7.08b Dominance: reduce pay-off matrix

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}

3 Summer and Haf play a zero-sum game. The pay-off matrix for the game is shown below. Haf 3

1. Show that the game has a stable solution.
   3
2. (i) State the value of the game for Summer. 3 (b) (ii) State the play-safe strategy for each player.

7 Kez and Lui play a zero-sum game. The game does not have a stable solution. The game is represented by the following pay-off matrix for Kez. 7

1. State, with a reason, why Kez should never play strategy \(\mathbf { K } _ { \mathbf { 2 } }\) 7
2. \(\quad\) Kez and Lui play the game 20 times.
   Kez plays their optimal mixed strategy.
   Find the expected number of times that Kez will play strategy \(\mathbf { K } _ { \mathbf { 3 } }\) Fully justify your answer.

6 Xander and Yvonne are playing a zero-sum game. The game is represented by the pay-off matrix for Xander. \begin{table}[h] \end{table} 6

1. Show that the game has a stable solution.
   6
2. State the play-safe strategy for each player. Play-safe strategy for Xander is \(\_\_\_\_\) Play-safe strategy for Yvonne is \(\_\_\_\_\) 6
3. The game that Xander and Yvonne are playing is part of a marbles challenge. The pay-off matrix values represent the number of marbles gained by Xander in each game. In the challenge, the game is repeated until one player has 24 marbles more than the other player. Explain why Xander and Yvonne must play at least 3 games to complete the challenge.

6 Victoria and Albert play a zero-sum game. The game is represented by the following pay-off matrix for Victoria. 6

1. Find the play-safe strategies for each player.
   6
2. State, with a reason, the strategy that Albert should never play.
   6
3. (i) Determine an optimal mixed strategy for Victoria.
   [0pt] [5 marks]
   6 (c) (ii) Find the value of the game for Victoria.
   6 (c) (iii) State an assumption that must made in order that your answer for part (c)(ii) is the maximum expected pay-off that Victoria can achieve.

2 Jonathan and Hoshi play a zero-sum game.
The game is represented by the following pay-off matrix for Jonathan. The game does not have a stable solution.
Which strategy should Jonathan never play?
Circle your answer.
[0pt] [1 mark] \(\mathbf { J } _ { \mathbf { 1 } }\) \(\mathbf { J } _ { \mathbf { 2 } }\) \(\mathbf { J } _ { \mathbf { 3 } }\) \(\mathbf { J } _ { \mathbf { 4 } }\)

6. A two person zero-sum game is represented by the pay-off matrix for player A, shown above.

1. Explain, with justification, why this matrix may be reduced to a \(3 \times 3\) matrix by removing option S from player A's choices.
2. Verify that there is no stable solution to the reduced game. Player A intends to make a random choice between options \(\mathrm { Q } , \mathrm { R }\) and T , choosing option Q with probability \(p _ { 1 }\), option R with probability \(p _ { 2 }\) and option T with probability \(p _ { 3 }\) Player A wants to find the optimal values of \(p _ { 1 } , p _ { 2 }\) and \(p _ { 3 }\) using the Simplex algorithm. Player A formulates the following linear programme, writing the constraints as inequalities. Maximise \(P = V\), where \(V =\) the value of original game + 3 $$\begin{aligned} \text { subject to } & V \leqslant 4 p _ { 1 } + 7 p _ { 2 } + 6 p _ { 3 } \\ & V \leqslant 8 p _ { 1 } + p _ { 3 } \\ & V \leqslant 6 p _ { 1 } + 4 p _ { 2 } + 3 p _ { 3 } \\ & p _ { 1 } + p _ { 2 } + p _ { 3 } \leqslant 1 \\ & p _ { 1 } \geqslant 0 , p _ { 2 } \geqslant 0 , p _ { 3 } \geqslant 0 , V \geqslant 0 \end{aligned}$$
3. Explain why \(V\) cannot exceed any of the following expressions $$4 p _ { 1 } + 7 p _ { 2 } + 6 p _ { 3 } \quad 8 p _ { 1 } + p _ { 3 } \quad 6 p _ { 1 } + 4 p _ { 2 } + 3 p _ { 3 }$$
4. Explain why it is necessary to use the constraint \(p _ { 1 } + p _ { 2 } + p _ { 3 } \leqslant 1\) The Simplex algorithm is used to solve the linear programming problem.
   Given that the optimal value of \(p _ { 1 } = \frac { 7 } { 11 }\) and the optimal value of \(p _ { 3 } = 0\)
5. calculate the value of the game to player A .
   (3) Player B intends to make a random choice between options \(\mathrm { X } , \mathrm { Y }\) and Z , choosing option X with probability \(q _ { 1 }\), option Y with probability \(q _ { 2 }\) and option Z with probability \(q _ { 3 }\)
6. Determine the optimal strategy for player B, making your working clear.

