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 below a justified utility-compatible additive information cost — skip that study; with nonlinear utility evaluate terminal costs before deciding

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

Quick Reference

Primitive Index

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

Formal Supporting Theory

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

Misuse Boundaries

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

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

Expected utility (EU, #1) does not require literal parallel ensembles: it ranks lotteries from stated probabilities and a utility function. The practical failure occurs when a one-period payoff model is reused for a repeated, path-dependent, multiplicative, or ruin-constrained process without modeling wealth state and survival. In those settings an arithmetic expected return can be positive while long-run log growth is negative. Model the dynamic process directly; expected utility with an appropriate state-dependent utility can still be coherent, while expected log growth/Kelly is one specific objective rather than a universal replacement (Peters, 2019; Kelly, 1956).

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

• Does the payoff compound or depend on the path? Model wealth/state transitions and absorbing boundaries before evaluating the choice. Compute expected-log or time-average growth only when long-run growth is the stated objective; otherwise evaluate the selected terminal or path-dependent utility and survival/drawdown constraints on that process.
• 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 #1 and #6: state whether the objective is terminal expected utility, survival probability, drawdown control, or long-run growth. Apply Kelly only when growth optimality, repeated comparable opportunities, and credible probabilities match the decision.

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).

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:

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.

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)

Anti-Patterns

All VoI cost gates below assume utility-compatible additive costs; otherwise recalculate expected utility with costs/delay in terminal outcomes.

Composition Recipes

Should we run this experiment?

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

1. Define utility, affected population and decision horizon. EVPI is the cost-free upper bound (#4); compare it to an additive study cost only on the same utility scale (e.g., risk-neutral net money).
2. Compute EVSI for the specific study design — account for noise and sample size (#4).
3. Compare study versus immediate-action expected utility including cost and delay. With nonlinear utility, incorporate costs into terminal outcomes before applying utility; do not subtract pounds or hours from utility EVSI. Optimize over discrete sample sizes, no study and endpoints; marginal gain=cost is only a differentiable interior condition.
4. If the decision maker exhibits ambiguity aversion over the prior distribution, apply minimax regret (#3) as a robustness check alongside EU.

Worked example: Choose between an outside action worth 0 and an action paying +1 or -1 with equal probability. The current value is 0 and perfect information is worth 0.5. A symmetric signal that identifies the payoff correctly 75% of the time produces posterior means +0.5 and -0.5, so the posterior-optimal expected value is 0.25 and EVSI is 0.25. Here utility is linear in payoff and cost is in those same payoff units; run the study only when its cost is below 0.25. EVPI exceeding cost merely leaves open the possibility that a study is worthwhile; it does not approve a particular study. Likewise, a percentage reduction in posterior variance is not a percentage of EVPI. Compute EVSI from the signal likelihoods and posterior-optimal actions.

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.

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.

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): define states, priors, actions, utilities and the call's signal likelihoods; compute EVSI from posterior-optimal actions and compare with utility-compatible cost. Confidence alone does not imply zero information value: prior (.999,.001), risky utilities (1,−100), safe (0,0) give current EU .899 and EVPI .1. Perfect observation costing .01 is worthwhile. An uninformative signal instead has EVSI=0. EVPI only supplies a cost-free upper bound: below additive cost it can exclude a call, above cost it does not approve that call. Apply the same logic to retrieval and model-tier selection; evaluate terminal outcomes when utility/cost is nonlinear.
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.

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 EVSI (#4) using its answer likelihoods, priors and action utilities; use EVPI only as an upper bound unless the answer reveals the full relevant state. 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. Compare with utility-compatible asking cost. Paper-specific EVPI-based surrogate scores are not exact EVSI; EVPI-scored clarification cut question count 1.5–2.7 times with higher ambiguous-task coverage on the study-specific ClarifyBench evaluation (Suri et al., arXiv:2511.08798, §7 and Table 2).
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. Model timing as part of VoI. Rework cost can reduce a question's value after execution begins, but the decay curve depends on task, ambiguity type, benchmark, and model. Gulati et al. (arXiv:2605.07937) tested 84 forced-injection variants across three benchmarks and four models in more than 6,000 runs; their 10% and 50% injection positions are experimental grid points, not universal deadlines.
4. Budget asking explicitly and measure locally. Front-load high-impact goal questions when rework is costly, but keep asking available whenever a possible answer changes the safe or authorized action. The paper's natural-asking results used 300 sessions; the reported 52% over-asking figure applies to GPT-5.2 on 100 TAC sessions, not to frontier models as a class.

