Your SaaS Is an Insurance Product: A Modeling Framework

An LLM application often sells or internally allocates several service products: a small or premium model, a short or long token cap, and possibly multiple posted prices. The operational decision is not merely which model answers a prompt. A price changes purchase probability, a token cap changes both user value and the tail of resource consumption, and accepted requests compete for shared compute and premium-model capacity. Demand and output length are initially uncertain, while an offline model may provide useful but imperfect predictions. We formulate sequential pricing and admission with stochastic resource consumption. Each arriving request belongs to an observable segment. The platform chooses a product--price pair or makes no offer; purchase, revenue, and resource use are then random. An offline predictor supplies a uniform, validated error radius for every segment--product cell. We propose Prediction-Clipped UCB (PCUCB), which intersects the offline prediction interval with an online confidence interval, evaluates products using resource shadow prices, and reserves a sample-path envelope before commitment. The prior gives a fast start when accurate, while online learning protects the platform when predictions are coarse. The analysis is modular. On a simultaneous confidence event, regret against a buffered fluid benchmark is bounded by a pacing term plus the cumulative diameter of the intersected intervals. For $J$ segment-product cells and prediction radius $\varepsilon$, this yields $$\widetilde O\left( \sqrt{T}+(1+\barΛ) \min\{T\varepsilon,\sqrt{JT}\} \right),$$ where $\barΛ$ bounds operational shadow prices. Thus the algorithm smoothly interpolates between an almost full-information regime and learning from scratch. Hard feasibility holds on every sample path through reservation envelopes.

We study a large language model (LLM) service in which a provider chooses a per-token price and a default reasoning-token allocation, while a user may accept the default, customize the allocation, or exit. Larger allocations can improve accuracy but increase token cost and latency. We model this interaction as a Stackelberg game and derive the user's unique optimal customized allocation in closed form. For any price, the acceptable defaults form either an empty set or a compact interval. We characterize the provider's optimal default through a three-regime rule, reduce equilibrium computation to a one-dimensional price optimization, and prove the existence of the equilibrium. We further show that defaults affect the implemented reasoning allocation only when users value the convenience of avoiding customization; otherwise, every service-providing outcome implements the user's optimal customized allocation. Experiments with two compact open-weight reasoning models on five mathematics and science benchmarks support the accuracy-token model and show how model and task characteristics determine equilibrium prices, defaults, and reasoning allocations.

Autonomous AI agents increasingly issue side-effect-bearing actions: database mutations, refunds, payments, external commitments. We propose the Actuarial Action Interface (AAI), a deterministic runtime contract that prices each such action against a contractually fixed safe default under a time-consistent risk mapping, and gates execution against a per-boundary reserve capital budget. We then develop the Authority Frontier, an evaluation primitive measuring how much autonomous authority the runtime releases at each level of reserve capital. The framework provides (i) a deterministic quote-bind-commit protocol with toll-bounded capability tokens; (ii) a universal seven-class action taxonomy mapping heterogeneous tool calls to comparable authority units; (iii) replay determinism and pathwise reserve coverage under alpha-spending; (iv) cross-domain normalization via full reserve demand C_full and capital metrics Capital@k. We instantiate AAI across four agentic environments (database mutation, customer-service refund, and the public tau-bench retail and airline tool-use traces) and report a live Postgres panel in which three Azure-hosted models propose actions through the same contract. The frontier exhibits a common low-reserve refusal and intermediate-release pattern across domains, with saturation only where the budget grid reaches full reserve demand; required reserve capital varies by 22x (Capital@50 from 289 to 6457). The framework does not force domains into the same shape; it surfaces each domain's actuarial geometry. In the live panel the contract prevents realized loss across all three models at low budget while differing in underwriting persistence under denial: model identity is an actuarial underwriting variable. The contribution is a benchmark-ready evaluation framework for runtime actuarial control of autonomous-agent side effects.