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REVIEW 3 major objections 4 minor 39 references

The geometry of moral decision making

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Pith's one-line read Moral and legal decision making, the paper claims, is one variational problem whose two terms are utilitarianism, $E_p[U]$, and deontology, the penalty $\frac{1}{\beta} R(p,q)$.

desk verdict A clean recapitulation of bounded rationality dressed as a moral theory, with a legal example that has a sign error making the deontic term a reward rather than a penalty. read the letter →

arxiv 2501.08865 v1 pith:LPXGOAAP submitted 2025-01-15 cs.IT math.ITphysics.data-an

Roland M. Friedrich This is my paper · ORCID
classification cs.ITmath.ITphysics.data-an MSC 94A1794A3462B1191B06
keywords boundedrationalitydeontologyutilitarianismratedistortioninformationgeometryBoltzmann-GibbsdistributionMarkovkernelsconstitutionalrights
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper tries to establish that resource-bounded moral and legal decision making is a single optimization problem, not a clash between two incompatible theories. The decision maker chooses the policy $p$ that maximizes expected utility $E_p[U]$ minus the scaled divergence penalty $\frac{1}{\beta} R(p,q)$, where the expected-utility term is the utilitarian component and the penalty, anchored to a prior $q$ over the permitted actions, is the deontological component. The optimum is always a Boltzmann-Gibbs distribution, interpolating from pure rule-following at $\beta \to 0$ to pure consequence-maximizing at $\beta \to \infty$, and the coupling constant $\beta$ is declared a free parameter to be fixed by the legislator or the court. This would matter because it reduces an old philosophical dichotomy to a concrete quantitative trade-off and gives one formal object, Eq. (78), in which the restriction of constitutional rights appears as a constrained second-best optimization with the authority's discretion built in as the free parameter.

What carries the argument

The central object is the variational objective of Eq. (78), $\max_p \left( E_p[U] - \frac{1}{\beta} R(p,q) \right)$, in which $q$ is a prior over actions, $R$ is a divergence-based regularizer, and $\beta$ is the inverse temperature. That objective carries the argument because its support condition makes the prior the encoding of the legal or moral code — only actions in the support of $q$ are eligible — and the divergence term makes rule-following a graded pull rather than a hard constraint. Solving it yields the Boltzmann-Gibbs distribution, an exponential family on the probability simplex, and varying $\beta$ moves the solution along the $e$-geodesic through $q$. The constraint-robust variant replaces the fixed prior with a source distribution and organizes the same trade-off through the rate-utility function, whose slope is $1/\beta$ and whose tangency points with constant-mutual-information surfaces define the utility expansion path.

What would settle it

The framework predicts that, for fixed utility assignments, observed choices follow the Boltzmann-Gibbs form $p(i) \propto q_i e^{\beta u_i}$ for some prior $q$ and weight $\beta \geq 0$. That prediction is falsifiable by behavioral data: fit $\beta$ and $q$ to the response frequencies of a large set of moral-dilemma decisions with known utilities, and the claim fails if the residuals are systematic — for example, if choices persistently violate the ratio property of Luce's choice axiom, or if no single fitted pair $(q, \beta)$ reproduces the frequencies across variations of the dilemma. In the legal domain, the applied model predicts discontinuous switches between permitted restrictions as $\beta$ crosses thresholds; observing smooth, history-dependent restriction patterns that no piecewise-constant $\beta$ can reproduce would falsify the application.

