A causal modeling perspective on decision theory

Abstract: Decision theory provides a formal framework for how agents should make choices under uncertainty, drawing on ideas from philosophy, probability, and causality. Despite significant progress, the field still lacks a unified modeling language, and key concepts - such as the distinction between subjective and objective elements, or what it means for a decision theory to perform well - are often left implicit. This can make it difficult to evaluate and compare competing theories, particularly in controversial cases. In this paper, we address these issues by introducing a formal framework for decision theory based on nonparametric structural equation models (NPSEMs), a well-established tool in causal inference. NPSEMs provide a unified foundation for representing agents, counterfactuals, and causal relationships, allowing for unambiguous definitions of EDT and CDT. Building on this foundation, we propose a novel decision theory - personal decision theory - which instructs agents to maximize a subjective model of their own counterfactual utility. We introduce a formal performance metric based on hypothetical interventions that enforce a given decision theory across a population - such as might be achieved through education or policy -- and show that, under certain assumptions, personal decision theory is optimal with respect to this metric. Throughout, we use the smoking lesion problem as a running example and conclude with a formal analysis of Newcomb's problem. Our aim is to provide decision theory with a clearer modeling language and firmer evaluative ground, thereby enabling more rigorous comparisons and facilitating conceptual progress in the field.

• The paper introduces a formal NPSEM framework that clearly distinguishes between objective data-generating models and subjective internal models.
• It rigorously formalizes EDT, CDT, and a new personal decision theory, demonstrating that PDT can outperform others under intervention-based metrics.
• The model resolves classical puzzles like the smoking lesion and Newcomb’s problem via counterfactual reasoning and analysis of agent heterogeneity.

A Causal Modeling Perspective on Decision Theory

Introduction

This paper advances the field of decision theory by articulating a formal modeling framework that leverages nonparametric structural equation models (NPSEMs). The author identifies persistent ambiguities within decision theory regarding the formalization of agents, the distinction between subjective and objective elements, and the lack of a principled performance metric for comparing decision theories. Crucially, the work provides a rigorous, uniform mathematical apparatus to define classical decision theories—evidential decision theory (EDT) and causal decision theory (CDT)—and introduces a new theory, personal decision theory (PDT), within the NPSEM framework.

Formalization via Nonparametric Structural Equation Models

NPSEMs, widely accepted in causal inference, represent each variable as a function of its parents and an exogenous stochastic component, capturing both causal structure and agent heterogeneity. The paper demonstrates that standard decision-theoretic scenarios, such as the smoking lesion problem and Newcomb's problem, can be precisely cast as NPSEMs. Agents are associated with specific realizations of exogenous error terms, capturing both inter-agent variability and the generation of choices and outcomes.

Counterfactual reasoning, central to both CDT and interventionist perspectives, is naturally handled in NPSEMs by the modular modification of structural equations—a fact that eliminates ambiguities endemic to prior literature.

Agent Modeling, Subjectivity, and Performance Metrics

A major contribution is the explicit distinction between the objective (true) data-generating model and each agent's potentially subjective internal model. This separation permits formal definitions of EDT and CDT agents: EDT agents select acts maximizing their subjectively modeled conditional expected utility, $E(U\mid A=a; M)$, and CDT agents select acts maximizing their subjectively modeled counterfactual expected utility, $E(U(a); M)$. The model further introduces endogenous variables encoding both the agent's decision theory ($D$) and subjective modeling assumptions ($M$), supporting explicit discussion of how decision theory causally interacts with agent action.

To compare decision theories, the work proposes an objective population-level metric: the mean utility achieved by hypothetical interventions that enforce a given decision theory across a population, denoted $E[U(d)]$ for decision theory $d$. This metric circumvents previous ad hoc comparisons and supports principled evaluation.

Personal Decision Theory and Its Optimality

The paper defines personal decision theory (PDT), under which agents maximize their own subjective expected counterfactual utility: $$\arg\max_a E[U(a, \boldsymbol{\varepsilon}) \mid \boldsymbol{\varepsilon}; M]$$, where $\boldsymbol{\varepsilon}$ encodes the agent's exogenous characteristics. PDT generalizes CDT and EDT to settings where acts may have highly heterogeneous causal effects across agents.

Given correct subjective models and the absence of direct causal effects from $D$ to $U$ not mediated by $E(U(a); M)$0, the paper rigorously demonstrates that PDT is optimal with respect to the intervention-based performance metric. Formally, for any competing decision theory $E(U(a); M)$1, $E(U(a); M)$2, with strict improvement whenever the population exhibits effect heterogeneity that CDT cannot exploit due to its focus on population-level averages. CDT is only optimal among theories enforcing constant action across agents.

These claims are validated via proof and example, notably with the smoking lesion problem—where PDT outperforms CDT whenever utility maximization requires individualized (heterogeneous) policy.

Application to Newcomb's Problem

Newcomb's problem serves as a particularly incisive test case due to the complexity of counterfactual dependencies involved. The proposed NPSEM precisely encodes the roles of the agent's act ($E(U(a); M)$3), the predictor's forecast ($E(U(a); M)$4), the agent’s characteristics ($E(U(a); M)$5), and the agent’s adopted decision theory ($E(U(a); M)$6). The model addresses feedback between the decision theory and the predictor via a direct arrow from $E(U(a); M)$7 to $E(U(a); M)$8.

Under the intervention-based performance metric, the analysis reveals that the optimality of a decision theory (EDT, CDT, PDT) pivots on the structural effect of $E(U(a); M)$9 on $D$0—i.e., the extent to which choice of decision procedure is causally accessible to the predictor. If the decision theory has substantial influence on the predictor's accuracy, EDT may outperform CDT and PDT. If not, CDT and PDT are optimal and behave identically in this context. The analysis, made formal by the NPSEM, rigorously clarifies ambiguous informal debates and demonstrates under what parametric regimes exotic decision-theoretic phenomena arise.

Theoretical and Practical Implications

This work delivers substantial clarifications and provides a rigorous language for discussing agents, their beliefs, underlying causal mechanisms, and the evaluation of decision algorithms. By making explicit the distinction between subjective and objective components and introducing decision theories as endogenous variables, the paper resolves longstanding confusions regarding agency, evidence, and causality in the formal literature.

Practically, the NPSEM framework is positioned to support automated reasoning, mechanized evaluation of decision theories, and application to domains—policy, economics, AI agent design—where effect heterogeneity or complex structural dependencies are present. The personalized, interventionist approach of PDT is especially salient for settings in which maximizing average utility is insufficient or inappropriate due to agent-level diversity.

Theoretically, the methods suggest clear directions for extending decision theory into high-dimensional, dynamic, or agent-interactive regimes, presenting an avenue for convergence between statistical causal inference and philosophical theories of rationality.

Conclusion

By supplying decision theory with a precise, causal modeling foundation—NPSEMs—and formalizing the evaluative, agency, and subjective elements that arise in utility-driven choice, this work enables more incisive analysis and comparison of decision rules. The introduction of personal decision theory articulates the unique optimality of individualized utility maximization under plausible structural assumptions, while the rigorous handling of nontrivial problems such as Newcomb’s offers a template for the analysis of further decision-theoretic puzzles.

Adoption of the proposed framework promises to elevate conceptual precision in the study of rational action, causal reasoning, and agent modeling, and to underpin richer future generalizations in theory and practice.