Workflow

1. Identify the decision structure: risky choice, ambiguous probabilities, sequential learning, or multi-objective ranking.
2. Use the Decision Checklist to select the applicable primitive(s).
3. For a small finite action-by-state matrix, copy a case from data/finite-state-decision-fixtures.json, edit its input, and run python3 scripts/decision_calculator.py model.json. Omit prior for minimax regret alone; add explicit signal_likelihoods and study_cost for exact finite-signal EVSI. Read the input, calculation, and interpretation contract before using the result.
4. Open the per-primitive playbook in assets/templates/decision-theory/ for the full definition, inputs, outputs, failure modes, and worked example.
5. For compound decisions, use the Composition Recipes to stack primitives.
6. Verify inputs: probability estimates, utility function parameters, and criteria weights are the most common failure points.
7. Disclose assumptions explicitly before acting on any MCDA ranking or EU calculation.

ASCII Flow

Single-agent decision under uncertainty
  -> Define actions, states, outcomes, and constraints
  -> Classify structure: risk, ambiguity, sequential learning, or multi-criteria
  -> Select primitive and open playbook
  -> Elicit probabilities, utilities, regret, or weights
     +-- inputs weak -> run sensitivity or value-of-information check
     +-- inputs usable -> compute recommendation
  -> Report action, assumptions, uncertainty, and decision boundary

Navigation

• Practical completion contract and known-answer controls: references/practical-contract.md.

• Per-primitive playbooks: assets/templates/decision-theory/ (one file per primitive)

• Composition guide and selection matrix: assets/templates/decision-theory/README.md

• Formal theory map: references/formal-theory-map.md

• Patterns, scenarios, and traps: references/patterns-scenarios-traps.md

• Domain-agnostic primitives overview, anti-patterns by decision structure, and checklist: references/primitives-overview.md

• Finite-state calculator contract and interpretation: references/finite-state-calculator.md

• Exact calculator fixtures: data/finite-state-decision-fixtures.json

• Sources: data/sources.json

Related Skills

This skill is a self-contained foundations primitive. Cross-link only to other foundations-* skills when a task requires joint coverage (e.g., foundations-game-theory for multi-agent settings, foundations-causal-inference for causal identification before decision framing).

