Decision Theory Foundations

11 canonical decision-theory primitives for decisions under uncertainty. Each primitive is a formal tool with defined inputs, outputs, and failure modes. Primitives are domain-agnostic: the same expected-utility calculation that gates a product launch gates a capital investment; the same EVPI formula that sizes a market research study sizes a pre-launch pilot.

When to Apply

Apply decision-theory when:

• Single irreversible call under uncertainty (launch / kill / restructure)
• Value-of-information question — "is the next experiment worth running?"
• Real-options framing — staged investment with kill criteria
• Multi-criteria choice with explicit weights (MCDA, AHP)
• Multi-armed bandit allocation between treatments under regret minimisation

Skip and use simpler alternatives when:

• Decision is reversible and low-cost — just try it; analysis paralysis costs more than the wrong choice
• Multiple agents with strategic interaction — use foundations-game-theory
• Causal "did X cause Y" question — use foundations-causal-inference
• A clear oracle exists (test suite, KPI threshold) — use the oracle
• All candidate options are dominated by one option on every criterion — no decision-theory needed
• EVPI is much smaller than the cost of acquiring info — skip the study and decide now

Contents

• Quick Reference
• Primitive Index
• Formal Supporting Theory
• Misuse Boundaries
• When Expected-Value Reasoning Breaks Down
• Elicitation Failure Modes
  • Machine-Elicited Probabilities
• Decision Checklist
• Anti-Patterns
• Composition Recipes
• Workflow
• ASCII Flow
• Related Skills
• Fact-Checking

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Quick Reference

#	Primitive	When to Reach For It
1	Expected Utility (EU)	Ranking risky options when outcomes are commensurable
2	Bayesian Decision	Updating action after observing evidence; minimizing posterior expected loss
3	Minimax Regret	Adversarial or ambiguous probability; Savage-style robustness
4	Value of Information	Deciding whether to run an experiment, study, or pilot
5	Multi-Criteria Decision Analysis	Ranking options on incommensurable objectives
6	Risk Aversion	Adjusting EU for concave utility; certainty-equivalent pricing
7	Real Options	Valuing flexibility: defer, expand, or abandon
8	Prospect Theory	Predicting or correcting human choice under risk
9	Ellsberg and Allais Paradoxes	Diagnosing EU violations under ambiguity and certainty effects
10	Multi-Armed Bandit	Sequential exploration–exploitation under uncertainty
11	Stochastic Dominance	Distribution-level ranking without specifying a utility function

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Primitive Index

Each primitive has a full playbook (definition, when to use, inputs, outputs, failure modes, worked example, sources).

#	Primitive	Failure Mode It Addresses
1	Expected Utility	Choosing options by raw expected value, ignoring risk
2	Bayesian Decision	Acting on prior beliefs without updating on available evidence
3	Minimax Regret	Paralysis or overconfidence under deep uncertainty
4	Value of Information	Running experiments whose cost exceeds their decision value
5	Multi-Criteria Decision Analysis	Collapsing incommensurable objectives into a single number without disclosure
6	Risk Aversion	Ignoring the difference between expected value and certainty equivalent
7	Real Options	Treating irreversible decisions as if they were reversible
8	Prospect Theory	Prescriptive models failing to predict or explain actual human choice
9	Ellsberg and Allais Paradoxes	Applying EU where ambiguity aversion or certainty effects dominate
10	Multi-Armed Bandit	Fixed allocation ignoring the value of exploration
11	Stochastic Dominance	Comparing distributions only at their means

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Formal Supporting Theory

Theory Area	Use When	Applied Primitives It Grounds
Expected utility axioms	Need normative ranking under known probabilities	#1, #6, #11
Bayesian decision theory	Need posterior expected loss, Bayes risk, or decision rules after evidence	#2, #4
Robust decision criteria	Need action under ambiguity, adversarial states, or unclear probabilities	#3, #9
Information economics	Need to decide whether evidence is worth buying	#4
Multi-attribute utility	Need transparent tradeoffs across incommensurable goals	#5
Real options theory	Need irreversibility, deferral, expansion, or abandonment value	#7
Descriptive decision theory	Need to predict human deviations from EU	#8, #9
Sequential learning theory	Need exploration-exploitation allocation	#10

Use references/formal-theory-map.md when the task needs theorem assumptions, estimand boundaries, or a normative-vs-descriptive split.

