Functional Decision Theory

• Functional Decision Theory is a normative framework that intervenes on an agent’s decision function to maximize expected utility, accounting for both causal and logical dependencies.
• FDT effectively resolves classical dilemmas like Newcomb’s Problem and Parfit’s Hitchhiker by strategically adjusting the decision policy rather than merely the physical act.
• Empirical and simulation studies demonstrate that FDT outperforms CDT and EDT in game-theoretic and evolutionary settings through superior coordination and predictive accuracy.

Functional Decision Theory (FDT) is a normative framework for instrumental rationality that prescribes maximizing expected utility by intervening on the agent's decision function itself, rather than on the physical act or by conditioning on one's action. FDT is designed to account for both logical and causal dependencies, enabling agents to consistently achieve high utility in decision-theoretic and game-theoretic problems that have proven problematic for both Causal Decision Theory (CDT) and Evidential Decision Theory (EDT). FDT's core innovation is to treat the agent's policy as a fundamental node in a mechanized causal or logical model and to compute expected utility by evaluating counterfactuals corresponding to different outputs of this policy function (Yudkowsky et al., 2017, MacDermott et al., 2023, Topper, 2020).

1. Theoretical Motivation and Distinction from CDT/EDT

FDT was introduced to resolve the canonical failures of CDT and EDT in classic dilemmas such as Newcomb’s Problem, the Smoking Lesion, and Parfit’s Hitchhiker (Yudkowsky et al., 2017). CDT prescribes choosing actions based on their direct, physical effects by performing Pearl-style "do"-interventions on the action node, which fails in decision scenarios involving logical correlations (e.g., predictors or perfect copies). EDT chooses actions by conditioning on the action and updating beliefs accordingly, which leads to spurious correlations and “managing the news,” and fails in cases like the Smoking Lesion problem.

FDT takes a fixed-function view: the agent models her own decision procedure as a mathematical function $f: S \to A$ mapping input states to actions. She then seeks the output $a^*$ of $f$ such that the world in which this function outputs $a^*$ yields maximal expected utility. The key formalism is:

$$a^* = \arg\max_{a \in A} \mathbb{E}[U(s, f(s) = a)]$$

where the expectation is taken over counterfactual worlds where the "decision function" outputs $a$. The crucial distinguishing feature is that the agent's own source code (decision procedure) is treated as a variable capable of logical dependencies with other parts of the environment or other agents (Yudkowsky et al., 2017, MacDermott et al., 2023).

2. Formal Structure and Mechanized Causal Models

FDT is most explicitly defined in the language of mechanized causal Bayesian networks (MCBN), where:

• There is an object-level decision node $D$ and a mechanism (policy) node $\tilde{D}$, such that $D = \delta(\tilde{D})$
• Predictor mechanisms ($\tilde{P}$), predictor outputs ($a^*$0), and utility nodes ($a^*$1) are included
• Logical-causal edges encode that predictors can access the agent's source code or that twin agents compute using the same procedure

The FDT agent selects among policies $a^*$2 by maximizing:

$a^*$3

The intervention $a^*$4 severs any incoming arrows into the policy node but leaves outgoing logical dependencies intact (e.g., $a^*$5 remains), allowing predictors or twins to remain logically coupled to the agent's choice. FDT thus evaluates the expected consequences of different policy outputs under this logical-structural intervention (MacDermott et al., 2023).

3. Comparative Taxonomy of Decision Theories

FDT is taxonomically situated alongside EDT and CDT within a classification based on two axes: (A) the type of dependency (Evidential, Causal, Functional) and (B) whether the agent is updateful (conditions on observations) or updateless (does not condition on observations).

Theory	Dependency Type	Conditioning	Utility Computed As
Updateful EDT	Evidential	Obs_D	$a^*$6
Updateful CDT	Causal	Obs_D	$a^*$7
Updateful FDT	Functional	Obs_D	(rare; logical-causal, observed)
Updateless EDT	Evidential	Policy	$a^*$8
Updateless CDT	Causal	Policy (do)	$a^*$9
Updateless FDT	Functional	Policy (do)	$f$0

Classic FDT corresponds to the updateless, functional cell of this taxonomy. It differs from EDT by not simply updating on the agent's action and from CDT by intervening on the policy node in a logical, rather than merely physical, causal model (MacDermott et al., 2023).

