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An Axiomatic Foundation for Decisions with Counterfactual Utility
Pith reviewed 2026-05-08 15:14 UTC · model grok-4.3
The pith
Expected counterfactual utility satisfies the von Neumann-Morgenstern axioms when preferences are defined over all potential outcomes rather than realized ones alone.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
By defining preferences directly on the extended domain of all potential outcomes, expected counterfactual utility satisfies the von Neumann-Morgenstern axioms and therefore admits a coherent preference representation. Menu-dependent and context-dependent projections then connect these preferences back to the space of realized outcomes. This framework reconciles inconsistencies highlighted by the Russian roulette example and resolves the Allais paradox. An additional axiom is derived that is necessary and sufficient to reduce counterfactual utilities to standard utilities on the same potential outcome space, and an axiomatic foundation is given for additive counterfactual utilities, which in
What carries the argument
The extension of the von Neumann-Morgenstern preference domain to the space of all potential outcomes under every possible decision, together with the menu-dependent and context-dependent projections that recover preferences over realized outcomes.
If this is right
- Counterfactual preferences admit representation as maximization of expected counterfactual utility without violating the standard rationality axioms.
- Inconsistencies previously noted in the Russian roulette example are reconciled by the larger preference domain.
- The Allais paradox is resolved because the counterfactual framework permits consistent representations of the observed choice patterns.
- An additional axiom is required and sufficient to reduce counterfactual utilities to ordinary utilities defined on the same potential outcome space.
- Additive counterfactual utilities satisfy a necessary and sufficient condition for point identification and the results continue to hold when potential outcomes are stochastic.
Where Pith is reading between the lines
- If the claim is correct, standard economic models could incorporate regret or harm-avoidance terms directly while retaining transitivity and other consistency requirements.
- The projections imply that the same underlying counterfactual preference may appear different depending on the menu or context in which realized outcomes are observed.
- Further analysis could test whether the additional reduction axiom holds in repeated choice settings or when agents receive partial information about counterfactuals.
Load-bearing premise
That the von Neumann-Morgenstern axioms can be imposed directly on the extended space of potential outcomes without introducing new inconsistencies that would prevent coherent projections onto realized outcomes.
What would settle it
A concrete set of choices over potential outcomes that satisfies the extended axioms yet produces intransitive or inconsistent preferences when projected to the realized outcome space, or empirical patterns in Allais-style lotteries that cannot be recovered from any menu-dependent projection of counterfactual utilities.
read the original 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.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper extends the von Neumann-Morgenstern (vNM) expected-utility framework from realized outcomes to the larger space of all potential outcomes (including counterfactuals under every alternative decision). It asserts that a preference relation ≽ on this extended domain satisfies the standard vNM axioms, yielding a representation by expected counterfactual utility. Menu-dependent and context-dependent projection operators are then defined to recover rankings over realized outcomes; the framework is claimed to resolve the Russian-roulette inconsistency and the Allais paradox. Additional results include an axiom that reduces counterfactual utilities to ordinary vNM utilities on the same space and an axiomatic characterization of additive counterfactual utilities that are point-identified. The results are stated to hold whether potential outcomes are deterministic or stochastic.
Significance. If the technical claims are correct, the work supplies a parameter-free axiomatic route to incorporating regret, harm avoidance, and other counterfactual considerations inside an expected-utility representation. The absence of ad-hoc parameters and the explicit treatment of both deterministic and stochastic cases are strengths. The attempted reconciliation of the Allais paradox and the Russian-roulette example would be of interest to decision theorists and behavioral economists if the projection step is shown to preserve the required properties.
major comments (2)
- [§4] §4 (menu-dependent projections): the claim that the induced preference relation on realized lotteries remains transitive and satisfies independence is not established. Because the projection operators are explicitly allowed to depend on the menu and on the set of counterfactuals, the same realized outcome can receive different rankings when it appears in two menus whose counterfactual sets differ. No argument is given that this menu-dependence cannot produce cycles or violations of independence on the realized domain, which is load-bearing for the assertion that the framework delivers a coherent preference representation.
- [§3] §3 (extended domain and vNM axioms): while the text asserts that completeness, transitivity, continuity, and independence hold on the space of all potential outcomes, the verification that the definition of counterfactual utility does not introduce new inconsistencies (for example, with the continuity axiom when counterfactuals are stochastic) is not supplied in sufficient detail to confirm that the representation theorem applies without additional restrictions.
minor comments (2)
- [§2] The notation for potential outcomes versus realized outcomes should be introduced with a single, self-contained definition early in the paper rather than piecemeal.
- [§5] The additional axiom that reduces counterfactual utilities to standard utilities is stated informally; a formal statement with the exact mathematical condition would improve readability.
