Question 6 of 10: Bayesian Decision Analysis — Credit-Extension Decision
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
National Exams — December 2018 — 17-Ind-A1 Operations Research. Three-hour, open-book exam (any non-communicating calculator permitted); the paper totals 170 marks across 10 questions and only 100 marks are required, so a candidate would normally answer a subset — all ten are solved below for completeness.
Reference texts: Hillier & Lieberman, Introduction to Operations Research (11th ed., McGraw-Hill) — linear programming, the simplex method & sensitivity analysis/duality (ch. 3–4/6), integer programming & branch and bound (ch. 12), queueing theory (ch. 17), decision analysis (ch. 15), computer simulation (ch. 20). Nahmias, Production and Operations Analysis — single-period (newsvendor) and multi-period (dynamic lot-sizing / Wagner–Whitin) inventory models.
Given. Prior: $P(\text{Poor})=0.2,\ P(\text{Avg})=0.5,\ P(\text{Good})=0.3$. Payoffs if credit is extended: $-\$15{,}000$ (Poor), $+\$10{,}000$ (Avg), $+\$20{,}000$ (Good); not extending always nets $0. Service cost $5,000. Track record (rows = rating given, columns = true category):
Given data — credit-rating service accuracy, P(rating | actual)
Rating given
Actual: Poor
Actual: Average
Actual: Good
Poor
50%
40%
20%
Average
40%
50%
40%
Good
10%
10%
40%
Find. (a) EMV-optimal action without the service. (b) EMV-optimal action given a "poor" rating, and its expected profit. (c) Whether the $5,000 service is worth buying (compare its expected value of sample information, EVSI, to its cost).
Approach. (a)–(b) are direct expected-monetary-value (EMV) comparisons, (b) using Bayes' theorem to revise the prior on the reported rating. (c) requires the full pre-posterior analysis: the expected value WITH the sample information (averaging the best EMV action under every possible rating, weighted by that rating's marginal probability) compared against the value without it, net of the $5,000 fee.
(a) EMV without the service.
$$EMV(\text{extend})=0.2(-15{,}000)+0.5(10{,}000)+0.3(20{,}000)=-3{,}000+5{,}000+6{,}000=\boxed{\$8{,}000}$$
Since $EMV(\text{extend})=\$8{,}000>EMV(\text{not extend})=\$0$: extend credit; expected profit $8,000.
(b) Bayes' revision on a "poor" rating. Marginal probability of a poor rating (law of total probability down the "Poor" row):
$$P(\text{rating=Poor})=0.5(0.2)+0.4(0.5)+0.2(0.3)=0.10+0.20+0.06=0.36$$
Posterior via Bayes' theorem, $P(\text{actual}\mid\text{Poor rating})=\dfrac{P(\text{Poor rating}\mid\text{actual})\,P(\text{actual})}{0.36}$:
Posterior distribution given a "poor" rating
Actual
Prior
Posterior | rated Poor
Poor
0.20
0.10/0.36 = 0.2778
Average
0.50
0.20/0.36 = 0.5556
Good
0.30
0.06/0.36 = 0.1667
(b) Re-compute EMV(extend) under this posterior:
$$EMV(\text{extend}\mid\text{Poor rating})=0.2778(-15{,}000)+0.5556(10{,}000)+0.1667(20{,}000)=\boxed{\$4{,}722.22}$$
Still $>\$0$, so still extend credit even after a poor rating — the report shifts the odds but not enough to flip the decision; expected profit $4,722.22.
(c) Repeat the posterior EMV comparison for the other two possible ratings. By the same Bayes' method (marginal probabilities $P(\text{Avg rating})=0.4(0.2)+0.5(0.5)+0.4(0.3)=0.45$ and $P(\text{Good rating})=0.1(0.2)+0.1(0.5)+0.4(0.3)=0.19$, which correctly sum with 0.36 to 1.00):
$$EMV(\text{extend}\mid\text{Avg rating})=\$8{,}222.22,\qquad EMV(\text{extend}\mid\text{Good rating})=\$13{,}684.21$$
Every posterior EMV is positive — the optimal action is "extend credit" regardless of what the service reports.
(c) Compute the expected value of sample information (EVSI) by weighting each rating's best EMV by its marginal probability, and comparing to the no-information EMV:
$$EV(\text{with report, gross})=0.36(4{,}722.22)+0.45(8{,}222.22)+0.19(13{,}684.21)=1{,}700+3{,}700+2{,}600=\$8{,}000$$
$$EVSI=EV(\text{with report})-EV(\text{no info})=8{,}000-8{,}000=\boxed{\$0}$$
$$\text{Net value of the service}=EVSI-\text{fee}=0-5{,}000=\boxed{-\$5{,}000\ \Rightarrow\ \text{do NOT buy the service}}$$