4. Canonical Examples and Prescriptions

FDT is explicitly designed to outperform CDT and EDT in structured dilemmas characterized by logical correlation. The following examples are illustrative (Yudkowsky et al., 2017, MacDermott et al., 2023, Topper, 2020):

Newcomb’s Problem:

• Predictor fills Box B iff predicts one-boxing; 99% predictor accuracy.
• FDT computes expected utilities under two counterfactuals:
  • $f$1
  • $f$2
• Since $f$3, FDT prescribes one-boxing.

Smoking Lesion:

• Lesion causes both craving for smoking and cancer; smoking does not causally affect cancer.
• FDT intervenes on its decision node; since the cancer node does not depend on the decision function, it prescribes to smoke, achieving higher expected utility than EDT.

Parfit’s Hitchhiker:

• Survival depends on predictor's expectation that agent will pay if rescued.
• FDT, by treating the "pay" action as determined by its decision function (logically mirrored in the predictor’s simulation), prescribes to pay upon rescue, ensuring survival.

Prisoner's Dilemma with Twins:

• With a perfect copy (twin), FDT propagates the intervention to the twin; both cooperate, yielding higher payoffs compared to CDT (which defects against its clone).

Stag Hunt:

• In a population of FDT agents, logical coupling ensures Pareto-optimal coordination on Stag, outperforming CDT in similar settings.

5. Algorithmic Skeleton and Implementation Notes

FDT can be represented algorithmically by iterating through all possible outputs of the decision function, constructing the corresponding counterfactual world for each, and computing the expected utility:

$f$5

Implementation considerations include encoding the logical ("quine") node for the decision function and accurately modeling subjunctive/causal graphs involving logical dependencies and predictors (Yudkowsky et al., 2017, MacDermott et al., 2023).

6. Evolutionary and Game-Theoretic Performance

Simulation studies embedding FDT, CDT, and EDT agents in evolutionary environments confirm that FDT agents outperform rivals in mixed populations. In repeated games like PD and Stag Hunt, FDT invades and dominates under replicator dynamics whenever logical coupling ("twin correlation") and predictor accuracy exceed 0.5, with convergence rates above 95% prevalence for FDT in typical parameter settings. FDT achieves superior coordination and maximal expected utility in symmetric and predictor-driven dilemmas (Topper, 2020).

Key theoretical results:

• In populations with twin-correlation $f$4, FDT strictly outperforms CDT and EDT in PD.
• In Newcomb/Parfit scenarios, FDT one-boxes or pays when predictor accuracy exceeds 0.5.
• FDT consistently coordinates on payoff-dominant equilibria in symmetric coordination games, unlike CDT (Topper, 2020).

7. Limitations and Open Problems

While FDT is normatively and empirically robust across a wide class of dilemmas, several limitations and unresolved issues remain (Yudkowsky et al., 2017):

• Modeling and learning subjunctive (logical/counterpossible) dependencies requires a richer formalism than standard causal graphs.
• A fully satisfactory theory of logical counterfactuals is needed to formalize how counterpossibles are evaluated in practice.
• The computational tractability of self-referential interventions remains nontrivial.
• Open questions include learning subjunctive structure from data, representation theorems for FDT, the existence of “fair” dilemmas where FDT might underperform, and the correct bounded-rationality approximations for real-world agents.

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Functional Decision Theory upgrades expected-utility maximization by shifting the locus of intervention from physical actions to the outputs of fixed decision functions. In doing so, it remedies the canonical failures of CDT and EDT in Newcomb-type and related dilemmas, and demonstrates evolutionary robustness and game-theoretic superiority in simulation and analytic settings (Yudkowsky et al., 2017, MacDermott et al., 2023, Topper, 2020).