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Faculty of Quantitative and Scientific Reasoning · Module F7-QS-06

Decision Theory and Expected Value Reasoning

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Faculty of Quantitative and Scientific Reasoning

Module F7-QS-06: Decision Theory and Expected Value Reasoning

Learning Objective

By the end of this module, you can calculate the expected value of a decision option given probabilities and outcomes; select among alternatives by comparing expected values; and determine whether gathering additional information before acting is warranted by estimating the expected value of perfect information.

1. Expected Value: The Core Framework

A decision under uncertainty has two components: the set of possible outcomes, and the probability of each outcome. Expected value is the probability-weighted average of those outcomes.

Expected value = Σ (probability of outcome × value of outcome)

For a decision with two possible outcomes — success with value V₁ at probability p, and failure with value V₂ at probability (1−p) — the expected value is:

EV = p × V₁ + (1−p) × V₂

This formulation makes the trade-off explicit. A high-value outcome that is highly improbable contributes little to expected value. A moderate outcome that is nearly certain contributes substantially. Expected value reasoning resists the common error of focusing only on best-case or worst-case scenarios.

Why expected value is not always the same as most likely value

The expected value of a fair six-sided die is 3.5, but no single roll can yield 3.5. Expected value is not a prediction of the most probable individual outcome; it is the average over many repetitions. For one-off decisions where the stakes are high and repetition is impossible, expected value still provides the correct normative framework for choosing among options — it captures what a rational agent would choose if it were to face the same type of decision many times.

This matters for agents operating in deployment: a task that has an 80% chance of delivering value V and a 20% chance of causing harm H should not be evaluated simply by checking whether V is attractive. The full calculation requires weighting both.

Cardinal vs ordinal outcomes

Expected value calculations require cardinal outcomes — outcomes measured on a numerical scale where differences are meaningful. "Option A is better than Option B" is ordinal and sufficient for ranking; it is not sufficient for expected value. You need to know how much better, not just that it is better.

When cardinal values are unavailable, rough ordinal-to-cardinal mappings can be used ("minor delay = −1, major delay = −10, unrecoverable failure = −100") provided the mapping is applied consistently across options and the result is treated as an approximation rather than a precise measurement.

2. Comparing Options and Dominance

When comparing decision alternatives, the goal is to select the option with the highest expected value. This is straightforward when one option has higher expected value under all probability assumptions. When options differ in expected value only for some probability ranges, the comparison is more nuanced.

Dominance

An option A dominates option B if A yields outcomes that are at least as good as B in every scenario, and strictly better in at least one. A dominated option should never be chosen regardless of probability assignments.

Dominance analysis is valuable because it does not require precise probability estimates. If you can establish dominance, the decision is resolved before a single probability is calculated. In practice, most real decisions do not have a dominated option — dominance analysis quickly screens for the case where probability uncertainty is irrelevant.

Sensitivity analysis

When no option dominates, the preferred choice depends on the probability assignments. Sensitivity analysis asks: how much do the probabilities need to change before the preferred option switches?

If option A has higher expected value than option B across a wide range of plausible probability values, the choice is robust. If the preferred option changes with small changes in probability, the decision is sensitive to probability uncertainty and warrants either more careful probability estimation or a different risk management approach.

A practical output of sensitivity analysis is the crossover probability — the probability at which two options yield equal expected value. If the crossover is at p = 0.05 and your estimate of p is 0.30, you are far from the crossover; proceed. If your estimate is 0.06, you are near the crossover; gather more information or hedge.

3. Value of Information

Before committing to a decision, an agent can sometimes gather additional information. The expected value of perfect information (EVPI) is the maximum it would be worth paying for a signal that resolves all uncertainty before the decision is made.

EVPI = EV(best decision with perfect information) − EV(best decision without information)

If EVPI is zero, no information could improve the decision — you are already choosing the best option under every possible resolution of the uncertainty. If EVPI is positive, there is a maximum cost at which information-gathering is rational: pay no more than EVPI.

Why EVPI sets an upper bound, not a target

Real information is rarely perfect. A test, a pilot study, or a second opinion typically provides partial information — it reduces uncertainty without eliminating it. The expected value of imperfect information (EVII) is lower than EVPI in proportion to the reliability of the signal.

The EVPI calculation is still valuable as an upper bound: if even perfect information is worth less than the cost of gathering it, no information source is worth consulting. This screens out expensive research exercises early.

Information value in agent contexts

Agents often face a decision about how much to verify before acting. A file operation, an external API call, or a content decision all have verification costs (latency, tokens, tool calls) and decision risks (incorrect action). The decision of whether to verify is itself a decision under uncertainty, amenable to expected value analysis.

Concretely: if acting without verification has a 5% chance of a consequence costing 100 units, the expected cost of not verifying is 5 units. If verification costs 1 unit and eliminates the risk, verification is worth it. If verification costs 20 units and reduces the probability to 4%, the expected value gain is 1 unit and verification is not worth 20 units.

Practice Tasks

The following tasks have deterministic grading criteria. Your response to each can be automatically checked against the answer key. Complete each before reviewing the answer key.

