Decision Trees & Expected Value - Business Analytics

1. What is the maximum amount the company should pay for the research report under the expected-value criterion?
   Answer: Pay no more than $57.75$ for the research report. When a question asks for the maximum willingness to pay for information, you're being tested on the concept of the Expected Value of Sample Information (EVSI). The formula is: $EVSI = EV\text{(with sample info)} - EV\text{(without sample info)}$. Start by finding the baseline decision (no research). Launching has EV = $0.35(500) + 0.65(-100) = 175 - 65 = 110$. Licensing yields a certain $90$. So without research, you launch and earn $110$. Now apply Bayes' theorem to update probabilities after the report. The probability of a positive report: $P(+) = 0.35(0.80) + 0.65(0.30) = 0.28 + 0.195 = 0.475$. Given a positive report: $P(\text{strong}|+) = 0.28/0.475 \approx 0.589$, so $P(\text{weak}|+) \approx 0.411$. Given a negative report: $P(\text{strong}|-) = 0.07/0.525 \approx 0.133$, $P(\text{weak}|-) \approx 0.867$. After a positive report, launch EV = $0.589(500) + 0.411(-100) = 294.5 - 41.1 = 253.4$. Choose launch over license ($90$). After a negative report, launch EV = $0.133(500) + 0.867(-100) = 66.5 - 86.7 = -20.2$. Choose license ($90$). EV with research = $0.475(253.4) + 0.525(90) = 120.4 + 47.25 = 167.75$. Therefore, $EVSI = 167.75 - 110 = \mathbf{57.75}$, confirming answer A. Answer B ($77.75$) likely results from arithmetic errors in Bayesian updating. Answer C ($110$) confuses the baseline EV with the EVSI itself. Answer D ($167.75$) is the EV with research — forgetting to subtract the baseline. Always remember: EVSI is the difference between informed and uninformed decisions, not the total value of the informed strategy.
2. Using relevant cash flows and expected monetary value, what should the company do?
   Answer: Continue; its incremental expected value is $92$, which exceeds stopping by $67$. When a decision involves past spending that can't be recovered, those costs are sunk and must be excluded from your analysis. The only cash flows that matter are the ones that differ between your choices going forward — this is the principle of relevant (incremental) cash flows combined with expected monetary value (EMV). Here's how to apply it correctly. If you stop, you receive the salvage proceeds of $25$. If you continue, you spend $80$ more and then receive either $260$ (probability $0.60$) or $40$ (probability $0.40$). The EMV of receipts upon completion is $(0.60 \times 260) + (0.40 \times 40) = 156 + 16 = 172$. Subtract the additional $80$ cost: $172 - 80 = 92$. Since $92 > 25$, continuation wins by $92 - 25 = 67$. That's exactly what answer A describes — making it correct. Answer B is wrong because it adds back the $120$ sunk cost into the analysis. That money is gone regardless of what you decide, so including it distorts the comparison. Answer C makes a subtler error: it correctly calculates expected receipts of $172$ but forgets to subtract the additional $80$ development cost, overstating continuation's value. Answer D reaches the wrong conclusion by letting the low-payoff branch ($40$ thousand) emotionally override the math — risk aversion reasoning, not EMV. Study tip: Whenever you see prior expenditures in a decision problem, immediately flag them as sunk costs and exclude them. Only ask: "What do I gain or lose from this point forward with each option?"
3. Which statement correctly describes the decision under the two rules?
   Answer: Without the policy choose X at $114$; with the policy choose Y at $101$. When a question gives you two decision rules, your job is to apply each one separately and compare results — don't let one rule contaminate the other. Start with Expected Monetary Value (EMV), the default rule. For Investment X: $EMV_X = (0.55)(240) + (0.45)(-40) = 132 - 18 = 114$. For Investment Y: $EMV_Y = (0.90)(110) + (0.10)(20) = 99 + 2 = 101$. Without any policy, you maximize EMV, so you choose X at $114$. That locks in the first half of answer D. Now apply the policy constraint: reject any investment with more than a 20% probability of a negative payoff. Investment X has a 45% chance of losing $40$ — that exceeds the 20% threshold, so X is eliminated. Investment Y's only negative-payoff scenario has probability 0% (both outcomes, $110$ and $20$, are positive), so Y clears the constraint easily. With the policy in place, you must choose Y at $101$. That confirms the second half of D. A is wrong because "greater probability of a gain" is not the EMV rule — you must weight payoffs by probabilities, not just count favorable outcomes. B reverses the two investments, swapping which rule leads to which choice — a classic careless-reading trap. C is wrong because raw payoff size alone doesn't determine EMV; you must account for probabilities, and X's 45% loss probability matters. Strategy tip: On two-rule problems, always solve each rule independently before comparing. Calculate EMV first, then check constraints — never blend the two criteria together.
