Question

Systronics is considering introducing a new solvent that is biodegradable and that they believe has a 60% chance of being successful. If this product is successful, Systronics expects the net profit will be $50 million. If unsuccessful the net loss is expected to be $20 million.

1. Set up the payoff table on a spreadsheet and calculate the expected payoff. Which is the optimal decision using Bayes’ decision rule?
2. Calculate the expected value of perfect information.
3. Systronics has the option to try out the product in a test market. Past experience has shown that products that were ultimately successful in the broader market were approved during the test phase 80% of the time. Products that were found to be unsuccessful in the broader market had been approved in the testing phase only 15% of the time.
   1. Draw the decision tree Systronics could use to help them decide whether or not to test the new solvent in a test market and ultimately launch the new product.
   2. What is the Expected Monetary Value of the decision to test the new product and what is their recommended course of action based on the results of the market test?
   3. How much would Systronics be willing to pay for the market test?

Answer 1

(a)

	States of nature		Expected
Alternatives	Successful	Unsuccessful	payoff
Introduce	50	-20	22
Don't introduce	0	0	0
Probability	0.6	0.4
Opportunity loss	States of nature		Expected
Alternatives	Successful	Unsuccessful	OL
Introduce	0	20	8
Don't introduce	50	0	30
Probability	0.6	0.4

As per the Baye's decision rule, we will take the alternative which minimizes the expected opportunity loss i.e. to Introduce.

(b)

Expected value of perfect information = Expected opportunity loss = $8 million

(c)

Define events as:

IS = Test approved
IF = Test failed
S = Actual success in the broad market
F = Actual failure in the broad market

Given is Priors i.e. P(S) = 0.6 and P(F) = 0.4 and conditionals i.e. P(IS | S) = 0.80 and P(IS | F) = 0.15.

Use the Baye's rule using the following table to compute the posteriors.

IS
Prior		Conditional		Joint		Posterior
P(S)	0.6	P(IS|S)	0.8	P(S) x P(IS|S)	0.48	P(S|IS)	0.89
P(F)	0.4	P(IS|F)	0.15	P(F) x P(IS|F)	0.06	P(F|IS)	0.11
Total	1			P(IS)	0.54		1
IF
Prior		Conditional		Joint		Posterior
P(S)	0.6	P(IF|S)	0.2	P(S) x P(IF|S)	0.12	P(S|IF)	0.26
P(F)	0.4	P(IF|F)	0.85	P(F) x P(IF|F)	0.34	P(F|IF)	0.74
Total	1			P(IF)	0.46		1

(i)

Decision tree

(ii)

The optimal decision is to go for the test and then introduce if the test shows success; don't introduce otherwise.

(iii)

Expected monetary value = 22.80 million

(iv)

Willing ot pay = 22.80 - 22 = 0.80 million or less