Question 3 of 8: Decision Analysis with Sample Information — Pollution Patrol

Question 3: Decision Analysis with Sample Information — Pollution Patrol (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. Priors $P(L)=0.5,\ P(M)=0.4,\ P(H)=0.1$; patrol accuracy: $P(\text{report}=t\mid\text{true}=t)=0.8$, $P(\text{report}=r\mid\text{true}=t)=0.1$ for $r\ne t$ (symmetric confusion matrix); patrol fee $0.49M (sunk once investigation occurs); costs by true state, no-citation vs. citation ($M): (L: 0 vs 3), (M: 8 vs 5), (H: 25 vs 10).

Find. (a) The citation decision for each possible patrol report. (b) The citation decision with no investigation (prior only). (c) Whether maintaining the patrol system is worthwhile.

Approach. Use Bayes' rule to convert each patrol report into a posterior over the true pollution level, compute the expected cost of citing vs. not citing under that posterior, then compare the overall expected cost of investigating (weighted over reports, plus the patrol fee) against deciding on the prior alone.

1. Part (a) — posterior probabilities by Bayes' rule for each possible report $r$, $P(t\mid r)=\dfrac{P(r\mid t)P(t)}{\sum_{t'}P(r\mid t')P(t')}$:

   Posterior $P(\text{true}\mid\text{report})$ and expected cost ($M) of each action

   Report	$P(r)$	$P(L\mid r)$	$P(M\mid r)$	$P(H\mid r)$	E[no cite]	E[cite]	Decision
   Low	0.450	0.889	0.089	0.022	1.267	3.333	No citation
   Medium	0.380	0.132	0.842	0.026	7.395	4.868	Cite
   High	0.170	0.294	0.235	0.471	13.647	6.765	Cite

   $$\boxed{\text{Issue a citation on a Medium or a High report; do not cite on a Low report.}}$$
2. Part (b) — decision using the prior alone (no investigation), $E[\text{cost}]=\sum_t P(t)\cdot\text{cost}(t)$: $$E[\text{no cite}]=0.5(0)+0.4(8)+0.1(25)=3.2+2.5=\$5.70\text{M}$$ $$E[\text{cite}]=0.5(3)+0.4(5)+0.1(10)=1.5+2.0+1.0=\boxed{\$4.50\text{M}}$$ Since $4.50M < $5.70M, yes — without any investigation EC should issue citations everywhere (i.e. adopt a blanket citation policy), even though a per-region investigation would sometimes reveal Low is more likely.
3. Part (c) — expected cost of investigating (weighted over the three possible reports, using the Part-(a) optimal action for each, plus the $0.49M patrol fee): $$E[\text{cost}\mid\text{investigate}]=0.45(1.267)+0.38(4.868)+0.17(6.765)=0.570+1.850+1.150=\$3.570\text{M}$$ $$\boxed{E[\text{cost}\mid\text{investigate, incl. fee}]=3.570+0.490=\$4.060\text{M}}$$ Compare against the Part-(b) best no-investigation cost of $4.50M: $4.06M < $4.50M, a saving of $0.44M, so yes — EC should maintain the patrol system; the information it provides (letting EC skip citations on Low reports) is worth more than its $0.49M fee.