Simulated Author's Rebuttal
We thank the referee for the careful reading and constructive comments. The points raised identify places where the manuscript would benefit from additional formal detail. We address each major comment below and will incorporate the necessary revisions.
read point-by-point responses
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Referee: [§4] §4 (menu-dependent projections): the claim that the induced preference relation on realized lotteries remains transitive and satisfies independence is not established. Because the projection operators are explicitly allowed to depend on the menu and on the set of counterfactuals, the same realized outcome can receive different rankings when it appears in two menus whose counterfactual sets differ. No argument is given that this menu-dependence cannot produce cycles or violations of independence on the realized domain, which is load-bearing for the assertion that the framework delivers a coherent preference representation.
Authors: We agree that the current text asserts preservation of transitivity and independence on the realized domain but does not supply a complete formal argument. In the revision we will add a dedicated lemma and proof in §4 showing that the menu-dependent projections cannot induce cycles or independence violations. The argument proceeds by contradiction: any such violation on the realized domain would lift to a violation of the vNM axioms on the extended space of potential outcomes, contradicting the maintained assumption that the preference satisfies those axioms there. We will also note that while projections may differ across menus, they are required to be consistent with the fixed underlying ranking on the extended domain. revision: yes
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Referee: [§3] §3 (extended domain and vNM axioms): while the text asserts that completeness, transitivity, continuity, and independence hold on the space of all potential outcomes, the verification that the definition of counterfactual utility does not introduce new inconsistencies (for example, with the continuity axiom when counterfactuals are stochastic) is not supplied in sufficient detail to confirm that the representation theorem applies without additional restrictions.
Authors: We acknowledge that the verification of the vNM axioms, particularly continuity under stochastic counterfactuals, is stated at a high level rather than derived in full detail. The revised manuscript will include an expanded appendix that explicitly checks each axiom, with a separate subsection confirming that continuity is preserved when potential outcomes are stochastic. This will establish that the representation theorem applies directly on the extended domain without further restrictions. revision: yes
Circularity Check
No significant circularity; standard axiomatic extension to extended domain
full rationale
The paper defines an extended preference domain over all potential outcomes (including counterfactuals), imposes the standard vNM axioms directly on that domain, and derives the expected counterfactual utility representation as a consequence. This is a direct application of the vNM theorem to a larger set; the representation theorem holds by the same proof structure without any reduction of target quantities to fitted parameters, self-defined quantities, or prior self-citations. Menu- and context-dependent projection operators are subsequently defined as mappings from the extended preferences back to realized outcomes; these are constructive definitions rather than tautological re-labelings of inputs. No equations equate a 'prediction' to a fitted value by construction, and no uniqueness theorem or ansatz is imported from the authors' own prior work as a load-bearing step. The framework is self-contained against external benchmarks (the classical vNM axioms), yielding an independent axiomatic foundation.
Axiom & Free-Parameter Ledger
axioms (3)
- domain assumption Preferences are defined on the extended space of all potential outcomes rather than realized outcomes alone
- domain assumption The vNM axioms hold on this extended domain
- domain assumption Counterfactual preferences map onto realized outcomes via menu-dependent and context-dependent projections
Reference graph
Works this paper leans on
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[1]
Moreover, the utility functionuis unique up to positive affine transformation
there exists an outcome utilityu(d;y,x) =u(y d)that represents≿in the sense of Definition 1. Moreover, the utility functionuis unique up to positive affine transformation. S2 Formal Results for Section 6.4 We formalize and generalize the discussion in Section 6.4 on Gelman and Mikhaeil (2025)’s proposal to extend counterfactual utilities by switching the ...
2025
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[2]
Assumption 2 (Bounded Means)There existL < Usuch thatp d ∈I:= [L, U]ford∈ {0,1}
We impose the following assumptions. Assumption 2 (Bounded Means)There existL < Usuch thatp d ∈I:= [L, U]ford∈ {0,1}. Assumption 3 (Decision Monotonicity)For every fixedp 0 ∈I, the mapp 1 →Γ˜uExt(p0, p1)is non-decreasing onI. Similarly, for every fixedp 1 ∈I, the mapp 0 →Γ˜uExt(p0, p1)is non-increasing onI. Assumption 4 (Unique Crossing)Fixp 0 ∈I. (a) the...
2025
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[3]
We will verify Axiom 5
Since any standard utility is also a special case of counterfactual utilities onZ, Theorem 1 implies that the induced preference relation satisfies Axioms 1–4. We will verify Axiom 5. Let P d, Qd ∈∆(Z) satisfy the premise of Axiom 5 with point mass atD=dandX=x, and supposeP(Y(d) =y) =Q(Y(d) =y) for ally∈ Y. Then,PandQare degenerate inX, i.e., P(y,x ′) =P(...
