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Decision Analysis

Decision making under uncertainty from a payoff table of alternatives against states of nature: the classical criteria (maximax, maximin, minimax regret, Hurwicz, Laplace, maximum likelihood), the Bayes decision rule on the prior probabilities and the expected value of perfect information; with a likelihood table for a test, survey or sample, Bayes' theorem gives the posterior probabilities, the best decision after each outcome, the expected value of sample information, its efficiency and whether the test is worth its cost; the decision tree is drawn and rolled back, the best decision is traced against one prior probability with the crossover points, the risk profiles of the leading alternatives are plotted, and a risk tolerance turns the payoffs into exponential utilities with certainty equivalents and risk premiums. Written for strategy, legal, medical, policy and project-risk work.

  • Criteria table: maximax, maximin, Laplace, Hurwicz, maximum likelihood, EMV and minimax regret, with the choice under each

  • Bayes decision rule, EV with perfect information and EVPI

  • Test information: joint, marginal and posterior probabilities by Bayes' theorem, EVSI, efficiency, net value against the test cost

  • Rolled-back decision tree, sensitivity of the best decision to a prior, risk profiles, exponential utility and certainty equivalents

Decision Analysis

Unit System: /

Payoff Table


Test, Survey or Sample (optional)


Criteria and Risk Attitude




Criteria, Bayes' Rule, Value of Information, Decision Trees and Utility | Method page
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Decision Analysis

Decision Analysis: Criteria, Bayes' Rule, Value of Information, Decision Trees and Utility

States, alternatives and payoffs

A decision under uncertainty has a set of alternatives (the courses of action open to the decision maker), a set of states of nature (the possible conditions, exactly one of which will turn out to be true, outside the decision maker's control) and a payoff for every pair: the consequence, in one unit, of taking that alternative when that state obtains. The payoff table lists the alternatives as rows and the states as columns. Payoffs may be profits, to be maximized, or costs, to be minimized; the tool negates a cost table, runs the same arithmetic and reports costs back with their sign, so every statement below is written for profits and holds for costs with best and worst exchanged. When the decision maker can attach a probability to each state, the prior probabilities complete the problem; they sum to one. The default case is the Goferbroke Company example of Hillier and Lieberman: drill for oil on a tract (700 if oil, −100 if dry, in thousands of dollars) or sell the land (90 either way), with a prior probability of oil of 0.25.

The classical criteria

Without probabilities, or when the decision maker distrusts them, several rules have been proposed, each defensible in its own circumstances and none right in general.

  • Maximax takes the alternative whose best payoff is largest. It is the optimist's rule; it ignores every outcome but the most favourable, and is defensible when the downside is survivable and the opportunity cannot be repeated.
  • Maximin (Wald) takes the alternative whose worst payoff is largest: it guarantees the best floor. It is the pessimist's rule and the right one when the worst case is ruinous or an adversary chooses the state.
  • Minimax regret (Savage 1951) forms the regret, or opportunity loss, of each cell as the best payoff in that state less the cell, and takes the alternative whose largest regret is smallest. It protects against hindsight: no state can make the decision look worse than that regret. It can change its choice when an irrelevant alternative is added, which is its standing criticism.
  • Hurwicz weights each alternative's best payoff by a coefficient of optimism α and its worst by 1 − α; α = 1 is maximax, α = 0 maximin. It is a compromise whose α has no objective basis.
  • Laplace (equally likely, the principle of insufficient reason) averages the payoffs over the states, which is the expected value with equal priors. It is defensible only when nothing distinguishes the states.
  • Maximum likelihood looks only at the most probable state and takes the best payoff there. It discards the other states entirely, however large their payoffs, and is defensible only when one state is far more probable than the rest.

The Bayes decision rule

With prior probabilities the expected monetary value of an alternative is the probability-weighted sum of its payoffs, EMV = Σ P(state) × payoff(alternative, state), and the Bayes decision rule takes the alternative with the largest EMV (smallest expected cost). It is the rule a risk-neutral decision maker should follow for a decision that will be repeated, or that is small against the decision maker's wealth, because over many such decisions the average result tends to the EMV. For a one-off decision with large stakes the risk attitude matters, which is the subject of the utility section.

Expected value of perfect information

Suppose a perfect forecast could tell which state will obtain before the decision is taken. The decision maker would then choose the best alternative in each state, and the expected payoff with perfect information is EVPI = Σ P(state) × max over the alternatives of payoff(alternative, state). The expected value of perfect information is EVPI = EVPI − best EMV. No forecast, test, survey or study can be worth more than EVPI, so it is the first number to compute before buying information: when EVPI is small, nothing is. In the Goferbroke example EVPI = 0.25 × 700 + 0.75 × 90 = 242.5 and EVPI = 242.5 − 100 = 142.5.

Bayes' theorem on the likelihood table

A real test, survey or sample is imperfect. Its reliability is stated as a likelihood table, P(outcome | state): how likely each outcome of the test is in each state (every row sums to one). The joint probability of a state and an outcome is P(state) P(outcome | state); the probability of an outcome is the sum of its column over the states; and Bayes' theorem gives the posterior probability of each state once an outcome has been seen,

P(state | outcome) = P(state) P(outcome | state) / P(outcome).

For the Goferbroke seismic survey, P(favorable sounding | oil) = 0.6 and P(favorable | dry) = 0.2 give P(favorable) = 0.25 × 0.6 + 0.75 × 0.2 = 0.3 and P(oil | favorable) = 0.15/0.3 = 0.5, against P(oil | unfavorable) = 0.1/0.7 = 1/7.

