| Exam Board | OCR |
|---|---|
| Module | D2 (Decision Mathematics 2) |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear Programming |
| Type | Game theory LP formulation |
| Difficulty | Standard +0.3 This is a standard textbook exercise in game theory LP formulation from D2. It requires systematic application of a well-defined algorithm (converting payoff matrix to LP) with multiple routine steps, but no novel insight or problem-solving. The presence of negative values adds a minor complication (requiring matrix transformation), but this is a standard taught technique. Slightly above average difficulty due to the multi-part nature and need for careful bookkeeping. |
| Spec | 7.08a Pay-off matrix: zero-sum games7.08b Dominance: reduce pay-off matrix7.08c Pure strategies: play-safe strategies and stable solutions7.08d Nash equilibrium: identification and interpretation7.08e Mixed strategies: optimal strategy using equations or graphical method7.08f Mixed strategies via LP: reformulate as linear programming problem |
| \cline { 3 - 5 } \multicolumn{2}{c|}{} | \(B\) | |||
| \cline { 2 - 5 } \multicolumn{2}{c|}{} | I | II | III | |
| \multirow{3}{*}{\(A\)} | I | 6 | - 4 | - 1 |
| \cline { 2 - 5 } | II | - 2 | 5 | 3 |
| \cline { 2 - 5 } | III | 5 | 1 | - 3 |
\begin{enumerate}
\item The payoff matrix for player $A$ in a two-person zero-sum game with value $V$ is shown below.
\end{enumerate}
\begin{center}
\begin{tabular}{ | c | c | c | c | c | }
\cline { 3 - 5 }
\multicolumn{2}{c|}{} & \multicolumn{3}{c|}{$B$} \\
\cline { 2 - 5 }
\multicolumn{2}{c|}{} & I & II & III \\
\hline
\multirow{3}{*}{$A$} & I & 6 & - 4 & - 1 \\
\cline { 2 - 5 }
& II & - 2 & 5 & 3 \\
\cline { 2 - 5 }
& III & 5 & 1 & - 3 \\
\hline
\end{tabular}
\end{center}
Formulate this information as a linear programming problem, the solution to which will give the optimal strategy for player $B$.\\
(a) Rewrite the matrix as necessary and state the new value of the game, $v$, in terms of $V$.\\
(b) Define your decision variables.\\
(c) Write down the objective function in terms of your decision variables.\\
(d) Write down the constraints.\\
\hfill \mbox{\textit{OCR D2 Q1 [8]}}