1970
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[4]
Then, by definition Γ˜u(p ′ 0, p1)<0 for allp 1 ∈I
=U+ 1. Then, by definition Γ˜u(p ′ 0, p1)<0 for allp 1 ∈I. By Assumption 3, p0 →Γ˜u(p0, p1) is non-increasing, implying Γ˜u(p′′ 0, p1)≤Γ˜u(p′ 0, p1)<0. Hence,ϕ 0(p′′
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[5]
Sinceϕ 0(p0)≥L−1 for allp 0 ∈I, it follows thatϕ 0(p′ 0)≤ϕ 0(p′′ 0)
=L−1. Sinceϕ 0(p0)≥L−1 for allp 0 ∈I, it follows thatϕ 0(p′ 0)≤ϕ 0(p′′ 0)
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[6]
Ifϕ 0(p′′
Supposeϕ 0(p′ 0)∈[L, U], ϕ 0(p′′ 0)∈ {L−1, U+ 1}. Ifϕ 0(p′′
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[7]
Now supposeϕ 0(p′′
=U+ 1 then triviallyϕ 0(p′ 0)≤ ϕ0(p′′ 0). Now supposeϕ 0(p′′
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[8]
By Assumption 3 andp ′ 0 < p ′′ 0, we must have Γ˜u(p′ 0, p1)≥Γ˜u(p ′′ 0, p1)>0
=L−1, then by definition Γ˜u(p ′′ 0, p1)>0 for allp 1 ∈I. By Assumption 3 andp ′ 0 < p ′′ 0, we must have Γ˜u(p′ 0, p1)≥Γ˜u(p ′′ 0, p1)>0. Hence, Γ˜u(p ′ 0, p1)>0 for allp 1 ∈Iwhich impliesϕ 0(p′
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[9]
=L−1, contradictingϕ 0(p′ 0)∈[L, U]
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[10]
Note that sinceϕ 0(p′ 0), ϕ0(p′′ 0)∈[L, U], we have Γ˜u(p′ 0, ϕ0(p′ 0)) = 0 and Γ˜u(p′′ 0, ϕ0(p′′ 0)) = 0
andϕ 0(p′′ 0)∈[L, U]. Note that sinceϕ 0(p′ 0), ϕ0(p′′ 0)∈[L, U], we have Γ˜u(p′ 0, ϕ0(p′ 0)) = 0 and Γ˜u(p′′ 0, ϕ0(p′′ 0)) = 0. Suppose for contradiction thatϕ 0(p′ 0)> ϕ 0(p′′ 0). Assumption 3 fur- ther implies 0 = Γ˜u(p′′ 0, ϕ0(p′′ 0))≤Γ˜u(p′′ 0, ϕ0(p′ 0))≤Γ˜u(p′ 0, ϕ0(p′ 0)) = 0 such that Γ˜u(p′′ 0, ϕ0(p′ 0)) = 0. By Assumption 4(a) this imposesϕ 0(p′
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[11]
This proves thatϕ 0 is non-decreasing
=ϕ 0(p′′ 0), yielding a contradiction. This proves thatϕ 0 is non-decreasing. Finally, we show that Γ˜u(p0, p1) andϕ 1(p1)−ϕ 0(p0) have the same sign. Fixp 0 ∈I. We consider the following cases
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[12]
This directly impliesϕ 1(p1)−ϕ 0(p0) =p 1 −(L−1)>0 for allp 1 ∈I
Supposeϕ 0(p0) =L−1. This directly impliesϕ 1(p1)−ϕ 0(p0) =p 1 −(L−1)>0 for allp 1 ∈I. Moreover, by construction ofϕ 0, we must have Γ˜u(p0, p1)>0 for allp 1 ∈I
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[13]
In this caseϕ 1(p1)−ϕ 0(p0) =p 1 −(U+ 1)<0 for allp 1 ∈I
Supposeϕ 0(p0) =U+ 1. In this caseϕ 1(p1)−ϕ 0(p0) =p 1 −(U+ 1)<0 for allp 1 ∈I. As in the previous case, the construction ofϕ 0 implies Γ˜u(p0, p1)<0 for allp 1 ∈I
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[14]
By construction, Γ˜u(p 0, ϕ0(p0)) = 0
Supposeϕ 0(p0)∈[L, U]. By construction, Γ˜u(p 0, ϕ0(p0)) = 0. Ifp 1 < ϕ 0(p0), thenϕ 1(p1)− ϕ0(p0)<0. By Assumption 3, we must have Γ˜u(p0, p1)≤Γ˜u(p0, ϕ0(p0)) = 0. If Γ˜u(p0, p1) = 0, 34 this would give a second root, which violates Assumption 4. Hence, Γ˜u(p 0, p1)<0. The same argument applies to the casep 1 > ϕ0(p0). Ifp 1 =ϕ 0(p0), we trivially haveϕ ...
discussion (0)
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