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Misuse Boundaries

Misuse	Why It Is Wrong	Required Correction
Optimizing expected value for a risk-averse decision maker	EV ignores utility curvature and downside pain	Compute expected utility and certainty equivalent
Treating MCDA weights as objective truth	Weights encode stakeholder preferences	Disclose weights and run sensitivity analysis
Reporting an MCDA ranking without a rank-reversal test	Adding or dropping an irrelevant alternative silently reorders the result; an audit of 27 published MCDM pipeline/dataset combinations found recomposition consistency (RRT3) failing in ~48% of examples and transitivity (RRT2) in ~15%	Run the Wang–Triantaphyllou RRT1–RRT3 tests alongside weight sensitivity (Cabral et al., arXiv:2508.00129)
Running experiments without VoI	A study can be statistically interesting but decision-worthless	Compute EVPI/EVSI before funding research
Applying EU under deep ambiguity	Unknown probabilities violate the input contract	Use minimax regret, maximin, or ambiguity-aware criteria; or use Wasserstein DRRO when sample data on states are available
Treating bandits as free optimization	Exploration has opportunity cost and fairness/product constraints	Set regret budget, guardrails, and stopping rules
Comparing only means	Distribution tails and dominance can reverse decisions	Check stochastic dominance and downside risk
Treating a positive-EV recurring business (rake, spread, underwriting) as ergodic	Per-transaction EV is an ensemble statement; the operator lives one path and correlated exposures collapse into a single joint loss	Bound the worst joint loss against capital and separate collected from estimated edge — see finance-trading-investing

Check references/patterns-scenarios-traps.md before using outputs as decision authority.

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When Expected-Value Reasoning Breaks Down (Non-Ergodicity, Ruin Risk, and Kelly)

EU (#1) and certainty-equivalent (#6) reasoning implicitly average over an ensemble of parallel outcomes for a single decision. Repeated or leveraged bets compound multiplicatively instead — the ensemble average and the time average (the growth rate one actor actually experiences across repeated plays) diverge whenever there is a nonzero chance of an absorbing floor (ruin, bankruptcy, delisting, project death). This is the ergodicity-economics critique (Peters, 2019, Nature Physics): a bet with strictly positive expected value can still have a negative time-average growth rate once outcomes compound — no utility-curvature adjustment fixes this; the fix is switching from an ensemble average to a time average.

Expert checks before applying EU/CE to a repeated or leveraged decision:

• Does the payoff compound? If outcomes multiply (returns, survival odds, reputation, compounding debt) rather than add, compute the time-average growth rate, not the single-shot expectation.
• Kelly criterion (Kelly, 1956): for a repeated bet with a known edge, the growth-optimal wager fraction is f* = edge / odds (binary case: f* = p − q/b). Betting above Kelly reduces long-run growth even though each individual bet has positive EV. Full-Kelly is higher-variance than most real decision makers tolerate; fractional Kelly (e.g., half-Kelly) is the standard practitioner correction for parameter uncertainty and risk tolerance.
• Ruin is a constraint, not a tradeoff. Any state with an absorbing floor must be gated with a maximum-drawdown or survival constraint before the EU calculation — "the EV is positive" does not rescue a bet with non-negligible ruin probability.
• Use this alongside, not instead of, #1 and #6: EU/CE for single non-compounding decisions; add ergodicity/Kelly reasoning whenever the decision repeats, compounds, or has an absorbing failure state.

Sources: Peters, O. (2019). "The ergodicity problem in economics." Nature Physics 15, 1216–1221. Kelly, J. L. (1956). "A New Interpretation of Information Rate." Bell System Technical Journal 35(4).

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Elicitation Failure Modes

Formal primitives are only as good as the probabilities, utilities, and weights fed into them. The most common failures are in elicitation, not in the arithmetic:

Elicitation Trap	What Goes Wrong	Correction
Anchoring the first number	Whoever states a probability or weight first anchors the group; later "adjustments" under-correct	Elicit independently before group discussion (Delphi-style); aggregate afterward
False-precision point estimates	A single-point probability hides genuine uncertainty about the probability itself	Elicit ranges or a 10/50/90 percentile distribution; calibration-train the elicitor where the decision is high-stakes
Analysis paralysis	Teams keep requesting more studies or precision past the point where the information can change the action	Compute EVPI (#4) before approving further elicitation; stop and decide once EVPI is below the cost of refinement
Weights presented as objective	MCDA (#5) weights are framed as model output rather than negotiated stakeholder preference	Disclose weight provenance and run sensitivity analysis; treat weights as an input to be negotiated, not a discovered fact
Stated risk tolerance vs. revealed risk tolerance	Survey-elicited utility/risk-aversion parameters diverge from what the same stakeholder actually does under real stakes	Cross-check elicited CARA/CRRA parameters (#6) against revealed past choices (insurance, past bets) where available
Ambiguity flattened into a probability	An unknown probability is silently converted to 50/50 or a base rate, hiding ambiguity aversion	Run the Ellsberg/Allais diagnostic (#9) first; do not treat "unknown" as "known and uniform"

