Challenging +1.8 This is a game theory problem requiring understanding of play-safe strategies, stability, dominance, and LP formulation for mixed strategies. While the individual components (finding row/column minima/maxima, comparing strategies, solving a simplified LP) are mechanical, the problem requires synthesis across multiple game theory concepts and careful interpretation of the LP setup. The simplex reduction and final optimization require multi-step reasoning beyond standard textbook exercises, placing it well above average difficulty.
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
A
B
C
X
2
- 3
- 4
Y
0
1
3
Z
- 2
2
4
Determine play-safe strategies for each player.
Show that the game is not stable.
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 \(m \geqslant 0 , x \geqslant 0 , y \geqslant 0 , z \geqslant 0\)
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 \(m \geqslant 0 , x \geqslant 0\)
Solve this problem to find the optimal values of \(x , y\) and \(z\) and the corresponding value of the game to player 1.
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
\begin{center}
\begin{tabular}{ c | r | r | r }
& A & B & C \\
\hline
X & 2 & - 3 & - 4 \\
\hline
Y & 0 & 1 & 3 \\
\hline
Z & - 2 & 2 & 4 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Determine play-safe strategies for each player.
\item Show that the game is not stable.
\end{enumerate}\item 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 $m \geqslant 0 , x \geqslant 0 , y \geqslant 0 , z \geqslant 0$
\item 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 $m \geqslant 0 , x \geqslant 0$
\item Solve this problem to find the optimal values of $x , y$ and $z$ and the corresponding value of the game to player 1.
\end{enumerate}
\hfill \mbox{\textit{OCR Further Discrete 2023 Q7 [12]}}