7. A two person zero-sum game is represented by the pay-off matrix for player A shown above. It is given that \(k\) is an integer.

1. Show that Q is the play-safe option for player A regardless of the value of \(k\). Given that Z is the play-safe option for player B ,
2. determine the range of possible values of \(k\). You must make your working clear.
3. Explain why player B should never play option X. You must make your reasoning clear. Player A intends to make a random choice between options \(\mathrm { Q } , \mathrm { R }\) and S , choosing option Q with probability \(p _ { 1 }\), option R with probability \(p _ { 2 }\) and option S with probability \(p _ { 3 }\) Player A wants to find the optimal values of \(p _ { 1 } , p _ { 2 }\) and \(p _ { 3 }\) using the Simplex algorithm.
   Given that \(k > - 4\), player A formulates the following objective function for the corresponding linear program. $$\text { Maximise } P = V \text {, where } V = \text { the value of the original game } + 4$$
4. 1. Formulate the constraints of the linear programming problem for player A. You should write the constraints as equations.
   2. Write down an initial Simplex tableau, making your variables clear. The Simplex algorithm is used to solve the linear programming problem. It is given that in the final Simplex tableau the optimal value of \(p _ { 1 } = \frac { 7 } { 37 }\), the optimal value of \(p _ { 2 } = \frac { 17 } { 37 }\) and all the slack variables are zero.
5. Determine the value of \(k\), making your method clear.

Andrew (\(A\)) and Barbara (\(B\)) play a zero-sum game. This game is represented by the following pay-off matrix for Andrew. $$A \begin{pmatrix} 3 & 5 & 4 \\ 1 & 4 & 2 \\ 6 & 3 & 7 \end{pmatrix}$$

1. Explain why this matrix may be reduced to $$\begin{pmatrix} 3 & 5 \\ 6 & 3 \end{pmatrix}$$ [8]
2. Hence find the best strategy for each player and the value of the game.

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]

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

	Strategy	\(S_1\)	\(S_2\)	\(S_3\)
\multirow{4}{*}{Janet}	\(J_1\)	2	7	6
	\(J_2\)	5	5	1
	\(J_3\)	4	3	8
	\(J_4\)	1	6	4

1. Explain why Janet should never play strategy \(J_4\) [1 mark]
2. Janet wants to maximise her winnings from the game. She defines the following variables. \(p_1 = \) the probability of Janet playing strategy \(J_1\) \(p_2 = \) the probability of Janet playing strategy \(J_2\) \(p_3 = \) the probability of Janet playing strategy \(J_3\) \(v = \) the value of the game for Janet Janet then formulates her situation as the following linear programming problem. Maximise \(P = v\) subject to \(2p_1 + 5p_2 + 4p_3 \geq v\) \(7p_1 + 5p_2 + 3p_3 \geq v\) \(6p_1 + p_2 + 8p_3 \geq v\) and \(p_1 + p_2 + p_3 \leq 1\) \(p_1, p_2, p_3 \geq 0\)
   1. Complete the initial Simplex tableau for Janet's situation in the grid below. [2 marks]

      \(P\)	\(v\)	\(p_1\)	\(p_2\)	\(p_3\)			value

   2. Hence, perform one iteration of the Simplex algorithm, showing your answer on the grid below. [2 marks]

      \(P\)	\(v\)	\(p_1\)	\(p_2\)	\(p_3\)			value

3. Further iterations of the Simplex algorithm are performed until an optimal solution is reached. The grid below shows part of the final Simplex tableau.

   \(p_1\)	\(p_2\)	value
   1	0	\(\frac{1}{12}\)
   0	1	\(\frac{1}{2}\)

   Find the probability of Janet playing strategy \(J_3\) when she is playing to maximise her winnings from the game. [1 mark]