Benedikt Koch

Benedikt Koch, Kosuke Imai, Tomasz Strzalecki

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.

Benedikt Koch, Kosuke Imai

Many researchers have applied classical statistical decision theory to evaluate treatment choices and learn optimal policies. However, because this framework is based solely on realized outcomes under chosen decisions and ignores counterfactual outcomes, it cannot assess the quality of a decision relative to feasible alternatives. For example, in bail decisions, a judge must consider not only crime prevention but also the avoidance of unnecessary burdens on arrestees. To address this limitation, we generalize standard decision theory by incorporating counterfactual losses, allowing decisions to be evaluated using all potential outcomes. The central challenge in this counterfactual statistical decision framework is identification: since only one potential outcome is observed for each unit, the associated counterfactual risk is generally not identifiable. We prove that, under the assumption of strong ignorability, the counterfactual risk is identifiable if and only if the counterfactual loss function is additive in the potential outcomes. Moreover, we demonstrate that additive counterfactual losses can yield treatment recommendations, which differ from those based on standard loss functions when the decision problem involves more than two treatment options. One interpretation of this result is that additive counterfactual losses can capture the accuracy and difficulty of a decision, whereas standard losses account for accuracy alone. Finally, we formulate a symbolic linear inverse program that, given a counterfactual loss, determines whether its risk is identifiable, without requiring data.