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Extended reading notes

Core claim

The central claim is that a resource-bounded decision maker resolves the conflict between deontology and utilitarianism inside a single variational objective, Eq. (78): the optimal policy maximizes expected utility $E_p[U]$ minus the scaled penalty $\frac{1}{\beta} R(p,q)$, where the expected-utility term is the utilitarian component and the regularizer $R$, anchored to a prior $q$ whose support is the set of permitted actions, is the deontological component. The solution is the Boltzmann-Gibbs distribution, $p^*_\beta(i) \propto q_i e^{\beta u_i}$, an exponential-family weighting of actions by utility that interpolates between pure rule-following at $\beta \to 0$ and unrestricted utility maximization at $\beta \to \infty$, tracing an $e$-geodesic through the prior as the weight is swept. In the constraint-robust version with a source distribution over world states, the same trade-off is organized by a rate-utility function whose slope at every point is $1/\beta$, and the optimal kernels solve self-consistent equations of rate-distortion type. The author's stated point is that neither moral theory determines the coupling constant: it remains a free parameter for the legislative or judicial authority, and that free parameter is the formal locus of the margin of discretion.

Load-bearing premise

The argument stands on the interpretive identification of rule-following (deontology) with a divergence penalty against a prior over permitted actions — a modeling choice that the paper asserts rather than derives from moral theory, legal texts, or data — together with the external supply of the penalty weight $\beta$.

Editorial extensions

If this is right

  • If the framework is right, every moral choice problem is specified by the prior $q$, the regularizer $R$, and the coupling $\beta$; no separate moral theory is needed beyond these ingredients.
  • The optimal moral policy is never a pure rule or a pure maximizer in the interior regime: it is the Boltzmann-Gibbs distribution, and the two classical theories are recovered only in the limits $\beta \to 0$ and $\beta \to \infty$.
  • In the legal application, a restriction of a fundamental right is justified exactly when the public utility gain exceeds the disutility of the restriction, and the model predicts that the chosen restriction switches discontinuously as the authority's weight $\beta$ crosses critical thresholds.
  • Because the constraint-robust problem is formally a rate-distortion problem, coarse-graining the space of states is governed by the data-processing inequality: abstraction cannot increase the information a decision can carry, so hierarchical and simplified moral reasoning follows from the same objective.
  • Autonomous agents implementing Eq. (78) inherit a tunable deontology: their rule-following behavior is controlled by a single externally supplied constant rather than by a hard-coded rule list.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial extension: with $\beta$ treated as a quantity to be fitted rather than legislated, the framework becomes behaviorally testable — one could estimate $\beta$ and $q$ from observed choice frequencies in moral dilemmas and ask whether a single pair transfers across situations, a check the paper does not perform.
  • Editorial extension: because the constraint-robust problem is formally a rate-distortion problem, moral or legal vagueness can be read as a compression phenomenon — coarse representations of a situation cost less information, so an 'optimal vagueness' would trade decision accuracy against coding cost, a consequence the paper leaves implicit.
  • Editorial consequence of the mapping: if deontology is a regularizer, then disagreements between rule-based and consequentialist moral theories are disagreements about the support of the prior and the value of the coupling constant — a philosophical dispute relocated onto two numbers that the paper hands to the legislative authority.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. This manuscript proposes that bounded-rational moral and legal decision making is captured by a variational objective in which expected utility plays the role of utilitarianism and a divergence/regularization term plays the role of deontology. After introducing Markov kernels, co-sheaves, and information-geometric tools, the paper derives the Boltzmann-Gibbs solution for the multiplier-robust control problem, studies a rate-utility version of the constraint robust-control problem, and applies the framework to a model of constitutional-right restrictions. The central claim is Eq. (78): the optimal policy maximizes E_p[U] minus (1/beta)R(p,q), where R is interpreted as a deontic regularizer and beta is a free coupling constant to be fixed by a legislative or judicial authority.

