 A commutative group \(G\) has order 18. The elements \(a , b\) and \(c\) have orders 2, 3 and 9 respectively.

1. Prove that \(a b\) has order 6 .
2. Show that \(G\) is cyclic.

2 The elements of a group \(G\) are polynomials of the form \(a + b x + c x ^ { 2 }\), where \(a , b , c \in \{ 0,1,2,3,4 \}\). The group operation is addition, where the coefficients are added modulo 5 .

1. State the identity element.
2. State the inverse of \(3 + 2 x + x ^ { 2 }\).
3. State the order of \(G\). The proper subgroup \(H\) contains \(2 + x\) and \(1 + x\).
4. Find the order of \(H\), justifying your answer.

8 A multiplicative group \(Q\) of order 8 has elements \(\left\{ e , p , p ^ { 2 } , p ^ { 3 } , a , a p , a p ^ { 2 } , a p ^ { 3 } \right\}\), where \(e\) is the identity. The elements have the properties \(p ^ { 4 } = e\) and \(a ^ { 2 } = p ^ { 2 } = ( a p ) ^ { 2 }\).

1. Prove that \(a = p a p\) and that \(p = a p a\).
2. Find the order of each of the elements \(p ^ { 2 } , a , a p , a p ^ { 2 }\).
3. Prove that \(\left\{ e , a , p ^ { 2 } , a p ^ { 2 } \right\}\) is a subgroup of \(Q\).
4. Determine whether \(Q\) is a commutative group.

Let \(V\) be the subspace of \(\mathbb { R } ^ { 4 }\) spanned by $$\mathbf { v } _ { 1 } = \left( \begin{array} { l } 1 \\ 2 \\ 0 \\ 2 \end{array} \right) , \quad \mathbf { v } _ { 2 } = \left( \begin{array} { r } - 2 \\ - 5 \\ 5 \\ 6 \end{array} \right) , \quad \mathbf { v } _ { 3 } = \left( \begin{array} { r } 0 \\ - 3 \\ 15 \\ 18 \end{array} \right) \quad \text { and } \quad \mathbf { v } _ { 4 } = \left( \begin{array} { r } 0 \\ - 2 \\ 10 \\ 8 \end{array} \right) .$$

1. Show that the dimension of \(V\) is 3 .
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2. Express \(\mathbf { v } _ { 4 }\) as a linear combination of \(\mathbf { v } _ { 1 } , \mathbf { v } _ { 2 }\) and \(\mathbf { v } _ { 3 }\).
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3. Write down a basis for \(V\).
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   \(\_\_\_\_\) Let \(\mathbf { M } = \left( \begin{array} { r r r r } 1 & - 2 & 0 & 0 \\ 2 & - 5 & - 3 & - 2 \\ 0 & 5 & 15 & 10 \\ 2 & 6 & 18 & 8 \end{array} \right)\).
4. Find the general solution of \(\mathbf { M x } = \mathbf { v } _ { 1 } + \mathbf { v } _ { 2 }\).
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   \(\_\_\_\_\) The set of elements of \(\mathbb { R } ^ { 4 }\) which are not solutions of \(\mathbf { M x } = \mathbf { v } _ { 1 } + \mathbf { v } _ { 2 }\) is denoted by \(W\).
5. State, with a reason, whether \(W\) is a vector space.
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10 The vectors \(\mathbf { b } _ { 1 } , \mathbf { b } _ { 2 } , \mathbf { b } _ { 3 } , \mathbf { b } _ { 4 }\) are defined as follows: $$\mathbf { b } _ { 1 } = \left( \begin{array} { c } 1 \\ 0 \\ 0 \\ 0 \end{array} \right) , \quad \mathbf { b } _ { 2 } = \left( \begin{array} { c } 1 \\ 1 \\ 0 \\ 0 \end{array} \right) , \quad \mathbf { b } _ { 3 } = \left( \begin{array} { c } 1 \\ 1 \\ 1 \\ 0 \end{array} \right) , \quad \mathbf { b } _ { 4 } = \left( \begin{array} { c } 1 \\ 1 \\ 1 \\ 1 \end{array} \right) .$$ The linear space spanned by \(\mathbf { b } _ { 1 } , \mathbf { b } _ { 2 } , \mathbf { b } _ { 3 }\) is denoted by \(V _ { 1 }\) and the linear space spanned by \(\mathbf { b } _ { 1 } , \mathbf { b } _ { 2 } , \mathbf { b } _ { 4 }\) is denoted by \(V _ { 2 }\).

1. Give a reason why \(V _ { 1 } \cup V _ { 2 }\) is not a linear space.
2. State the dimension of the linear space \(V _ { 1 } \cap V _ { 2 }\) and write down a basis. Consider now the set \(V _ { 3 }\) of all vectors of the form \(q \mathbf { b } _ { 2 } + r \mathbf { b } _ { 3 } + s \mathbf { b } _ { 4 }\), where \(q , r , s\) are real numbers. Show that \(V _ { 3 }\) is a linear space, and show also that it has dimension 3 . Determine whether each of the vectors $$\left( \begin{array} { l } 4 \\ 4 \\ 2 \\ 5 \end{array} \right) \quad \text { and } \quad \left( \begin{array} { l } 5 \\ 4 \\ 2 \\ 5 \end{array} \right)$$ belongs to \(V _ { 3 }\) and justify your conclusions.

5 Each day that Adam is at work he carries out one of three tasks A, B or C. Each task takes a whole day. Adam chooses the task to carry out on each day according to the following set of three rules.

1. If, on any given day, he has worked on task A then the next day he will choose task A with probability 0.75 , and tasks B and C with equal probability.
2. If, on any given day, he has worked on task B then the next day he will choose task B or task C with equal probability but will never choose task A .
3. If, on any given day, he has worked on task C then the next day he will choose task A with probability \(p\) and tasks B and C with equal probability.
   1. Write down the transition matrix.
   2. Over a long period Adam carries out the tasks \(\mathrm { A } , \mathrm { B }\) and C with equal frequency. Find the value of \(p\).
   3. On day 1 Adam chooses task A . Find the probability that he also chooses task A on day 5 .
   Adam decides to change rule 3 as follows.
   If, on any given day, he has worked on task C then the next day he will choose tasks \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) with probabilities \(0.4,0.3,0.3\) respectively.
4. On day 1 Adam chooses task A. Find the probability that he chooses the same task on day 7 as he did on day 4 .
5. On a particular day, Adam chooses task A. Find the expected number of consecutive further days on which he will choose A. Adam changes all three rules again as follows.
   • If he works on A one day then on the next day he chooses C .
   • If he works on B one day then on the next day he chooses A or C each with probability 0.5.
   • If he works on C one day then on the next day he chooses A or B each with probability 0.5 .
   • Find the long term probabilities for each task.

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 \(B\).