Question

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A city in Ohio is considering replacing its fleet of gasoline powered cars with electric cars. The manufacturer of the electric cars claims that this municipality will experience significant cost savings over the life of the fleet if it chooses to pursue this conversion. If the manufacturer is correct, the city will save about $1.5 million. If the new technology employed within the electric cars is faulty as some critics suggest, it will cost the city $675,000. A third possibility is that less serious problems will arise and the city will break even in the conversion. A consultant hired by the city estimates the probabilities of these three outcomes are 0.30, 0.30, and 0.40, respectively. The city has an opportunity to implement a pilot program that would indicate the potential cost or savings resulting from the switch to electric cars. The pilot program involves renting a small number of electric cars for three months and running them under typical conditions. This program would cost the city $75,000. The city’s consultant believes that the results of the pilot program would be significant but not conclusive. She submits the following compilation of probabilities based on the experience of other cities to support her contention.

Savings		Loss		Breakeven
Indicates Saving	0.6		0.1		0.4
Indicates Loss	0.1		0.4		0.2
Indicates Breakeven	0.3		0.5		0.4

For example, the first column of her table indicates that given that the conversion to electric cars actually results in a savings, the conditional probabilities that the pilot program will indicate that the city saves money, loses money, and breaks even are 0.6, 0.1, and 0.3. What actions should the city take to maximize its expected savings? When should it run the pilot program, if ever? (Note: If you set up the input section of your spredsheet in the right way, you will be able to perform all of the Bayes' rule calculations with a couple copyable formulas.)

Answer 1

Implementation of Baye's Rule

Indicated savings (IS)
Prior		Conditional		Joint		Posterior
P(S)	0.3	P(IS|S)	0.60	P(S) x P(IS|S)	0.18	P(S|IS)	0.486
P(L)	0.3	P(IS|L)	0.10	P(L) x P(IS|L)	0.03	P(L|IS)	0.081
P(B)	0.4	P(IS|B)	0.40	P(B) x P(IS|B)	0.16	P(B|IS)	0.432
Total	1			P(IS) =	0.37		1
Indicated loss (IL)
Prior		Conditional		Joint		Posterior
P(S)	0.3	P(IL|S)	0.10	P(S) x P(IL|S)	0.03	P(S|IL)	0.130
P(L)	0.3	P(IL|L)	0.40	P(L) x P(IL|L)	0.12	P(L|IL)	0.522
P(B)	0.4	P(IL|B)	0.20	P(B) x P(IL|B)	0.08	P(B|IL)	0.348
Total	1			P(IL) =	0.23		1
Indicated breakeven (IB)
Prior		Conditional		Joint		Posterior
P(S)	0.3	P(IB|S)	0.30	P(S) x P(IB|H)	0.09	P(S|IB)	0.225
P(L)	0.3	P(IB|L)	0.50	P(L) x P(IB|M)	0.15	P(L|IB)	0.375
P(B)	0.4	P(IB|B)	0.40	P(B) x P(IB|L)	0.16	P(B|IB)	0.400
Total	1			P(IB) =	0.40		1

Decision Tree

As per the optimality, the pilot is worthless at a cost of $75,000 as the expected value without the pilot is more.

The maximum cost of the pilot to make it effective will be = $208,500 + $75,000 - $247,500 = $36,000