Foundations Decision Theory

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

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)

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).

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

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.

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): 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.

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.

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. Open the per-primitive playbook in assets/templates/decision-theory/ for the full definition, inputs, outputs, failure modes, and worked example.
4. For compound decisions, use the Composition Recipes to stack primitives.
5. Verify inputs: probability estimates, utility function parameters, and criteria weights are the most common failure points.
6. 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

• 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
• 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; 1.5–2.7x fewer questions at higher success. [Primitive #4; clarify-or-commit recipe]
• Clarification timing decay: Gulati, Gupta, Lumer, Sen, Subbiah (arXiv:2605.07937, May 2026). ~6,000 runs, 4 dimensions, 3 benchmarks, 4 frontier models. Goal-clarification value decays by ~10% of execution, input by ~50%; asking past the midpoint underperforms never asking. [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 audited published pipelines. [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

Before applying this skill on a non-trivial task, read learnings.consolidated.md in this directory (and learnings.md if present).

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.