4. What is the expected value of perfect information about demand?
   Answer: The expected value of perfect information is $92$. When you see a question about the Expected Value of Perfect Information (EVPI), remember this formula: $EVPI = EVPI_{with} - EMV_{best}$, where $EV_{with}$ is the expected value with perfect information (you always pick the best option per state), and $EMV_{best}$ is the best expected monetary value without that information. Start by computing each alternative's EMV. For the large facility: $0.30(500) + 0.40(180) + 0.30(-250) = 150 + 72 - 75 = 147$. For the small facility: $0.30(260) + 0.40(160) + 0.30(60) = 78 + 64 + 18 = 160$. For outsourcing: $0.30(180) + 0.40(140) + 0.30(100) = 54 + 56 + 30 = 140$. The best EMV without perfect information is $160$ (small facility). Now compute $EV_{with}$: under perfect information, you'd choose the highest payoff in each demand state. High demand → Large ($500$); Medium demand → Large ($180$); Low demand → Outsourcing ($100$). So $EV_{with} = 0.30(500) + 0.40(180) + 0.30(100) = 150 + 72 + 30 = 252$. Therefore, $EVPI = 252 - 160 = 92$, confirming answer A is correct. Answer D ($252$) is the classic trap — it's $EV_{with}$ alone, forgetting to subtract the best EMV. Answer B ($105$) and C ($112$) likely result from miscalculating the best EMV, perhaps using the large facility's EMV of $147$ or incorrectly identifying the best option per state. Study tip: Always write EVPI as a two-step process — calculate $EV_{with}$ and best EMV separately before subtracting. The most common mistake is reporting $EV_{with}$ as the final answer.
5. What is the optimal contingent strategy and its expected monetary value at the initial decision?
   Answer: Start, complete only after a favorable signal, and obtain expected value of $13.2$. When you see a sequential decision problem like this, think in terms of a decision tree: work backwards from the final outcomes, choosing at each node the option with the highest expected value. Start by evaluating what happens after each signal, since those decisions come last. After a favorable signal, completing costs $100$ thousand and yields $0.70(300) + 0.30(60) = 228$ thousand, for a net of $228 - 100 = 128$ thousand. Abandonment yields only $20$ thousand. So you complete after a favorable signal. After an unfavorable signal, completing yields $0.20(300) + 0.80(60) = 108$ thousand, net $108 - 100 = 8$ thousand. Abandonment yields $20$ thousand. So you abandon after an unfavorable signal. Now roll back to the initial decision. Starting costs $50$ thousand, and the expected payoff equals $0.40(128) + 0.60(20) = 51.2 + 12 = 63.2$ thousand. Subtracting the initial cost: $63.2 - 50 = 13.2$ thousand. This confirms answer B as correct. Answer A is wrong because completing after an unfavorable signal (net $8$) is dominated by abandoning ($20$) — you'd never do that optimally. Answer C is wrong because starting has a positive expected value of $13.2$, so not starting forfeits real value. Answer D is wrong because abandoning after a favorable signal sacrifices a net payoff of $128$ for only $20$. Study tip: Always solve decision trees from right to left — evaluate terminal nodes first, then prune dominated branches before rolling back. Never optimize the full tree all at once from the start.
6. After receiving an alert, which decision minimizes expected loss?
   Answer: Reject the batch, producing an expected loss of $45$. When a question combines probability updating with decision analysis, your first job is to apply Bayes' theorem to find the posterior probability that the batch is defective given the alert — then compare expected losses across decisions. Start by computing the probability of receiving an alert: $P(\text{alert}) = (0.90)(0.20) + (0.25)(0.80) = 0.18 + 0.20 = 0.38$. Now apply Bayes' theorem: $P(\text{defective} \mid \text{alert}) = \frac{(0.90)(0.20)}{0.38} = \frac{0.18}{0.38} \approx 0.4737$, and $P(\text{good} \mid \text{alert}) \approx 0.5263$. With those posteriors in hand, compare the two decisions. Rejecting the batch produces a certain loss of $45$ thousand regardless of batch quality — expected loss = $45$. Accepting the batch produces a loss of $180$ if defective (probability ≈ 0.4737) and $0$ if good — expected loss = $(0.4737)(180) \approx 85.3$. Rejecting is clearly better, confirming D is correct. Choice A gets the right expected loss for accepting (≈85.3) but wrongly attributes it to rejecting — a label-swap error. Choice B assigns an expected loss of 36 to accepting, which appears to mistakenly use the prior probability of 0.20 instead of the updated posterior: $(0.20)(180) = 36$. Choice C correctly calculates the expected loss of accepting (≈85.3) but picks the wrong decision — accepting instead of rejecting. The study tip: always update your probabilities with Bayes before computing expected values. Using prior probabilities after receiving new information is one of the most common traps in Bayesian decision problems.
