NECO 2023 · Paper 2 · Q15

(a)
The payoff matrix of players $A$ and $B$ in a game is $\begin{pmatrix} 3 & 4 \\ 5 & 1 \end{pmatrix}$ . Find the (i) maximin value; (ii) minimax value; (iii) mixed strategies of $A$ and $B$ ; (iv) value of the game.
Show the answer
(i) 3; (ii) 4; (iii) $A$ : $(\frac45, \frac15)$ , $B$ : $(\frac35, \frac25)$ ; (iv) $\frac{17}{5} = 3.4$
(b)
The scores of 5000 students follow a normal distribution with mean 72 and variance 25. Determine the number of students who obtained scores between 80 and 84.

Worked solution (try it first)

Let $A$ play row 1 with probability $p$ : $3p + 5(1 - p) = 4p + 1(1 - p)$ , so $5 - 2p = 1 + 3p$ and $p = \frac45$ .
$A$ plays $\left(\frac45, \frac15\right)$ .
Let $B$ play column 1 with probability $q$ : $3q + 4(1 - q) = 5q + 1(1 - q)$ , so $4 - q = 1 + 4q$ and $q = \frac35$ .
$B$ plays $\left(\frac35, \frac25\right)$ .

Value $= 3\left(\frac45\right) + 5\left(\frac15\right)$
$= \frac{17}{5}$
$= 3.4$ .

$\sigma = \sqrt{25} = 5$ .
$z_1 = \dfrac{80 - 72}{5} = 1.6$ and $z_2 = \dfrac{84 - 72}{5} = 2.4$ .
$P = \Phi(2.4) - \Phi(1.6)$
$= 0.9918 - 0.9452$
$= 0.0466$ , so about $0.0466 \times 5000 \approx 233$ students.