Decision Theory and Artificial Intelligence

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Artificial intelligence supports a decision only through a feasible action and its reward. We develop a common framework linking action spaces, reward functions, statistical estimation, and costly information acquisition. We characterize minimal output contracts for specified reward families, decompose decision regret, and connect regression error to routing regret under capacity constraints. An acquisition-output contract characterizes which compressed representations preserve every nonnegative-price routing decision and bounds the loss when they do not. A routing-geometry dichotomy distinguishes discrete decisions, where information can be most valuable near an action boundary, from continuous quadratic sizing, where precision gains can favor the strongest signals. Delay, position limits, and conditional refinement determine when these patterns change. The framework accommodates supervised learning (SL), unsupervised learning (UL), reinforcement learning (RL), large language models (LLMs), and typed decision models such as JEV. SL estimates outcome statistics or action values; UL constructs representations whose usefulness depends on the reward-relevant information they preserve; RL learns policies or action values for sequential rewards. LLMs generate candidate answers or judgments, while JEV supplies probabilities over specified categories, propositions, or rubric levels. These training paradigms and output interfaces enter a common decision calculation: what each output identifies, how estimation error affects the selected action, and whether another source's expected benefit exceeds its fee and delay cost. Quantile calibration, refinement perturbation bounds, and selective-observation identities connect the framework to evaluation. Synthetic calculations verify the analytical formulas and expose the role of outcome noise. A retrospective study of 27,241 recorded LLM cases evaluates cost-aware replacement routing, isolating routing targets with matched learners and reporting representation sensitivity, transfer reversals, finite-sample screens, and hindsight ceilings. The results specify the output, acquisition rule, and evaluation target required by a decision problem.