Significance. If the moral identification in Eq. (78) were established, the paper would offer a compact mathematical bridge between information-theoretic bounded rationality, moral theory, and legal doctrine. The variational derivations in Section 5 are standard and mostly correct: the KKT conditions yield the Gibbs distribution, and the convexity/concavity statements about the rate-utility function in Section 6 follow familiar arguments. The paper is also honest that beta is not determined internally. However, the central 'deontology as regularization' claim is not derived but assumed by labelling R as deontic, the legal model in Section 7 contains a sign inconsistency, and the paper provides no falsifiable prediction or independent constraint on q, R, or beta. The mathematical framework is therefore not yet the moral theory the title promises.

major comments (3)
  1. [§7, Eq. (77) and the definition of D(d)] The legal application has a sign inconsistency with the main objective. With D(d(p,q)) = d_max - D_KL(p||q), the objective in (77) becomes E_p[U] - (1/beta)(d_max - D_KL(p||q)) = E_p[U] - d_max/beta + (1/beta)D_KL(p||q). Since the constant term does not affect the argmax, the optimization is equivalent to maximizing E_p[U] + (1/beta)D_KL(p||q), so the KL term rewards departure from the prior instead of penalizing it. This contradicts the sign of the deontic term in Eqs. (25) and (78), where R is subtracted as a cut-off protecting the support of q. In addition, D(d) = d_max - d is affine rather than strictly convex, contrary to the paper's own condition that the disutility function be a strictly decreasing convex function of d. As written, Section 7 does not instantiate 'deontology as regularization'; it instantiates the opposite of Eq. (78).
  2. [§7 and Eq. (78)] The identification of the divergence penalty with deontology is an interpretive assumption rather than a derived result. The label 'deontic' is attached to R(p,q) in Eq. (78), and the support of q is described as a deontological cut-off, but no argument from deontological ethics, legal texts, or behavioral data establishes that a moral system's content is represented by a prior q and a regularizer R. Section 7 selects the disutility function from three ad-hoc 'basic types (not to scale)' without doctrinal derivation. The paper's own statement that beta 'will remain a free parameter to be determined by the competent legislative or judicial authority' concedes that the framework has no independent predictive content for the central mapping. Unless the mapping is constrained by a separate theory or by testable predictions, Eq. (78) is a relabeling of a known bounded-rationality optimization problem, not a discovery about deontology.
  3. [§6, Eqs. (47)–(53)] The transition from the stated optimization problem (47) to the minimization in (48) and (52) is not an equivalence. Problem (47) maximizes over the prior kernel κ, while (48) fixes K and minimizes D_KL(P⋊K || P⋊κ) over κ; these are different variational problems. For a fixed K, the minimizer κ* = K, or q* = K_*P in the constant-kernel case, does not in general maximize the free-energy expression in (47), which is driven toward priors concentrated on high-utility actions. Consequently the rate-utility problem (53) and the subsequent concavity analysis are not consequences of the stated starting point. This does not invalidate the Section 5 derivation of the Gibbs solution, but it undermines the claimed generalization in Section 6 unless an additional equivalence argument is supplied.
minor comments (4)
  1. [§1, Eq. (1)] The displayed objective in Eq. (1) uses a minimization with a plus sign in front of (1/beta)D_KL(p||q), whereas Eqs. (25) and (78) use maximization with a minus sign; this inconsistency should be corrected.
  2. [§1, after Eq. (1)] The text says 'The expression is referred to as multiplier robust-control problem []' with an empty citation; the reference should be filled in.
  3. [§6.3, Definitions 6.1–6.2] The existence and uniqueness of the utility expansion path satisfying (74) and the contraction path satisfying (76) are not proved; the claimed disjointness and reflection symmetry of the two paths should be stated as a proposition with explicit hypotheses.
  4. [Figure 9 caption] The caption says the z-axis shows F_beta[p] as a function of temperature 1/beta, but also lists a prior q = (0.7,0.2,0.1) in Delta_2 and a utility vector U = (7,5); the dimensional mismatch and the precise role of the displayed curve should be clarified.

Circularity Check

1 steps flagged · score 6.0 of 10

The central “deontology is a regularizer” claim is stipulated by labeling R in Eq. (78) as “deontic”; the Section 7 legal example then fails to instantiate even that sign.

