The CEO of McBurger considers opening a new restaurant with two size options: la

Question

The CEO of McBurger considers opening a new restaurant with two size options: large model (Large) and small model (Small). She anticipates two possible states of peak-hour demand at the new location: high demand (H) with a probability of 0.75 and low demand (L) with a probability of 0.25. The CEO uses the expected value criterion for decision making and she developed the original decision tree below.

A market research firm called BurgerMarket can give McBurger a survey about whether demand will be high or low. The survey result on the demand will be either favorable (f) or unfavorable (u). The CEO would like to determine the expected value of the survey and calculated:
P(f) = 0.6,  P(u) = 0.4
P(H|f) = 0.9,  P(L|f) = 0.1,  P(H|u) = 0.7,  P(L|u) = 0.3
What is the expected value (in $1,000) of the new decision tree with the survey alternative added?

Explanation / Answer

Ans:Expected value without survey:

Case I:Large

Expected value=12*0.75-6*0.25=7.5

Case II:Small

Expected value=10*0.75+2*0.25=8

Calculate the posterior probabilities:

P(f/H)=P(H/f)*P(f)/[P(H/f)*P(f)+P(H/u)*P(u)]

=0.9*0.6/[0.9*0.6+0.7*0.4]

=0.54/[0.54+0.28]

=0.54/0.82

P(f/H)=0.658

P(u/H)=1-0.658=0.342

P(f/L)=P(L/f)*P(f)/[P(L/f)*P(f)+P(L/u)*P(u)]

=0.1*0.6/[0.1*0.6+0.3*0.4]

=0.06/[0.06+0.12]

=0.06/0.18

P(f/L)=0.333

P(u/L)=1-0.333=0.667

Case I:Large

Expected payoff=0.75*[0.658*12-0.342*6]+0.25*[12*0.667-6*0.333]

=4.383+1.501

=5.884

Case II: Small

Expected payoff=0.75*[0.658*10+0.342*2]+0.25*[10*0.667+2*0.333]

                        =5.448+1.834

                        =7.282

Expected value in Case II(Small) is higher than Case I(Large).

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