Decision Making under Uncertainty and Simulation

7.1 Foundations of Decision Analysis: The Goferbroke Company Problem

Managers often decide in uncertain environments. This section establishes a shared language: alternatives, states of nature, prior probabilities and payoff tables.

Four Typical Decision Problems

Case: The Goferbroke Company

• A company is considering oil-well development in an unexplored area. Geological consultants report that the site has a probability of 1/4 of containing oil.
• Drilling and exploration would cost about USD 100,000; if oil is found, estimated net proceeds are about USD 800,000.
• Another oil company offers USD 90,000 to acquire the site.

The Payoff Table and Five Key Terms

Units: USD thousands.

• Decision maker the person or team responsible for the decision.
• Alternative the available choices (here, drill or sell).
• State of nature a condition determined by uncontrolled random factors (oil or dry).
• Prior probability an initial estimate of how likely each state is.
• Payoff the outcome from combining one alternative with one state, quantified in money.

7.2 Decision Criteria and Decision Trees

How the Four Criteria Work

No criterion is inherently “right”—each represents a different risk attitude. Management often uses EMV because it incorporates probabilities, but EMV assumes risk neutrality, an assumption challenged in Section 7.4.

Decision Trees

A decision tree shows the sequence of decisions and random events graphically and applies the Bayes criterion:Square nodes are decision nodes, with branches for alternatives;circular nodes are event nodes, with branches for possible states.

Expected Value of Perfect Information (EVPI)

Suppose the true state could be known before deciding: drill if oil (700), sell if dry (90), giving expected payoff 0.25×700 + 0.75×90 = 242.5. This exceeds the no-information EMV of 100 by 142.5. The difference is the expected value of perfect information—an upper bound on the value of any information.

7.3 Bayesian Updating: Seismic Testing and Posterior Probabilities

Prior probabilities are often subjective and approximate. A company can pay for tests or research to update them more accurately; the updated probabilities are posterior probabilities.

Case: Seismic Testing at Goferbroke

The company can pay USD 30,000 for detailed seismic testing to improve its estimate of the probability of oil. The test has two results:FSS(favourable, indicating a higher chance of oil) and USS(unfavourable, indicating a low chance).

7.4 Utility and Risk: Why People Reject Higher-Expected-Value Gambles

So far we have assumed that expected monetary value is the correct decision criterion. This assumption fails in many settings. Choose between:

Most people choose B even though A has a higher expected value. The reason is risk aversion a utility function translates monetary amounts into a scale that reflects subjective preference.

• If two alternatives have equal expected utility the decision maker is indifferent between them.
• When decisions use utility, the Bayes criterion substitutes utility for monetary payoffs.
• The decision that maximises expected utility is optimal.

The Equivalent-Lottery Method

1. Set U(maximum payoff) = 1 and U(minimum payoff) = 0.
2. To determine the utility of an intermediate payoff M, offer two alternatives: A₁ gives the maximum with probability p and the minimum with probability 1−p; A₂ gives M for certain.
3. Ask: for what p are you indifferent? Then U(M) = p.

For the Goferbroke problem, Max’s indifference probabilities correspond to the following table (USD thousands):

Notice the curve: utility rises quickly from −130 to 90 and barely changes from 670 to 700.This is what risk aversion looks like.

7.5 Monte Carlo Simulation: Turning Uncertainty into Distributions

Decision trees handle only a few discrete states. Real uncertainty is often continuous—sales, price and cost are all examples. Use simulation: represent every uncertain quantity with a distribution and sample thousands of times to see the distribution of outcomes.

Case: Annual Profit from a New Product Launch

Core Simulation Ideas: Why Repeated Sampling Produces an Answer

Monte Carlo simulation does not solve a closed-form equation. It assigns a probability distribution to each uncertainty, repeatedly samples a set of values and calculates one outcome, and approximates the outcome distribution through thousands of repetitions. It is named after the casinos of Monaco because it relies on random sampling.

It works because of the law of large numbers: with enough repetitions, the sample average converges toward the true average. This is the same idea from earlier units—one draw is random, but the average of many draws is stable.

1. Define the distributions: each uncertain quantity needs a distribution and parameters supported by historical data, industry experience or expert judgement.
2. Draw one sample: sample one value for each uncertainty, substitute them into the model and calculate one outcome—a single “trial.”
3. Repeat thousands of times: repeat step 2 N times to obtain N outcomes.
4. Summarise the distribution: characterise the N outcomes with means, standard deviations, percentiles and probabilities.

How to read the results. A single average is insufficient; examine at least four things together:

• Mean: the expected profit—but it hides volatility.
• Standard deviation: the size of fluctuations. If it exceeds the mean, outcomes are highly unstable.
• Loss probability(or probability of missing any target): the share of outcomes below zero, directly corresponding to the managerial question “could we lose money?”
• Percentiles: worst 5% and best 5% outcomes, describing tail risk and upside.

Inspect the histogram shape as well: two peaks often indicate two distinct situations (for example, “demand takes off” and “demand fails”), making a single average less meaningful.

How many runs are needed? Around 1,000 runs are usually enough to reveal the distribution’s general shape; use several thousand when tail conclusions such as the worst 5% matter. More runs stabilise results, but improvement diminishes beyond a point.

Case: Annual Profit from a New Product Launch

How to Do It in Excel

7.6 The Value of AI Information: Using EVSI to Decide Whether to Buy

Treat an AI forecast as additional information: it can improve accuracy, but is usually imperfect and costly. EVSI determines the maximum value of that information.

1. Estimate AI accuracy and error patterns: how often is it correct, and how does it fail (false dry or false oil)?
2. Use EVSI to calculate the upper bound on information value: EVSI = expected payoff with information − expected payoff without information.
3. Compare with cost: if cost exceeds EVSI, the information is not worth relying on; if cost is below EVSI, it is worth buying.

This is one of the unit’s most practical management conclusions:calculate the value of information before deciding whether to buy it. For the seismic test above, EVSI is 53 (USD thousands), cost is 30 and net value is 23—worth doing.

7.7 Summary and Assignment

• Three tools for decisions under uncertainty:decision criteria, decision trees and expected value.
• New information can update probabilities through Bayes’ theorem, but it has a price: compare EVSI with cost.
• Monte Carlo simulation turns distributions into observable outcomes.
• Means can mislead: loss probability and worst-case outcomes matter too.
• Utility functions explain why people reject gambles with higher expected value.

Appendix: Interactive Demonstrations and Classroom Exercises

Data source: payoff table, test conditional probabilities and simulation parameters from 单元7_决策与仿真练习.xlsx; every expected value, posterior, EVSI and simulation result is calculated on the page.

Unit Glossary (English–Chinese)