Zero-sum game optimal mixed strategy

4 Euan and Wai Mai play a zero-sum game. Each is trying to maximise the total number of points that they score in many repeats of the game. The table shows the number of points that Euan scores for each combination of strategies.

1. Explain what the term 'zero-sum game' means.
2. How many points does Wai Mai score if she chooses \(X\) and Euan chooses \(A\) ?
3. Why should Wai Mai never choose strategy \(Z\) ?
4. Delete the column for \(Z\) and find the play-safe strategy for Euan and the play-safe strategy for Wai Mai on the table that remains. Is the resulting game stable or not? State how you know. The value 3 in the cell corresponding to Euan choosing \(D\) and Wai Mai choosing \(X\) is changed to - 5 ; otherwise the table is unchanged. Wai Mai decides that she will choose her strategy by making a random choice between \(X\) and \(Y\), choosing \(X\) with probability \(p\) and \(Y\) with probability \(1 - p\).
5. Write down and simplify an expression for the expected score for Wai Mai when Euan chooses each of his four strategies.
6. Using graph paper, draw a graph showing Wai Mai's expected score against \(p\) for each of Euan's four strategies and hence calculate the optimum value of \(p\).

6. 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.

2. 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 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.

8 Daryl and Clare play a zero-sum game. The game is represented by the following pay-off matrix for Daryl. Clare

7 Avon and Roj play a zero-sum game. The game is represented by the following pay-off matrix for Avon. 7 (c)

1. Find the optimal mixed strategy for Avon.
   7
2. Find the value of the game for Avon.
   

7 (d) Roj thinks that his best outcome from the game is to play strategy \(\mathbf { R } _ { \mathbf { 2 } }\) each time. Avon notices that Roj always plays strategy \(\mathbf { R } _ { \mathbf { 2 } }\) and Avon wants to use this knowledge to maximise his expected pay-off from the game. Explain how your answer to part (c)(i) should change and find Avon's maximum expected pay-off from the game. \includegraphics[max width=\textwidth, alt={}, center]{59347089-ea4a-4ee6-b40e-1ab78aa7cdc3-16_2490_1735_219_139}

Emma and Freddie play a zero-sum game. This game is represented by the following pay-off matrix for Emma. \(\begin{pmatrix} -4 & -1 & 3 \\ 2 & 1 & -2 \end{pmatrix}\)

1. Show that there is no stable solution. [3]
2. Find the best strategy for Emma and the value of the game to her. [8]
3. Write down the value of the game to Freddie and his pay-off matrix. [3]

(Total 14 marks)

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

	Y
	\(Y_1\)	\(Y_2\)
\multirow{2}{*}{X}	\(X_1\)	\(-2\)	4
	\(X_2\)	6	\(-1\)

1. Explain why the game does not have a saddle point. [3 marks]
2. Find the optimal strategy for
   1. player X, [8 marks]
   2. player Y.
3. Find the value of the game. [2 marks]