Two bookstores are competing for customers. Both bookstores can decide to offer

Question

Two bookstores are competing for customers. Both bookstores can decide to offer discounts to attract more customers. Bookstore-A has a 30% probability of offering a discount. The probability that Bookstore-B will offer a discount is unknown, and is represented by P. The payoffs for the bookstores depending on whether discounts are offered are listed as follows. Bookstore-A finds that the expected payoffs are the same whether the discount is offered or not. Bookstore-B also realizes that its expected payoff is identical whether the discount is offered or not. Find payoff value X and probability level P.

                                     Bookstore-B

                                     Discount No Discount

Bookstore-A Discount (100, 30) (40, 70)

                 No Discount (60, 50) (80, X)

explain step by step?

Explanation / Answer

Let Probability that bookstore B will offer discount =P

Case-1: Bookstore A offers discount

Payoff of A =100*Probability that B offers discount+40* Probability that B doesn’t offers discount

Payoff of A =100*P+40*(1-P)

Case-1: Bookstore A doesn’t offers discount

Payoff of A =60*Probability that B offers discount+80* Probability that B doesn’t offers discount

Payoff of A =60*P+80*(1-P)

Since Bookstore A finds that the expected payoffs are the same whether the discount is offered or not.

100*P+40*(1-P) = 60*P+80*(1-P)

100P+40-40P=60P+80-80P

60P+40=-20P+80

80P=40

P=0.5

Calculation of X

Case-1: Bookstore B offers discount

Payoff of B =30*Probability that A offers discount+70* Probability that a doesn’t offers discount

Payoff of B =30*0.3+50*(1-0.3)=44

Case-1: Bookstore B doesn’t offers discount

Payoff of B =70*Probability that A offers discount +X* Probability that A doesn’t offers discount

Payoff of B =70*0.3+X(1-0.3)

Payoff of B =21+X(1-0.3)=21+X*0.7

Since Bookstore B finds that the expected payoffs are the same whether the discount is offered or not.

44=21+X*0.7

X= 32.85714

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