An Axiomatic Foundation for Decisions with Counterfactual Utility

Abstract: Counterfactual utilities evaluate decisions not only by the realized outcome under a given decision, but also by the counterfactual outcomes that would arise under alternative decisions. By generalizing standard utility frameworks, they allow decision-makers to encode asymmetric criteria, such as avoiding harm and anticipating regret. Recent work, however, has raised fundamental concerns about the coherence and transitivity of counterfactual utilities. We address these concerns by extending the von Neumann-Morgenstern (vNM) framework to preferences defined on the extended space of all potential outcomes rather than realized outcomes alone. We show that expected counterfactual utility satisfies the vNM axioms on this extended domain, thereby admitting a coherent preference representation. We further examine how counterfactual preferences map onto the realized outcome space through menu-dependent and context-dependent projections. This axiomatic framework reconciles apparent inconsistencies highlighted by the Russian roulette example in the statistics literature and resolves the well-known Allais paradox from behavioral economics. We also derive an additional axiom required to reduce counterfactual utilities to standard utilities on the same potential outcome space, and establish an axiomatic foundation for additive counterfactual utilities, which satisfy a necessary and sufficient condition for point identification. Finally, we show that our results hold regardless of whether individual potential outcomes are deterministic or stochastic.

• The paper establishes that expected counterfactual utility maximization is a coherent extension of the vNM utility theory.
• It formalizes decision frameworks where utilities depend on the full vector of potential outcomes and tackles identification issues, particularly for additive utilities.
• The approach reconciles ethical and behavioral considerations with standard decision theory by emphasizing the proper evaluative domain and context dependence.

Axiomatic Decision Theory with Counterfactual Utilities

Introduction and Motivation

This paper articulates a formal axiomatization for decision-making frameworks in which preferences are defined over counterfactual utility, namely, utility functions that depend on the full vector of potential outcomes rather than realized outcomes alone. Classic frameworks, such as the von Neumann–Morgenstern (vNM) expected utility theory, typically posit preferences over lotteries of realized outcomes. Counterfactual utilities arise naturally in settings—particularly in causal inference and medical or policy decision-making—where agents wish to incorporate asymmetric or ethically-driven considerations (e.g., aversion to harm, regret, or the Hippocratic principle) that reference potential, unrealized consequences.

Recent statistical and philosophical literature has questioned the coherence of such frameworks, noting apparent violations of transitivity or independence (e.g., money pumps, preference cycles) when utility functions depend jointly on actual and counterfactual outcomes. This paper systematically addresses these criticisms and clarifies the internal consistency of counterfactual utility maximization, showing that incoherence only arises from improper identification of the domain of preference—realized versus potential outcome spaces.

Formal Setup and Main Results

Extension of the vNM Framework

The authors formalize decision problems where the agent chooses $d \in D$ for each unit, knowing that the potential outcomes $(Y(0), ..., Y(K-1))$ and covariates $X$ exist—even though only $Y(d)$ is ever observed. They define utility functions $\tilde u(d; y_0, ..., y_{K-1}, x)$ on the extended space $Z = D \times Y^{D} \times X$, with policies $\pi$ that can condition on the entire vector of potential outcomes (oracle policies), but which in practice typically depend only on observed covariates.

Decisions are evaluated by their expected counterfactual utility, $$V_P(\pi; \tilde u) = E_{P^\pi}[\tilde u(D; Y(0), ..., Y(K-1), X)]$$, where $P$ is the (unknown) state of nature—i.e., a joint distribution over potential outcomes and covariates.

Axioms for Counterfactual Preferences

The key innovation is the direct application of vNM axioms—completeness, transitivity, independence, and continuity—to probability measures over the full potential outcome space $Z$ rather than the space of observed outcomes. This crucial domain choice implies that expected counterfactual utility always induces a coherent, weakly-ordered, transitive, and independent preference relation on $(Y(0), ..., Y(K-1))$0; the classic representation theorem applies without modification.

Contradictory Claims and Resolution

The paper directly refutes previous claims of incoherence or intransitivity, tracing them to projections onto the realized outcome space (i.e., considering only observed outcomes and ignoring the dependence of utility on unrealized potential outcomes). The distinction between preferences on $(Y(0), ..., Y(K-1))$1 and on realized outcomes is analytically central. The authors show that the vNM axioms are satisfied on $(Y(0), ..., Y(K-1))$2, but not generally when projecting induced preferences onto realized outcomes.

Relation to Standard Utilities and Additional Axioms

Standard (realized-outcome-only) utilities are shown to be a special case, characterized by the Irrelevance of Counterfactual Outcomes axiom: preferences between distributions are invariant to the configuration of the unchosen potential outcomes. Conversely, when this axiom fails, explicit concern for counterfactual outcomes is present and can be coherently handled within the extended framework.