Machine-Elicited Probabilities

Probability inputs increasingly come from an LLM rather than a human panel. Treat them as a calibrated-but-not-superhuman forecaster, and score them the same way you would score a person:

• Accuracy is close but not yet at parity. On ForecastBench (Forecasting Research Institute), human superforecasters led the best LLMs by 0.017 Brier points as of 2026-01-29, with extrapolated parity projected for November 2026 (95% CI Jan 2026 – Nov 2027). Machine forecasts are usable as one panel member; they are not yet a replacement for a calibrated human on a high-stakes prior.
• Overconfidence is directional, not uniform. Models skew overconfident on events they rate as likely, while staying reasonably calibrated in the low-probability tail. Discount high stated probabilities more than low ones.
• Verbalized confidence is not the model's probability. A stated "I'm 90% sure" diverges from both token-level likelihood and realized accuracy, and RLHF-style alignment training degrades calibration by rewarding confident phrasing. Score against outcomes; never take the sentence at face value.
• Aggregate rather than single-shot. The anchoring correction above applies unchanged: sample independently across prompts or models before pooling, rather than accepting one generation as the estimate.

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Decision Checklist

• Risky choice: Are outcomes probabilistic and commensurable? → EU (#1), check risk aversion (#6)
• Evidence available: Has new information arrived that should change the action? → Bayesian decision (#2)
• Ambiguous probabilities: Are likelihoods unknown or contested? → minimax regret (#3), check Ellsberg (#9)
• Experiment proposed: Does a study, pilot, or A/B test precede the decision? → VoI (#4) before approving it
• Multiple objectives: Are criteria incommensurable (cost, quality, speed, risk)? → MCDA (#5)
• Risk-averse stakeholders: Does the decision maker care about variance, not just mean? → risk aversion (#6), certainty equivalent
• Irreversible action: Does the option foreclose future choices? → real options (#7), option to defer
• Human choice involved: Are you predicting or nudging actual human behavior? → prospect theory (#8)
• EU anomalies present: Do choices violate independence or sure-thing principle? → Ellsberg or Allais (#9)
• Sequential decisions under uncertainty: Is exploration vs. exploitation the core tension? → MAB (#10)
• Distribution comparison needed: Compare options without assuming a specific utility function? → stochastic dominance (#11)

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Anti-Patterns

Anti-Pattern	Decision Theory Diagnosis	Fix
Running an experiment when EVPI < experiment cost	VoI ignored; the information cannot improve the decision enough to justify the cost	Compute EVPI before approving any study or pilot (#4)
Choosing the highest-expected-value option for a risk-averse decision maker	Conflating EV with EU under concave utility; CE < EV for risk-averse agents	Apply utility function and compute certainty equivalent (#6)
Treating MCDA weights as objective	AHP/TOPSIS weights embed subjective preferences; different weight schemes reverse rankings	Disclose weights, run sensitivity analysis on weight perturbations (#5)
Applying EU under Ellsberg-type ambiguity	Decision maker exhibits ambiguity aversion — unknown probabilities trigger non-EU behavior	Switch to minimax regret (#3) or maximin for robustness; flag the ambiguity (#9)
Sunk-cost fallacy: not abandoning a losing project	Irreversibility conflated with commitment; option to abandon ignored	Price the option to abandon using real-options logic (#7)
Fixing traffic to each variant before observing response	Ignores exploration value; foregone learning from early-stopping	Use Thompson sampling or UCB; regret scales with suboptimal arm pulls (#10)
Comparing options only at their mean outcomes	Mean may be identical while variance differs materially	Check FSD or SSD before concluding indifference (#11)
Using EU where loss aversion and probability weighting apply	EU predicts poorly for mixed gains/losses around a reference point	Use prospect theory value function and probability weighting for descriptive accuracy (#8)

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Composition Recipes

Should we run this experiment?

Context: A team proposes a study, pilot, or A/B test before making a decision.

1. Compute EVPI — the maximum value the perfect information could provide (primitive #4). If EVPI < study cost, skip the study.
2. Compute EVSI for the specific study design — account for noise and sample size (#4).
3. If EVSI > study cost, approve. Then apply EU (#1) + risk aversion check (#6) to the post-study decision: does the posterior expected utility exceed the certainty equivalent threshold of the decision maker?
4. If the decision maker exhibits ambiguity aversion over the prior distribution, apply minimax regret (#3) as a robustness check alongside EU.

Worked example: Decision: ship feature A or B. Current best estimate: A = $200k value, B = $180k. Uncertainty: P(B actually better) = 0.3; expected regret if wrong = $40k. EVPI = 0.3 × $40k = $12k. Proposed A/B test costs $30k + 6 weeks → EVPI < test cost, skip the test; just ship A. If variance were higher — say P(B better) = 0.6 and regret = $100k — then EVPI = 0.6 × $100k = $60k, which exceeds the $30k cost, so the test pays for itself. EVSI refinement: a study that reduces variance by 60% (e.g. smaller sample, noisier measurement) captures 0.6 · EVPI. In the first scenario: 0.6 × $12k = $7.2k → still below $30k cost, skip. In the second: 0.6 × $60k = $36k > $30k → approve the cheaper, noisier study rather than the full test.

