7 Player 1 and player 2 are playing a two-person zero-sum game.
In each round of the game the players each choose a strategy and simultaneously reveal their choice. The number of points won in each round by player 1 for each combination of strategies is shown in the table below. Each player is trying to maximise the number of points that they win.
Player 2 Player 1

1. 1. Determine play-safe strategies for each player.
   2. Show that the game is not stable.
2. Show that the number of strategies available to player 1 cannot be reduced by dominance. You must make it clear which values are being compared. Player 1 intends to make a random choice between strategies \(\mathrm { X } , \mathrm { Y } , \mathrm { Z }\), choosing strategy X with probability \(x\), strategy Y with probability \(y\) and strategy Z with probability \(z\).
   Player 1 formulates the following LP problem so they can find the optimal values of \(x , y\) and \(z\) using the simplex algorithm. Maximise \(M = m - 4\)
   subject to \(m \leqslant 6 x + 4 y + 2 z\) $$\begin{aligned} & m \leqslant x + 5 y + 6 z
   & m \leqslant 7 y + 8 z
   & x + y + z \leqslant 1 \end{aligned}$$ and \(m \geqslant 0 , x \geqslant 0 , y \geqslant 0 , z \geqslant 0\)
3. Explain how the inequality \(m \leqslant 6 x + 4 y + 2 z\) was formed. The problem is solved by running the simplex algorithm on a computer.
   The printout gives a solution in which \(\mathrm { x } + \mathrm { y } = 1\).
   This means that the LP problem can be reduced to the following formulation.
   Maximise \(M = m - 4\)
   subject to \(m \leqslant 4 + 2 x\)
   \(\mathrm { m } \leqslant 5 - 4 \mathrm { x }\)
   \(m \leqslant 7 - 7 x\)
   and \(m \geqslant 0 , x \geqslant 0\)
4. Solve this problem to find the optimal values of \(x , y\) and \(z\) and the corresponding value of the game to player 1.