Zero-sum game stable solution

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.

2. (a) Explain what the term 'zero-sum game' means. Two teams, A and B , are to face each other as part of a quiz.
There will be several rounds to the quiz with 10 points available in each round.
For each round, the two teams will each choose a team member and these two people will compete against each other until all 10 points have been awarded. The number of points that Team A can expect to gain in each round is shown in the table below. The teams are each trying to maximise their number of points.
(b) State the number of points that Team B will expect to gain each round if Team A chooses Noel and Team B chooses Rashid.
(c) Explain why subtracting 5 from each value in the table will model this situation as a zero-sum game.
(d) (i) Find the play-safe strategies for the zero-sum game.
(ii) Explain how you know that the game is not stable. At the last minute, Olive becomes unavailable for selection by Team A.
Team A decides to choose its player for each round so that the probability of choosing Mischa is \(p\) and the probability of choosing Noel is \(1 - p\).
(e) Use a graphical method to find the optimal value of \(p\) for Team A and hence find the best strategy for Team A. For this value of \(p\),
(f) (i) find the expected number of points awarded, per round, to Team A,
(ii) find the expected number of points awarded, per round, to Team B.

3. Two teams, A and B , each have three team members. One member of Team A will compete against one member of Team B for 10 rounds of a competition. None of the rounds can end in a draw. Table 1 shows, for each pairing, the expected number of rounds that the member of Team A will win minus the expected number of rounds that the member of Team B will win. These numbers are the scores awarded to Team A. This competition between Teams A and B is a zero-sum game. Each team must choose one member to play. Each team wants to choose the member who will maximise its score. \begin{table}[h] \end{table}

1. 1. Find the number of rounds that Team A expects to win if Team A chooses Mischa and Team B chooses Paul.
   2. Find the number of rounds that Team B expects to win if Team A chooses Noel and Team B chooses Qaasim. Table 1 models this zero-sum game.
2. 1. Find the play-safe strategies for the game.
   2. Explain how you know that the game is not stable.
3. Determine which team member Team B should choose if Team B thinks that Team A will play safe. Give a reason for your answer. At the last minute, Rashid is ill and is therefore unavailable for selection by Team B.
4. Find the best strategy for Team B, defining any variables you use.

5 In each round of a card game two players each have four cards. Every card has a coloured number.

• Player A's cards are red 1 , blue 2 , red 3 and blue 4.
• Player B's cards are red 1 , red 2 , blue 3 and blue 4 .

Each player chooses one of their cards. The players then show their choices simultaneously and deduce how many points they have won or lost as follows:

• If the numbers are the same both players score 0 .
• If the numbers are different but are the same colour, the player with the lower value card scores the product of the numbers on the cards.
• If the numbers are different and are different colours, the player with the higher value card scores the sum of the numbers on the cards.
• The game is zero-sum.
  1. Complete the pay-off matrix for this game, with player A on rows.
  2. Determine the play-safe strategy for each player.
  3. Use dominance to show that player A should not choose red 3 . You do not need to identify other rows or columns that are dominated.
  4. Determine, with a reason, whether the game is stable or unstable.

1 Arif and Bindiya play a game as follows.

• They each secretly choose a positive integer from \(\{ 2,3,4,5 \}\).
• They then reveal their choices. Let Arif's choice be \(A\) and Bindiya's choice be \(B\).
• If \(A ^ { B } \geqslant B ^ { A }\), Arif wins \(B\) points and Bindiya wins \(- 4 - B\) points.
• If \(A ^ { B } < B ^ { A }\), Arif wins \(- 4 - A\) points and Bindiya wins \(A\) points.

6 Sam is playing a computer game in which he is trying to drive a car in different road conditions. He chooses a car and the computer decides the road conditions. The points scored by Sam are shown in the table. Sam is trying to maximise his total points and the computer is trying to stop him.

1. Explain why Sam should never choose \(S _ { 1 }\) and why the computer should not choose \(C _ { 3 }\).
2. Find the play-safe strategies for the reduced 2 by 2 game for Sam and the computer, and hence show that this game does not have a stable solution.
3. Sam uses random numbers to choose \(S _ { 2 }\) with probability \(p\) and \(S _ { 3 }\) with probability \(1 - p\).
   1. Find expressions for the expected gain for Sam when the computer chooses each of its two remaining strategies.
   2. Calculate the value of \(p\) for Sam to maximise his total points.
   3. Hence find the expected points gain for Sam.