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Feature roadmap ranking under multiple objectives

Context: A product team must rank features or bets across cost, reach, strategic value, and risk.

1. Enumerate criteria and elicit weights using AHP or direct assignment (#5). Document the weight provenance.
2. Score each option on each criterion. Run TOPSIS or weighted-sum to produce a ranking.
3. Apply sensitivity analysis: perturb each weight ±20% and observe rank stability. Surface rank-reversals to stakeholders.
4. For options with irreversible commitments, price the option to defer (#7) — deferral has value when uncertainty will resolve.
5. If the team is risk-averse, compute certainty equivalents (#6) for options with high-variance outcomes; a lower CE may reverse the MCDA ranking.

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Sequential resource allocation across uncertain alternatives

Context: Marketing budget, experiment slots, or engineering capacity must be allocated across options whose true performance is unknown.

1. Frame as a multi-armed bandit (#10): each option is an arm with an unknown reward distribution.
2. Choose a policy: Thompson sampling for Bayesian updating on observed rewards; UCB for frequentist regret guarantees.
3. Before the first pull, compute EVPI (#4) to bound the total value of optimal learning — this caps the budget worth spending on exploration.
4. After sufficient observations, check stochastic dominance (#11): if one arm FSD-dominates all others, reallocate fully to it regardless of remaining regret budget.
5. Apply risk aversion (#6) if the decision maker penalizes downside variance: a risk-averse CE may favor a lower-mean but lower-variance arm earlier than pure regret minimization would suggest.

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VoI gating for expensive LLM calls and bandit-driven model routing

Context: An AI agent or orchestration layer must decide whether to invoke an expensive large model, run a retrieval step, or route a query to one of several LLM backends — each with different quality-cost profiles.

1. VoI gate before each costly call (#4): estimate EVPI for the decision the LLM call is meant to inform. If the agent's current context already implies a high-confidence action, skip the call — the information cannot change the decision. Apply this gate to retrieval steps (is the retrieved chunk likely to shift the answer?) and to model-tier selection (does this query warrant the 175B model over the 7B?).
2. Bandit-driven model routing (#10): treat each LLM backend (or prompt variant) as a bandit arm with unknown quality distribution per query class. Use Thompson sampling to learn the best arm per context cluster; a LinUCB-based policy achieves sublinear regret without predicting future prompts or accessing model internals, including under unstructured context evolution as users refine queries mid-session (Poon et al., arXiv:2506.17670).
3. Risk aversion on tail latency (#6): for SLA-sensitive paths, compute the certainty equivalent of the latency distribution — a risk-neutral mean-latency comparison may select a high-variance backend a risk-averse product cannot afford.
4. Stochastic dominance check before full reallocation (#11): once enough observations accumulate, verify that the preferred arm FSD-dominates alternatives across quality and cost dimensions before committing the full traffic budget.

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Clarify-or-commit: should the agent ask the user a question?

Context: An agent holds an ambiguous instruction and must decide whether to ask a clarifying question or proceed on its best reading. Each question costs user patience; a wrong assumption costs a wasted trajectory.

1. Score each candidate question by EVPI (#4), not by how uncertain the agent feels. The value of a question is the expected improvement in the action, so a question whose answers all lead to the same next step has zero value however uncertain the agent is. Penalize by an asking cost to suppress redundant questions — EVPI-scored clarification cut question count 1.5–2.7x at higher task success than uncertainty-threshold baselines (Suri et al., arXiv:2511.08798).
2. Separate specification uncertainty from model uncertainty. Only the first is fixable by asking. Ambiguity about what the user wants is a question; ambiguity about whether the agent's own output is correct is a verification or retrieval step, and asking the user will not resolve it.
3. Treat EVPI as decaying with trajectory position — this is the main departure from single-shot VoI. Question value is not stationary over a long-horizon task: goal-level clarification decays to baseline value after roughly the first 10% of execution, while input-level clarification stays useful to about the 50% mark. Past the midpoint, asking performs worse than never asking, because the cost of rework already sunk exceeds the information gain (Gulati et al., arXiv:2605.07937, ~6,000 runs across 4 models).
4. Budget the asking rate explicitly. Frontier models fail this in both directions — over-asking in 52% of sessions or suppressing questions entirely. Front-load goal questions before acting, allow input questions mid-trajectory, and commit after the midpoint rather than asking late.

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Workflow