10 Kira and Julian play a zero-sum game that does not have a stable solution. Kira has three strategies to choose from: \(\mathbf { K } _ { 1 } , \mathbf { K } _ { 2 }\) and \(\mathbf { K } _ { 3 }\)
To determine her optimal mixed strategy, Kira begins by defining the following variables:
\(v =\) value of the game for Kira
\(p _ { 1 } =\) probability of Kira playing strategy \(\mathbf { K } _ { 1 }\)
\(p _ { 2 } =\) probability of Kira playing strategy \(\mathbf { K } _ { 2 }\)
\(p _ { 3 } =\) probability of Kira playing strategy \(\mathbf { K } _ { 3 }\)
Kira then formulates the following linear programming problem. $$\begin{array} { l l } \text { Maximise } & v
\text { subject to } & 7 p _ { 1 } + p _ { 2 } + 8 p _ { 3 } \geq v
& 3 p _ { 1 } + 7 p _ { 2 } + 2 p _ { 3 } \geq v
& 9 p _ { 1 } + 2 p _ { 2 } + 4 p _ { 3 } \geq v \end{array}$$ and $$\begin{array} { r } p _ { 1 } + p _ { 2 } + p _ { 3 } \leq 1
p _ { 1 } , p _ { 2 } , p _ { 3 } \geq 0 \end{array}$$ 10

1. 1. Explain why the condition \(p _ { 1 } + p _ { 2 } + p _ { 3 } \leq 1\) is necessary in Kira's linear programming problem. 10
2. (ii) Explain why the condition \(p _ { 1 } , p _ { 2 } , p _ { 3 } \geq 0\) is necessary in Kira's linear programming problem. 10
3. Julian has three strategies to choose from: \(\mathbf { J } _ { 1 } , \mathbf { J } _ { 2 }\) and \(\mathbf { J } _ { 3 }\) Complete the following pay-off matrix which represents the game for Kira.

   	Julian
   \cline { 2 - 5 }	Strategy	\(\mathbf { J } _ { 1 }\)	\(\mathbf { J } _ { 2 }\)	\(\mathbf { J } _ { 3 }\)
   \multirow{3}{*}{Kira}	\(\mathbf { K } _ { 1 }\)	7
   \cline { 2 - 5 }	\(\mathbf { K } _ { 2 }\)
   \cline { 2 - 5 }	\(\mathbf { K } _ { 3 }\)