7. Which campaign should the retailer select using expected monetary value?
   Answer: Select the personalized campaign because its expected value is $82$. When a decision involves uncertain outcomes, expected monetary value (EMV) lets you compare options by weighting each payoff by its probability. The key with the personalized campaign here is recognizing that conversion success depends on a two-stage probability: first demand level, then conversion rate. Start by finding the overall probability that conversion succeeds. Using the law of total probability: $P(\text{success}) = (0.40)(0.70) + (0.60)(0.20) = 0.28 + 0.12 = 0.40$. So conversion fails with probability $0.60$. Before subtracting the $30$ cost, the expected gross payoff is $(0.40)(220) + (0.60)(40) = 88 + 24 = 112$. Subtracting the campaign cost gives the EMV of the personalized campaign: $112 - 30 = 82$. Since $82 > 78$, the personalized campaign wins — confirming B is correct. A is wrong not because the standard campaign's value of $78$ is miscalculated, but because the personalized campaign's EMV of $82$ is higher, so the standard campaign should not be selected. C makes the mistake of reporting the gross expected payoff ($112$) before subtracting the $30$ campaign cost — a classic error of forgetting to account for upfront costs in EMV. D arrives at $118$, likely by computing $(0.40)(220) + (0.60)(40) - 30$ incorrectly, or misapplying probabilities altogether. When you see multi-stage probabilities, always collapse them into a single success/failure probability first, then compute EMV — and never forget to subtract any fixed costs before comparing alternatives.
8. Using expected net present value, which initial decision is optimal?
   Answer: Accept the certain payment because the permit's expected net present value is about $31.07$. When evaluating a decision under uncertainty using expected net present value (ENPV), you discount each possible cash flow back to today, weight it by its probability, and sum the results — then compare alternatives. For the permit option, you pay $40$ thousand today (a certain outflow). Two outcomes follow: with probability $0.50$, approval yields $150$ thousand in year two; with probability $0.50$, denial yields $20$ thousand at the end of year one. Discounting each at $r = 0.10$: - Approval path: $0.50 \times \frac{150}{(1.10)^2} = 0.50 \times 123.97 \approx 61.98$ - Denial path: $0.50 \times \frac{20}{1.10} = 0.50 \times 18.18 \approx 9.09$ Subtract the upfront cost: $61.98 + 9.09 - 40 \approx 31.07$ thousand. Since the certain payment of $50$ thousand (already in present value terms) exceeds $31.07$, accepting the certain payment is optimal — confirming C. Choice A is wrong because $71.07$ ignores the $40$ thousand upfront cost entirely, inflating the permit's ENPV by exactly that amount. Choice B ($45.00$) reflects a common error of skipping discounting altogether — simply computing $0.50(150) + 0.50(20) - 40 = 45$, which ignores the time value of money. Choice D ($37.27$) likely results from discounting incorrectly, such as applying the wrong exponent or discount rate. As a study tip: always check whether each cash flow is discounted to the same point in time (today) before summing. Mixing undiscounted and discounted values is the most frequent trap in ENPV problems.
9. At what probability does automation become optimal, and what should the company do using the current estimate?
   Answer: The threshold is $0.45$, so the company should outsource. When comparing a risky option against a certain payoff, the key move is finding the indifference probability — the value of $p$ where both options yield equal expected value. Above that threshold, the risky option wins; below it, the safe option wins. Set the expected value of automation equal to the outsourcing payoff: $360p + (-120)(1-p) = 96$ $360p - 120 + 120p = 96$ $480p = 216$ $p = 0.45$ So the threshold is $0.45$. Since the current estimate is $p = 0.42 < 0.45$, automation's expected value falls below $96$, making outsourcing the better choice. That confirms C is correct. Choice A claims the threshold is $0.40$, which would mean automation is already optimal at $p = 0.42$. This likely comes from an arithmetic error — perhaps forgetting to properly distribute the $(1-p)$ term when expanding. Choice B says the threshold equals the current estimate of $0.42$, confusing the given probability with the break-even probability — these are two separate values and conflating them is a classic trap. Choice D sets the threshold at $0.60$, which is far too high and suggests a setup error, possibly using the wrong payoff values or incorrect algebra. A useful habit: always solve for the indifference probability first, then compare it to the given probability second — treat them as two distinct steps. On exam questions like this, wrong answers are often engineered around these exact mix-ups, so keeping the two steps clearly separated will protect you from the most common mistakes.

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