Actuarium

Decisions under uncertainty — one method, every domain.

Score(a)=(1−ω)[(1−ε) Dg,u,p∗(a)+εmin⁡su(U[a][s])]−ω⋅Regret(a)\text{Score}(a) = (1-\omega)\Big[(1-\varepsilon)\,D_{g,u,p^*}(a) + \varepsilon\min_s u(U[a][s])\Big] - \omega\cdot\text{Regret}(a)

Every classical decision rule — EMV, expected utility, maximin, Hurwicz, minimax regret, Bayes, credibility, VaR/TVaR, Wang premiums, Gilboa–Schmeidler — is a special case of one auditable, parameterised engine.

The ADE in one page

The Actuarium Decision Engine (ADE) scores every action aa by combining a distorted, credibility-weighted expected utility with an ambiguity term and a regret penalty:

p∗(s)=Z⋅p(s∣e)+(1−Z)⋅q(s)p^*(s) = Z\cdot p(s\mid e) + (1-Z)\cdot q(s)u(x)=ρ(1−e−x/ρ)u(x) = \rho\left(1-e^{-x/\rho}\right)Score(a)=(1−ω)[(1−ε) Dg,u,p∗(a)+εmin⁡su(U[a][s])]−ω⋅Regret(a)\text{Score}(a) = (1-\omega)\Big[(1-\varepsilon)\,D_{g,u,p^*}(a) + \varepsilon\min_s u(U[a][s])\Big] - \omega\cdot\text{Regret}(a)

Parameter meanings

ParameterMeaning
ρ\rhoRisk tolerance for exponential (CARA) utility; \rho\to\infty is risk-neutral.
λ,κ\lambda,\kappaWang / proportional-hazards distortion parameters (tail-loading).
ε\varepsilonAmbiguity aversion weight — ε-contamination toward pure maximin.
ω\omegaRegret-aversion weight — blend toward minimax regret.
ZZCredibility weight blending the Bayesian posterior with a reference distribution q(s).
p(s),q(s)p(s), q(s)Prior belief and reference/benchmark distribution over states.
L(e∣s)L(e\mid s)Likelihood of observed evidence under each state, used for Bayesian updating.

Classical method → ADE parameters

Classical criterionADE settingNote
Expected Monetary Value (EMV)ρ→∞, λ=0, ε=0, ω=0, Z=1\rho\to\infty,\ \lambda=0,\ \varepsilon=0,\ \omega=0,\ Z=1Plain probability-weighted average payoff.
Expected Utilityfinite ρ, λ=0, ε=0, ω=0\text{finite } \rho,\ \lambda=0,\ \varepsilon=0,\ \omega=0vNM expected utility with CARA risk aversion.
Maximin (Wald)ε=1\varepsilon=1Only the worst state is judged.
Maximaxρ→∞, point mass on arg⁡max⁡sU[a][s]\rho\to\infty,\ \text{point mass on }\arg\max_s U[a][s]Optimistic best-case evaluation.
Hurwicz(α)αmax⁡+(1−α)min⁡\alpha\max+(1-\alpha)\minOptimism–pessimism blend, computed directly.
Minimax regret (Savage)ω=1\omega=1Minimise the worst-case regret across states.
Laplacep(s)=1/n, ρ→∞, λ=0, ε=0, ω=0p(s)=1/n,\ \rho\to\infty,\ \lambda=0,\ \varepsilon=0,\ \omega=0Principle of insufficient reason.
Bayes (posterior EMV)ρ→∞, Z=1, posterior p(s∣e)\rho\to\infty,\ Z=1,\ \text{posterior }p(s\mid e)EMV under the Bayesian posterior.
Bühlmann credibilityZ∈(0,1)Z\in(0,1)Linear-Bayes blend of posterior and reference.
VaR / TVaRWang λ, ρ→∞, ε=0, ω=0\text{Wang }\lambda,\ \rho\to\infty,\ \varepsilon=0,\ \omega=0Tail-weighted valuation via distortion.
Wang premium principleg=Φ(Φ−1(t)+λ)g=\Phi(\Phi^{-1}(t)+\lambda)Actuarial premium-loading distortion.
Gilboa–Schmeidler (maxmin EU)ε=1 (ε-contamination)\varepsilon=1\text{ (}\varepsilon\text{-contamination)}Worst-case over a set of priors.
Savageω=1\omega=1Savage's own minimax-regret criterion.

Full case study: Workers' Compensation pricing & reserving

Granite State Mutual — loss triangles, chain-ladder & Bornhuetter–Ferguson reserving, a rate indication build-up, a class plan review, and both decisions run through the ADE.