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      \section*{General Certificate of Education January 2006
      Advanced Level Examination} \section*{MATHEMATICS
      Unit Decision 2} MD02 \section*{Insert} Wednesday 18 January 20061.30 pm to 3.00 pm Insert for use in Questions 3 and 4.
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2 Alex and Roberto play a zero-sum game. The game is represented by the following pay-off matrix for Alex. \begin{table}[h] \end{table}

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

2 Stan and Christine play a zero-sum game. The game is represented by the following pay-off matrix for Stan. \begin{table}[h] \end{table}

1. Find the play-safe strategy for each player.
2. Show that there is no stable solution.
3. Explain why a suitable pay-off matrix for Christine is given by

3 Alex and Sam are playing a zero-sum game. The game is represented by the pay-off matrix for Alex. 3

1. Explain why the value of the game is 2
   3
2. Identify the play-safe strategy for each player.
   Each pipe is labelled with its upper capacity in \(\mathrm { cm } ^ { 3 } \mathrm {~s} ^ { - 1 }\) \includegraphics[max width=\textwidth, alt={}, center]{5a826f8b-4751-4589-ad0a-109fc5c821f2-04_620_940_450_550}

1 Deanna and Will play a zero-sum game.
The game is represented by the following pay-off matrix for Deanna. Which strategy is Deanna's play-safe strategy?
Circle your answer.
A
B
C
D

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

	I	II	III	IV
I	\(-4\)	\(-5\)	\(-2\)	4
II	\(-1\)	1	\(-1\)	2
III	0	5	\(-2\)	\(-4\)
IV	\(-1\)	3	\(-1\)	1

1. Determine the play-safe strategy for each player. [4]
2. Verify that there is a stable solution and determine the saddle points. [3]
3. State the value of the game to \(B\). [1]

In game theory explain what is meant by

1. zero-sum game, [2]
2. saddle point. [2]

(Total 4 marks)

Vaya and Wynne are playing a zero-sum game. The game is represented by the pay-off matrix for Vaya. \includegraphics{figure_6}

1. Find the play-safe strategies for Vaya and Wynne. Fully justify your answer. [4 marks]
2. Vaya and Wynne decide not to play their play-safe strategies. Deduce the best possible outcome for Wynne. [2 marks]

Bilal and Mayon play a zero-sum game. The game is represented by the following pay-off matrix for Bilal, where \(x\) is an integer.

	Mayon
	\(\mathbf{M_1}\)	\(\mathbf{M_2}\)	\(\mathbf{M_3}\)
\(\mathbf{B_1}\)	\(-2\)	\(-1\)	\(1\)
Bilal \quad \(\mathbf{B_2}\)	\(4\)	\(-3\)	\(1\)
\(\mathbf{B_3}\)	\(-1\)	\(x\)	\(0\)

The game has a stable solution.

1. Show that there is only one possible value for \(x\) Fully justify your answer. [6 marks]
2. State the value of the game for Bilal. [1 mark]

Ben and Jadzia play a zero-sum game. The game is represented by the following pay-off matrix for Ben.

	Jadzia
Strategy	X	Y	Z
A	-3	2	3
Ben B	6	0	-4
C	7	-1	1
D	6	-2	1

1. State, with a reason, which strategy Ben should never play. [1 mark]
2. Determine whether or not the game has a stable solution. Fully justify your answer. [3 marks]
3. Ben knows that Jadzia will always play her play-safe strategy. Explain how Ben can maximise his expected pay-off. [2 marks]

Daniel and Jackson play a zero-sum game. The game is represented by the following pay-off matrix for Daniel. Jackson

	Strategy	W	X	Y	Z
\multirow{4}{*}{Daniel}	A	3	\(-2\)	1	4
	B	5	1	\(-4\)	1
	C	2	\(-1\)	1	2
	D	\(-3\)	0	2	\(-1\)

Neither player has any strategies which can be ignored due to dominance.

1. Prove that the game does not have a stable solution. Fully justify your answer. [3 marks]
2. Determine the play-safe strategy for each player. [1 mark] Play-safe strategy for Daniel _______________________________________________ Play-safe strategy for Jackson ______________________________________________

Each day Alix and Ben play a game. They each choose a card and use the table below to find the number of points they win. The table shows the cards available to each player. The entries in the cells are of the form \((a, b)\), where \(a =\) points won by Alix and \(b =\) points won by Ben. Each is trying to maximise the points they win.

1. Explain why the table cannot be reduced through dominance no matter what values \(x\) and \(y\) have. [2]
2. Show that the game is not stable no matter what values \(x\) and \(y\) have. [2]
3. Find the Nash equilibrium solutions for the various values that \(x\) and \(y\) can have. [4]

The table shows the pay-off matrix for player \(A\) in a two-person zero-sum game between \(A\) and \(B\).

1. Find the play-safe strategy for player \(A\) and the play-safe strategy for player \(B\). Use the values of the play-safe strategies to determine whether the game is stable or unstable. [3]
2. If player \(B\) knows that player \(A\) will use their play-safe strategy, which strategy should player \(B\) use? [1]
3. Suppose that the value in the cell where both players use their play-safe strategies can be changed, but all other entries are unchanged. Show that there is no way to change this value that would make the game stable. [2]
4. Suppose, instead, that the value in one cell can be changed, but all other entries are unchanged, so that the game becomes stable. Identify a suitable cell and write down a new pay-off value for that cell which would make the game stable. [2]
5. Show that the zero-sum game with the new pay-off value found in part (iv) has a Nash equilibrium and explain what this means for the players. [3]