Question

3. Evaluate the following payoffs for the game pictured here: (a) ui(οι, Γ) for σ,-(ja, 1/4, 1/4, 1/4) (b) 112(oi, О) for σ,-(1/8, î/4, 1/43/8) 2 OA | 2,2 | 2,2 2,2 2, IA 4,2 1,3 IB 3,4 1,3

Answer 1

a)

P1 is playing mixed strategy $\small \sigma_1 = (1/4, 1/4, 1/4, 1/4)$

P2 is playing column I

$\small u_1 (\sigma_1, I) = \frac{1}{4}*2 + \frac{1}{4}*2 + \frac{1}{4}*4 + \frac{1}{4}* 3 = 11/4$

b)

P1 is playing mixed strategy $\small \sigma_1 = (1/8, 1/4, 1/4, 3/8)$

P2 is playing column O

$\small u_2 (\sigma_1, 0) = \frac{1}{8}*2 + \frac{1}{4}*2 + \frac{1}{4}*3+ \frac{3}{8}* 3 = \frac{21}{8}$

c)

P1 is playing mixed strategy $\small \sigma_1 = (1/4, 1/4, 1/4, 1/4)$

P2 is playing column $\small \sigma_2 = (1/3, 2/3)$

$\small u_1 (\sigma_1, \sigma_2) = \frac{1}{4}*\frac{1}{3}*2 + \frac{1}{4}*\frac{1}{3}*2 + \frac{1}{4}*\frac{1}{3}*4+ \frac{1}{4}*\frac{1}{3}* 3 + \frac{1}{4}*\frac{2}{3}*2 + \frac{1}{4}*\frac{2}{3}*2 + \frac{1}{4}*\frac{2}{3}*1+ \frac{1}{4}*\frac{2}{3}* 1$

$\small = \frac{2}{12} + \frac{2}{12} + \frac{4}{12}+ \frac{3}{12} + \frac{4}{12} + \frac{4}{12} + \frac{2}{12}+\frac{2}{12} = \frac{23}{12}$

d)

P1 is playing mixed strategy $\small \sigma_1 = (0, 1/3, 1/6, 1/2)$

P2 is playing column $\small \sigma_2 = (2/3, 1/3)$

$\small u_1 (\sigma_1, \sigma_2) = 0*\frac{2}{3}*2 + \frac{1}{3}*\frac{2}{3}*2 + \frac{1}{6}*\frac{2}{3}*4+ \frac{1}{2}*\frac{2}{3}* 3 +0*\frac{1}{3}*2 + \frac{1}{3}*\frac{1}{3}*2 + \frac{1}{6}*\frac{1}{3}*1+ \frac{1}{2}*\frac{1}{3}* 1$

$\small =0 + \frac{4}{9} + \frac{4}{9}+ 1 + 0+ \frac{2}{9}+\frac{1}{18}+ \frac{1}{6} = \frac{24}{18} = \frac{4}{3}$