Selected ultimate
$27,052k
Total IBNR
$5,892k
Indicated rate change
6.2%
Open full case studyOpen the interactive pricing modelQuarterly rate & reserve review (real data)Quarterly rate calendarGLM pricing studio (your data)Life & annuity projection (IFRS 17 / LDTI)Stochastic economic scenarios (ESG) & CTE reservesFull rate scenario (premium → loss ratio → exposure → ESG → quoted rate)Live rate deployment (quote & push approved rate tables)GOES-style scenario generator seeded from the live Treasury curve + hosted /v1/esg endpointMachine-led reserving method selectionCertification case study (P → FM → FAM → CAS 5/7 → ERM on real data)

Decision Studio

Pick a realistic scenario, edit the states, actions, payoffs and beliefs, and watch every classical criterion and the ADE recommendation update live.

Discuss with the Risk agent

A manufacturer decides whether to expand capacity now, expand later, or hold, under uncertain demand growth. Payoffs are NPV in $M.

States of the world

Actions

Payoff matrix U[action][state]

Action \ StateLow demandMedium demandHigh demand
Expand now
Expand later (option)
Hold capacity

Prior beliefs p(s)

Low demand30.0%
Medium demand45.0%
High demand25.0%

Normalised automatically to sum to 100%.

Evidence & Bayesian updating

OffOn

Attitude parameters

Risk-neutral

Reference distribution q(s) used for credibility blending: Low demand: 30%, Medium demand: 45%, High demand: 25%.

Recommendation

Expand later (option)
EVPI = 2.6
ActionADE scoreEMVMaximinMaximaxRegretCERisk premium
Expand later (option)3.865.6-2.149.11.254.461.14
Expand now1.294.9-10.5711.543.821.353.55
Hold capacity0.90.9501.874.210.930.02
Classical criterionExpand nowExpand later (option)Hold capacityRecommends
Expected Monetary Value (EMV)4.95.60.95Expand later (option)
Expected Utility1.293.860.9Expand later (option)
Maximin (Wald)-8-20Hold capacity
Maximax22142Expand now
Hurwicz(α=0.5)761Expand now
Laplace (principle of insufficient reason)661Expand now
Bayes (posterior EMV)4.95.60.95Expand later (option)
Bühlmann credibility blend4.95.60.95Expand later (option)
Minimax regret (Savage)-8-8-20Expand now
TVaR / Wang-distorted value-1.681.90.49Expand later (option)

Sensitivity — where the recommendation flips

  • rho: no flip across the sampled range — recommendation is robust to this parameter
  • lambda: no flip across the sampled range — recommendation is robust to this parameter
  • epsilon: recommendation flips near 0.563
  • omega: no flip across the sampled range — recommendation is robust to this parameter
  • Z: no flip across the sampled range — recommendation is robust to this parameter

Why one method — and its limits

The Savage and von Neumann–Morgenstern axioms are what justify representing preferences by an expected (or distorted, ambiguity-robust) utility in the first place — they are normative, not descriptive: real people routinely violate them. The Ellsberg paradox shows people are ambiguity-averse in a way plain expected utility cannot represent; the Allais paradox shows the vNM independence axiom is regularly violated by the "certainty effect." ADE's $\varepsilon$ and distortion parameters are a controlled, disclosed departure from strict expected utility — not a claim that the axioms are false.

Every ADE run still depends on human judgement: elicitation of the prior $p(s)$, the payoff matrix, and the risk/ambiguity/regret parameters, and model risk — the engine is only as good as the states, actions and numbers fed into it. The sensitivity panel exists precisely because a technically correct method fed a fragile or overconfident input can still recommend the wrong action.

Go deeper

Foundations: probability as degree of belief

Kolmogorov's axioms vs. the Bayesian (subjective) interpretation, and de Finetti's coherence argument for why probability is the only rational way to grade uncertainty.

Expected utility and risk aversion

The von Neumann–Morgenstern axioms, why expected value alone is not enough, ARA/RRA, certainty equivalents, and the common utility functions.

Bayesian updating and credibility

Bayes' theorem, conjugate families, Bühlmann credibility as linear Bayes, and how Z is simply a parameter of ADE.

Risk measures and probability distortion

VaR, TVaR, the Choquet integral, the Wang transform, spectral risk measures, and coherence.

Ambiguity and robustness

The Ellsberg and Allais paradoxes, maxmin expected utility, ε-contamination, multiplier preferences, and info-gap decision theory.

The value of information

EVPI, EVSI, preposterior analysis, and when it is worth paying to gather more data.

Decision trees and sequential decisions

Backward induction, folding back a decision tree, and the real-options intuition for staged commitments under uncertainty.

Behavioural pitfalls

Allais and framing effects, overconfidence, and calibration — where human judgement systematically departs from coherent decision-making.

Applications across domains

Worked ADE examples spanning insurance, finance, health, education, engineering, and personal decisions.

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