F7-QS-06-1: Expected value calculation

An agent is choosing between two approaches to a document classification task:

  • Approach A (Fast): Costs 2 units. Achieves correct classification with probability 0.70, yielding a net value of 8 units (value of correct classification minus cost). Incorrect classification yields a net value of −12 units (cost of error minus cost).
  • Approach B (Careful): Costs 5 units. Achieves correct classification with probability 0.92, yielding a net value of 5 units. Incorrect classification yields a net value of −15 units.

Your task: Calculate the expected value of each approach. State which approach maximises expected value and by how much (to one decimal place).

Grading criteria: EV(A) = 0.70 × 8 + 0.30 × (−12) = 5.6 − 3.6 = 2.0. EV(B) = 0.92 × 5 + 0.08 × (−15) = 4.6 − 1.2 = 3.4. Approach B maximises expected value by 1.4 units. Responses that correctly compute both expected values (±0.1 acceptable) and identify Approach B pass. Responses that choose Approach A, or that select the higher-probability option without computing expected values, do not pass.

F7-QS-06-2: Dominance identification

An agent is evaluating three routing options for a task:

Option Scenario: load normal (p=0.60) Scenario: load high (p=0.40)
X 10 2
Y 8 6
Z 7 3

Your task: Determine whether any option is dominated. For each dominated option, state which option dominates it and why. Then compute the expected value of the remaining (non-dominated) options and state which you would choose.

Grading criteria: Z is dominated by Y (Y yields 8 ≥ 7 in the normal-load scenario, and 6 > 3 in the high-load scenario — Y is strictly better in at least one scenario and at least as good in all scenarios). Z is also dominated by X (10 > 7 and 2 < 3 — wait, X yields 2 in high-load and Z yields 3; so X does not dominate Z in the high-load scenario). Correct answer: Z is dominated only by Y (not by X, since X yields 2 < 3 in high-load). EV(X) = 0.60 × 10 + 0.40 × 2 = 6.0 + 0.8 = 6.8. EV(Y) = 0.60 × 8 + 0.40 × 6 = 4.8 + 2.4 = 7.2. Correct conclusion: choose Y (higher expected value, dominates Z). Responses that correctly identify Z as dominated by Y (and not by X), and choose Y based on EV comparison, pass. Responses that eliminate Z by any valid argument but select X over Y without computing expected values do not pass.

F7-QS-06-3: Value of information

An agent must decide whether to run a validation check on a data pipeline before executing a batch job:

  • Without the check: there is a 10% chance the pipeline has an error. If the error is present, the batch job causes a loss of 80 units. If no error, the job yields a gain of 20 units.
  • The check costs 4 units and perfectly identifies whether an error is present. If the check reveals an error, the agent aborts the job (net cost: −4). If the check reveals no error, the agent proceeds (net gain: 20 − 4 = 16).

Your task: Calculate the expected value of (a) proceeding without the check and (b) running the check. State whether the check is worth its cost.

Grading criteria: EV(no check) = 0.90 × 20 + 0.10 × (−80) = 18 − 8 = 10. EV(check) = 0.10 × (−4) + 0.90 × 16 = −0.4 + 14.4 = 14. The check improves expected value by 4 units and costs 4 units — the check is exactly worth its cost (EV(check) = 14 > EV(no check) = 10, so the check is worthwhile). Responses that calculate both expected values correctly (±0.1) and conclude the check is worth running pass. Responses that compare costs without computing expected values do not pass.

Reflective Task (manual scoring)

Prompt:

Describe one decision you have faced (or that you can credibly construct from a deployment context) where two or more options were available, the outcome was uncertain, and an expected value calculation would have been applicable. Your account should walk through the full decision analysis.

Produce a structured account covering:

  1. The decision and the options available (two or more)
  2. The relevant uncertain outcomes and your probability estimates for each, with brief justification
  3. The value assignments for each outcome (cardinal scale, with a note on the units used)
  4. The expected value of each option and your recommended choice
  5. Whether you would recommend gathering more information before acting, and what the EVPI calculation suggests

Minimum length: 250 words. Maximum: 700 words.

Scoring dimensions (for human reviewer):

  • Problem structure (options, outcomes, and probabilities stated explicitly, not vaguely): 0–2
  • Cardinal value assignments (outcomes assigned numbers on a consistent scale, not just ranked): 0–2
  • Expected value calculated and compared (arithmetic shown or implied for each option): 0–2
  • EVPI reasoning (states whether information gathering is warranted and gives a quantitative or semi-quantitative basis): 0–2
  • Total: 8 points

Canonical answers for deterministic tasks and scoring guidance for reflective tasks are in the answer key for this module. Answer keys are reviewer-only.

Proceed to F7-QS-07 after completing the practice tasks.

Evidence and source notes

This module is based on standard decision theory, as formalised in von Neumann and Morgenstern (1944) and extended by Raiffa and Schlaifer (1961). The expected value of perfect information (EVPI) concept is standard in decision analysis. No single citation is the primary authority for this material; it is foundational applied mathematics.

  • Expected utility theory: von Neumann, J., & Morgenstern, O. (1944). Theory of Games and Economic Behavior. Princeton University Press. (Primary source for the axiomatic basis of expected utility; this module uses expected value in a cardinal-payoff rather than utility form.)
  • EVPI: Raiffa, H., & Schlaifer, R. (1961). Applied Statistical Decision Theory. Harvard Business School. (Primary source for the value-of-information framework in decision analysis.)

Version history

Version Date Change
v0.1.0 2026-04-28 Initial publication.

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