The paper introduces a further characterization of additive counterfactual utilities, wherein the expectation of the utility is point identified from observed data even under general randomization schemes. Additive utilities depend only on the marginals of the potential outcomes, and a corresponding Irrelevance of Counterfactual Correlation axiom completes the representation. The additive structure is both necessary and sufficient for identifiability without additional information. Only additive counterfactual utilities respect contextual irrelevance of the correlation among potential outcomes; more general counterfactual utilities may encode preferences depending on this dependence.

Consequences, Connections, and Behavioral Phenomena

Russian Roulette and the Hippocratic Principle

By examining the so-called "Russian roulette" critique, the authors establish that the apparent paradox—where an asymmetric utility (penalizing harm more than failure to benefit) leads to apparently suboptimal recommendations—simply reflects genuine, context-dependent preferences and does not imply irrationality. The formalism accommodates such asymmetries without logical contradiction.

Behavioral Economics: The Allais Paradox

Extending the approach to the Allais paradox, the authors show that counterfactual utilities can account for behavioral patterns traditionally viewed as violations of expected utility theory. For example, regret theory (where foregone outcomes influence realized utility) arises as a special case, and the generalized axiomatic framework subsumes such models while recovering transitivity when preferences are defined over the full potential outcome space.

Identification and Practical Implementation

The practical challenge with general counterfactual utilities is point identification: since only one component of the potential outcome vector is observed per unit, non-additive counterfactual utilities have unidentified expectations, and only partial bounds (via e.g. minimax or partial identification frameworks) are available.

Additive counterfactual utilities—those linear in the potential outcomes—are shown to be the maximal class for which the expected utility is identified from the observed marginals (in the absence of further structural assumptions or auxiliary data). This establishes precise, sharp limits on the applicability of counterfactual utility maximization for statistical decision-making in empirical contexts.

Mapping Preferences onto Realized Outcomes

The paper analyzes the projection of counterfactual utility-induced preferences onto the realized outcome space, distinguishing two regimes:

• Menu-Dependent Projection: Each decision scenario is treated with a possibly context-specific utility function and embedding, capturing menu effects and replicating models like regret theory. In this regime, revealed preference cycles and violations of Sen’s $(Y(0), ..., Y(K-1))$3 and $(Y(0), ..., Y(K-1))$4 conditions (WARP) can—and do—arise, reflecting the behavioral findings in the literature.
• Context-Dependent Projection: A fixed utility is used over the entire context; thus, while preferences remain transitive and compliant with rationality requirements over the finite set of lotteries, context dependence (analogous to menu dependence) can persist at the pairwise comparison level.

Notably, transitivity and vNM axioms are always satisfied on the extended space, even if they may fail after projection onto realized outcome space—explaining observed deviations in empirical choice data without logical contradiction at the axiomatic level.

Stochastic Potential Outcomes and Order of Evaluation

Responding to recent proposals (e.g., [Gelman_2025]) for incorporating stochastic potential outcomes—evaluating utility after integrating out potential outcome distributions—the authors prove that this does not fundamentally alter the axiomatic conclusions. The extension of the utility function to operate on expected values of potential outcomes is not unique; it may in fact collapse the preference framework to standard utilities, stripping away the asymmetric features that motivated counterfactual specifications. Furthermore, such extended utilities may violate the vNM independence axiom, unless the original utility is of additive form.

Implications and Future Directions

The results have substantial implications for the foundations of statistical decision theory with causal inputs:

• Subjectivity and Ethical Pluralism: The appropriate definition of utility—i.e., whether and how counterfactual outcomes enter—is ultimately an ethical and normative choice for the decision-maker, not the statistician. The framework validates the subjective and context-sensitive encoding of harm, regret, and other considerations.
• Limits of Identification: The sharp characterization of point identification for additive counterfactual utilities sets clear boundaries for empirical analysis and will guide future methodological advances aimed at learning optimal policies from finite data.
• Behavioral Decision Models: The embrace of menu and context dependence melds the formal utility-theoretic approach with the emphases of behavioral economics, offering tools for reconciling descriptive and normative paradigms in decision science.

Conclusion

The paper rigorously establishes that expected counterfactual utility maximization constitutes a coherent, axiomatizable theory of decision-making, provided preferences are correctly situated on the space of potential outcomes. Previous criticisms about incoherence or intransitivity are resolved via a formal understanding of the evaluative domain. Extensions to behavioral models, applications in high-stakes policy, and clarifications on the role of identification ground the work in both theory and practice. This foundation will inform both the ethical design of decision support tools (AI or otherwise) and the interpretative frameworks within causal inference and policy analysis.

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Reference:

"An Axiomatic Foundation for Decisions with Counterfactual Utility" (2605.05521)