Fact-Checking

• EU axioms and vNM theorem: von Neumann and Morgenstern (1944/1947). Theory of Games and Economic Behavior.
• Bayesian decision theory and Bayes risk: Raiffa and Schlaifer (1961). Applied Statistical Decision Theory.
• Minimax regret: Savage (1954). The Foundations of Statistics.
• Value of information (EVPI, EVSI): Raiffa and Schlaifer (1961); Howard (1966) "Information Value Theory."
• AHP: Saaty (1980). The Analytic Hierarchy Process.
• CARA/CRRA, certainty equivalent: Pratt (1964) "Risk Aversion in the Small and in the Large."
• Real options: Dixit and Pindyck (1994). Investment under Uncertainty.
• Prospect theory, probability weighting: Kahneman and Tversky (1979) "Prospect Theory: An Analysis of Decision under Risk."
• Loss aversion re-estimate: Brown, Imai, Vieider, and Camerer (2024). "Meta-Analysis of Empirical Estimates of Loss Aversion." Journal of Economic Literature 62(2), 485–516. Mean λ ≈ 1.955 [1.820, 2.102] across 607 estimates — supersedes the original λ ≈ 2.25 point estimate as the best current population value; both are contested. [Primitive #8]
• Ergodicity economics and Kelly criterion: Peters (2019) "The ergodicity problem in economics," Nature Physics 15; Kelly (1956) "A New Interpretation of Information Rate," Bell System Technical Journal 35(4). [Primitive #6; expected-value breakdown section]
• Ellsberg paradox: Ellsberg (1961) "Risk, Ambiguity, and the Savage Axioms."
• Allais paradox: Allais (1953) "Le comportement de l'homme rationnel devant le risque."
• Multi-armed bandit and UCB: Robbins (1952); Auer, Cesa-Bianchi, and Fischer (2002).
• Thompson sampling: Thompson (1933); Russo et al. (2018). "A Tutorial on Thompson Sampling."
• Stochastic dominance: Hadar and Russell (1969); Levy (1992) review.
• Lattimore and Szepesvári (2020). Bandit Algorithms.
• Constrained bandits (best-of-both-worlds): Bernasconi, Castiglioni, Celli (ICML 2025, PMLR 267:3877–3898). [Primitive #10]
• LLM-based PSRL: Arumugam and Griffiths (ICLR 2026). arXiv:2504.20997. [Primitive #10]
• Wasserstein DRRO: Fiechtner and Blanchet (2025). arXiv:2504.10796. [Primitive #3]
• Gen-WDRO: NeurIPS 2025 poster. [Primitive #3]
• Distributionally Robust Performative Optimization: Jia et al. (NeurIPS 2025). arXiv:2407.01344. [Primitives #3, #1]
• Online Decision-Focused Learning: Capitaine et al. (ICLR 2026). arXiv:2505.13564. [Primitive #4]
• DFL via Dual Surrogates: Rodriguez-Diaz et al. (NeurIPS 2025). arXiv:2511.04909. [Primitive #4]
• Multi-LLM selection via contextual bandits: Poon, Dai, Liu, Kong, Lui, Zuo (arXiv:2506.17670, June 2025). LinUCB routing across LLM backends with sublinear regret under unstructured context evolution. [Primitive #10; app-builder recipe]
• Decision-Centric Design for LLM Systems: Sun (arXiv:2604.00414, April 2026). Separates the decision layer from generation in LLM systems; formalizes VoI gating and clarify-or-commit tradeoffs as explicit decision problems. [Primitive #4; app-builder recipe]
• EVPI-scored agent clarification: Suri, Mathur, Lipka, Dernoncourt, Rossi, Manocha (arXiv:2511.08798, Nov 2025, rev. Apr 2026). SAGE-Agent; cost-penalized EVPI over candidate questions; 1.5–2.7 times fewer questions on ClarifyBench with higher ambiguous-task coverage (study-specific results, §7 and Table 2). [Primitive #4; clarify-or-commit recipe]
• Clarification timing: Gulati, Gupta, Lumer, Sen, Subbiah (arXiv:2605.07937, May 2026). More than 6,000 forced-injection runs across 84 variants, 3 benchmarks, and 4 models, plus 300 natural-asking sessions. Timing effects vary by benchmark, ambiguity dimension, and model; the tested 10%/50% positions are not general policy cutoffs. [Primitive #4; clarify-or-commit recipe]
• MCDA rank-reversal prevalence: Cabral et al. (arXiv:2508.00129, July 2025, rev. Aug 2026). Operationalizes Wang–Triantaphyllou RRT1–RRT3 in Scikit-Criteria; RRT3 fails in ~48% and RRT2 in ~14.8% of the study's 27 selected published pipeline/dataset combinations (§7.2 and §8; not population prevalence). [Primitive #5; misuse boundaries]
• LLM vs. superforecaster calibration: Bastani, Kučinskas, Reynolds (Forecasting Research Institute, ForecastBench). Superforecasters ahead by 0.017 Brier points as of 2026-01-29; extrapolated parity Nov 2026 (95% CI Jan 2026 – Nov 2027). Verify the current leaderboard before citing the gap — it is a moving number. [Elicitation Failure Modes]
• Numeric thresholds (e.g., EVSI formulas, CE approximations) should be verified against primary sources before citing in decisions.

Learnings Loop

When prior decisions or pitfalls are relevant, consult learnings.consolidated.md if present; use learnings.md only for needed history or as the available fallback. Otherwise skip both.

After applying it, if you encountered a pattern worth remembering, a mistake worth preventing, or a domain fact that surprised you, append one dated bullet to learnings.md via agents-skills-feedback-loop/scripts/append_learning.py. Do not modify SKILL.md itself.