Converting games to linear programming problemsAQA A-Level Further Maths: Flashcards

Exam questions on Converting games to linear programming problems

1. Rowan and Colin play a zero-sum game with pay-off matrix for Rowan $\begin{pmatrix} -3 & 2 & 1 \\ 1 & -1 & 0 \\ 2 & 0 & -2 \end{pmatrix}$. Rowan wants to convert the game into a linear programming problem to find his optimal mixed strategy, and first adds the same constant to every entry.
   State the effect of adding the constant on Rowan's optimal strategy and on the value of the game, giving a reason.2 marks
2. Rowan and Colin play a zero-sum game. After a constant has been added, Rowan's pay-off matrix is $\begin{pmatrix} 3 & 1 \\ 2 & 4 \\ 1 & 5 \end{pmatrix}$, with rows 1, 2, 3 for Rowan and columns 1, 2 for Colin. Rowan plays rows 1, 2, 3 with probabilities $p_1$, $p_2$, $p_3$, and $V$ is the value of this game. Rowan wishes to maximise $V$.
   Explain, in context, why $V$ must be no greater than Rowan's expected pay-off against each of Colin's columns.2 marks
3. Rowan and Colin play a zero-sum game with pay-off matrix for Rowan $\begin{pmatrix} -1 & 2 & 1 \\ 2 & -2 & 0 \\ 0 & 1 & -1 \end{pmatrix}$. Two is added to every entry, giving $\begin{pmatrix} 1 & 4 & 3 \\ 4 & 0 & 2 \\ 2 & 3 & 1 \end{pmatrix}$. Rowan plays rows 1, 2, 3 with probabilities $p_1$, $p_2$, $p_3$, and $V$ is the value of the adjusted game.
   Formulate Rowan's problem as a linear programming problem, in the form of an objective function and constraints, ready to be solved by the simplex algorithm.3 marks