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.

\((\text{i})\) 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?\((\text{ii})\) 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.

\((\text{iii})\) 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:

(You are\((\text{iv})\) Explain how \(9x + 3y + 11z\) was obtained.

The Simplex algorithm is used to solve the LP problem. The solution has \(x = \dfrac{7}{48}\), \(y = \dfrac{27}{48}\), \(z = \dfrac{14}{48}\).

\((\text{v})\) Calculate the optimal value of \(M\).

**If \(M=\dfrac{a}{b}\) for coprime \(a\) and \(b\), input \(a+b\) as your answer.**

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