Expected value of sample information and efficiency

After each outcome the decision maker applies the Bayes rule on the posterior probabilities and gets the best EMV for that outcome. Weighting those by the outcome probabilities gives the expected payoff with the test, EVSI, before the test's cost. The expected value of sample (or experiment) information is EVSI = EVSI − best EMV without the test: the most the test is worth. The test should be run when EVSI exceeds its cost, and the efficiency EVSI/EVPI, between 0 and 1, says how close the test comes to a perfect forecast. In the example EVSI = 0.3 × 300 + 0.7 × 90 = 153, EVSI = 53, the efficiency is 53/142.5 = 37 %, and with a survey cost of 30 the net value 23 says to run it.

Decision trees and rollback

A decision tree lays the problem out in time: a square node is a decision, its branches the alternatives; a circle is a chance node, its branches the states or test outcomes with their probabilities; the leaves carry the payoffs, with any cost incurred on the way (the test cost) deducted. The tree is solved backwards (rollback, or folding back): each chance node takes the probability-weighted value of its branches, each decision node takes the best branch and strikes out the rest, and the value at the root is the expected payoff of the optimal strategy. With a test, the root decision is whether to test; the test branch leads to a chance node on the outcomes, each followed by a decision on the posterior probabilities. The rollback table on the results page lists every node with its expected value, and the figure draws the tree when it has 60 leaves or fewer.

Sensitivity analysis

The prior probabilities are usually the least certain numbers in the table. The tool varies the prior of one chosen state from 0 to 1, scaling the other priors in proportion so that they still sum to one, and plots the EMV of every alternative, which is a straight line in that probability. The best decision changes where the top lines cross, and the crossover probabilities are reported: when the entered prior is far from the nearest crossover the decision is robust, when it is close the prior deserves more study (or a test). For the example, 700 p − 100 (1 − p) = 90 gives the crossover at p = 190/800 = 0.2375, close to the prior of 0.25, which is why the survey is worth buying.

Risk profiles

The risk profile of an alternative is the cumulative distribution of its payoff: the probability that the payoff is at or below each value. Two alternatives with similar EMVs may have very different profiles, and a profile lying to the right of another everywhere dominates it stochastically, whatever the risk attitude. The profile of the test strategy includes the test outcomes and the decisions taken after them.

Utility theory and the exponential utility function

A decision maker who would not stake the company on a fair coin is not maximizing expected money. Von Neumann and Morgenstern showed that consistent preferences over gambles are represented by a utility function u(x) such that gambles are ranked by expected utility. The exponential form u(x) = 1 − exp(−x/R) is the standard choice in practice (Howard 1988, Clemen and Reilly 2014): it has constant risk aversion, so the ranking of gambles does not depend on the decision maker's current wealth, and it has one parameter, the risk tolerance R, in the payoff unit. A small R means strong risk aversion; as R grows the ranking tends to the EMV ranking, and a blank R in the tool is the risk-neutral case.

Estimating R. The usual rule of thumb: R is about the largest stake x for which the decision maker would accept a gamble that wins x or loses x/2 with equal probability (the exponential form gives R ≈ 1.04 x for indifference). Howard (1988) found for the firms he studied that R was about 6 % of net sales, 100 to 150 % of net income, or one sixth of equity. Decision makers are often more risk averse than their position warrants, which is a reason to show the EMV and the utility result side by side.

Certainty equivalents and risk premiums

The certainty equivalent of a gamble is the sure amount the decision maker values equally: with EU its expected utility, CE = −R ln(1 − EU) for the exponential function. The risk premium EMV − CE is what the decision maker gives up to be rid of the gamble; it is positive for a risk-averse decision maker and grows with the spread of the payoffs and with 1/R. Alternatives are ranked by CE, which is the same as by expected utility but in money. For the example at R = 1,000 (thousands of dollars) the drilling gamble has EU = 0.25 (1 − e−0.7) + 0.75 (1 − e0.1) = 0.0470 and CE = 48.1 against the sure 90 of selling: the risk-averse decision maker sells although drilling has the higher EMV, a risk premium of 51.9. The tool also values the test strategy by expected utility, with the decisions after each outcome taken by expected utility, so that the worth of the test to this decision maker is a certainty equivalent as well.

Limits

The tool handles one decision, one set of states and at most one test taken before it (a single-stage experiment); sequential decisions, several tests or a test whose outcomes depend on an earlier decision need a hand-built tree. The states are assumed mutually exclusive and exhaustive, with the test's likelihoods depending on the state alone; correlated or continuous states are outside the scope. Payoffs are in one unit with no time value of money, and the utility function is the exponential form only. Posterior probabilities are computed exactly, so a textbook that rounds them (0.78 for 0.7778) will print slightly different expected values downstream.



References

Raiffa, H. (1968). Decision Analysis: Introductory Lectures on Choices under Uncertainty. Addison-Wesley, Reading, MA.

Hillier, F. S., and Lieberman, G. J. (2015). Introduction to Operations Research, 10th ed. McGraw-Hill, New York.

Clemen, R. T., and Reilly, T. (2014). Making Hard Decisions with DecisionTools, 3rd ed. South-Western, Mason, OH.

Howard, R. A. (1988). "Decision Analysis: Practice and Promise." Management Science, Vol. 34, No. 6, pp. 679–695.

Savage, L. J. (1951). "The Theory of Statistical Decision." Journal of the American Statistical Association, Vol. 46, No. 253, pp. 55–67.

Render, B., Heizer, J., and Munson, C. (2017). Operations Management: Sustainability and Supply Chain Management, 12th ed. Pearson, Boston.

Worked Examples

Published examples this tool reproduces; each page lists the inputs to enter and the printed answers.

All worked examples

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