Personal Decision Theory (PDT) Framework

• Personal Decision Theory (PDT) is a framework that personalizes decision-making under uncertainty using each agent’s subjective counterfactual utilities and NPSEM modeling.
• It leverages both experimental and observational data to derive tight bounds on individual treatment effects for optimized, tailored actions.
• PDT distinguishes itself from evidential and causal decision theories by prescribing individualized decision rules based on an agent’s unique exogenous variables.

Personal Decision Theory (PDT) is a framework for modeling and optimizing the choices of agents under uncertainty, based on each agent’s subjective counterfactual utility. Distinguished from both population-level and purely conditional approaches, PDT leverages modern causal modeling to provide rigorous individual-level prescriptions and clearly specified performance criteria. PDT has strong connections to the formalism of nonparametric structural equation models (NPSEMs) and offers sharp conceptual and mathematical distinctions from both evidential and causal decision theories.

1. Formal Preliminaries and Modeling Framework

PDT is naturally formulated within the NPSEM framework. An NPSEM consists of endogenous random variables $V_1,\ldots,V_k$ and exogenous (noise) variables $\epsilon_1,\ldots,\epsilon_k$ with a joint density $p(\epsilon)$, such that each $V_i$ is determined by a structural function $f_i$ of its parent variables and exogenous noise:

$V_i := f_i(\mathrm{pa}_i, \epsilon_i)$

An agent is specified by a particular instantiation $$\epsilon = (\epsilon_1,\ldots,\epsilon_k)\sim p(\epsilon)$$. Actions ($A$) and outcomes/utilities ($U$) are defined as particular variables in the model. Counterfactuals are obtained via interventions, e.g., the variable $U(a, \epsilon)$ denotes the agent’s utility if $\epsilon_1,\ldots,\epsilon_k$0 were externally set to $\epsilon_1,\ldots,\epsilon_k$1.

Decisions can be modeled in terms of a variable $\epsilon_1,\ldots,\epsilon_k$2 indicating the agent’s adopted decision theory, and $\epsilon_1,\ldots,\epsilon_k$3 representing the subjective model. Consistency holds: if $\epsilon_1,\ldots,\epsilon_k$4 for an agent, then $\epsilon_1,\ldots,\epsilon_k$5.

2. Definitions: Personalized vs. Population-Level Decision Rules

Personalized decision making targets the specific individual $\epsilon_1,\ldots,\epsilon_k$6 or type $\epsilon_1,\ldots,\epsilon_k$7. For an individual $\epsilon_1,\ldots,\epsilon_k$8:

• $\epsilon_1,\ldots,\epsilon_k$9: outcome for $p(\epsilon)$0 under treatment
• $p(\epsilon)$1: outcome for $p(\epsilon)$2 under control
• $p(\epsilon)$3 (Individual Treatment Effect)

Sub-population or population-based models operate via the conditional average treatment effect (CATE):

$p(\epsilon)$4

where $p(\epsilon)$5 are observed covariates. Population rules act based on $p(\epsilon)$6; personalized rules seek to optimize $p(\epsilon)$7 directly (Mueller et al., 2022).

3. Decision Rules in PDT, EDT, and CDT

The behavioral prescription of each theory can be formalized as follows, given a subjective model $p(\epsilon)$8:

Evidential Decision Theory (EDT)

$p(\epsilon)$9

EDT agents act to maximize conditional expected utility given the act.

Causal Decision Theory (CDT)

$V_i$0

CDT agents act to maximize expected utility under intervention.

Personal Decision Theory (PDT)

Each agent $V_i$1 acts by

$V_i$2

or equivalently,

$V_i$3

PDT tailors the action to the agent’s realization $V_i$4, leveraging her full subjective counterfactual model (Sjölander, 29 Jun 2026).

4. Causal Identification and Theoretical Guarantees

PDT’s key assumptions include:

• Consistency: $V_i$5, and $V_i$6, etc.
• RCT ideality: no unmeasured confounding, perfect compliance, no selection bias.
• Observational data may contain confounding, which can be exploited for individual-level variation.

For binary outcomes, define:

• $V_i$7 (Probability of Necessary and Sufficient causation: benefit)
• $V_i$8 (harm)
• $V_i$9

PDT’s optimal rule: treat if and only if $f_i$0 (Mueller et al., 2022).

Tian–Pearl bounds on $f_i$1 are given by:

$f_i$2

where $f_i$3 are RCT rates; $f_i$4 are observational frequencies. Analogous bounds apply for $f_i$5 (Mueller et al., 2022).

5. Performance Metrics and Optimality Criteria

A central contribution is the formalization of a performance metric based on hypothetical enforced-policy interventions. If a planner can force all agents to follow a fixed decision theory $f_i$6, the performance is:

$f_i$7

Under two assumptions—correct subjective models and no direct $f_i$8 effect—PDT is strictly optimal:

$f_i$9

for any decision theory $V_i := f_i(\mathrm{pa}_i, \epsilon_i)$0. Under the same assumptions, both EDT and CDT yield constant population actions (do not adapt to $V_i := f_i(\mathrm{pa}_i, \epsilon_i)$1), and CDT weakly dominates all constant-act theories (Sjölander, 29 Jun 2026).

6. Mixed Data: Experimental and Observational Integration

PDT leverages both experimental (RCT) and observational data. Observational confounding, rather than being dismissed as a nuisance, is used to sharpen or bound individual causal effects. Estimation proceeds by combining RCT arm rates with stratified observational counts, providing tight bounds or even point identification of $V_i := f_i(\mathrm{pa}_i, \epsilon_i)$2 and $V_i := f_i(\mathrm{pa}_i, \epsilon_i)$3. When $V_i := f_i(\mathrm{pa}_i, \epsilon_i)$4 is not point-identified, action selection may be “definite” or “ambiguous” based on the sign of the lower and upper bounds. This approach enables feature (covariate) discovery: if current covariates do not explain individual heterogeneity, further markers can be sought to sharpen individualized rules (Mueller et al., 2022).

7. Illustrative Problems and Practical Implications

Canonical examples include the smoking lesion problem and Newcomb’s problem. In the smoking lesion problem, EDT misleads due to conditioning on actions intertwined with unobserved risk factors; CDT yields the welfare-maximizing constant act; PDT allows for truly individualized action based on each agent’s realized $V_i := f_i(\mathrm{pa}_i, \epsilon_i)$5, strictly increasing aggregate utility if $V_i := f_i(\mathrm{pa}_i, \epsilon_i)$6 is heterogeneous.

For clinical applications, personalized rules differ sharply from population-based approaches. For instance, the Number Needed to Treat (NNT) is correctly $V_i := f_i(\mathrm{pa}_i, \epsilon_i)$7, not $V_i := f_i(\mathrm{pa}_i, \epsilon_i)$8. In decision support systems for precision medicine, finance, and other domains, the PDT framework sustains utility-based, risk-sensitive, and constraint-aware synthesis of policy, grounded in causal identification theory (Mueller et al., 2022).