  1. self definitional [Abstract; Section 7, Eq. (78) and surrounding summary text]
    "In essence, it can be succinctly summarised by analysing the components of the main object of our investigation, which is the following type of formula max_{p∈Δ(supp(q))} { Ep[U ]|utilitarian − 1/β R(p,q)|deontic } , (78) where R is a regularisation function and q is a Bayesian prior. … In contrast, R, with the help of the prior q and its support, serves as a cut-off and thus incorporates deontological considerations for the protection of individual rights, which expected utility cannot provide."

    The identification of deontology with regularization is not derived; it is inserted as the label “deontic” attached to R in the very formula presented as the paper’s conclusion. Eq. (25) already defines the same bounded-rationality objective max_p E_p[U] − (1/β)D_KL(p||q) using standard results, and the only new step at the summary is to call R the deontic term. No theorem, legal source, or data fixes that mapping, and the paper says the coupling constant is “ultimately up to a third party, such as the legislator”. The claimed finding is therefore equivalent to its own definition: R is named “deontic”, and the paper then reads “deontology is a regularizer” back out of Eq. (78).

full rationale

Sections 3–6 are a largely self-contained exposition of standard bounded-rationality and rate-distortion mathematics: the Gibbs/Boltzmann solution, the Legendre/geodesic discussion, and the rate-utility function are derived from stated optimization problems, and where results are cited (Mattsson–Weibull, Ortega–Braun, Genewein et al., Berger) they are used as ordinary background, not as a uniqueness theorem that forces the moral interpretation. The circularity lies in the philosophical wrapper. The abstract’s “we interpret deontology as a regularisation function” and Eq. (78)’s explicit labels make the central claim true by stipulation: R is called deontic, and the summary then presents the labeled objective as the paper’s insight. Because β is explicitly left to an external authority and no moral/legal dataset is used, there is no independent check that would let the labeling fail. The one concrete legal instantiation in Section 7 further weakens rather than supports the claim: with D(d(p,q)) := dmax − D_KL(p||q), the objective (77) is algebraically equivalent to maximizing E_p[U] + (1/β)D_KL(p||q), the opposite sign of the deontic penalty in (78). That is an internal-consistency/correctness problem rather than a circularity, but it confirms that the legal example is not an independent derivation of the deontology-as-regularization thesis. Overall: the mathematics is not circular; the central interpretive claim reduces by construction, giving a score of 6 rather than a higher score.

Assumptions & free parameters 5 free parameters · 7 assumptions · 2 invented entities

The ledger shows that the central contribution rests on the variational bounded-rationality objective inherited from prior work, plus the ad hoc identification of the penalty with deontology and an externally supplied beta and utility. The paper contributes mostly interpretation and geometric reformulation rather than new empirical constraints.

free parameters (5)
  • Inverse temperature beta (coupling constant)
    Appears in Eqs. (25), (65), and (77); controls the balance between expected utility and the divergence/deontology penalty. The paper explicitly states it must be determined by a legislative or judicial authority.
  • Prior q over actions = e.g., q=(0.7,0.2,0.1) in Figure 9
    Chosen for each application to represent the legal or moral system; it is an input, not determined by the model.
  • State-dependent utility function U(x,y) = e.g., U=(7,5) in Figure 9
    The expected-utility term in Eqs. (1), (46), and (53) is an external input from welfare economics or legal judgment; the paper does not measure it.
  • Disutility function D and divergence d = e.g., D(d)=dmax-DKL(p||q)
    Section 7 chooses D by hand from three 'basic types (not to scale)' and the example adopts a specific form. This choice determines the legal proportionality model and is not derived from doctrine or data.
  • Source probability P on world states = not specified, assumed given
    The constraint-robust setting in Section 6 assumes a source distribution P over states; it is an input inherited from rational inattention and information bottleneck theory.
assumptions (7)
  • domain assumption Finite discrete measurable spaces and Markov kernels encode all relevant uncertainty and actions.
    Sections 3 through 6 restrict to finite X and Y and use stochastic matrices; continuous or unbounded action spaces are not treated.
  • domain assumption The decision maker chooses a distribution p maximizing Ep[U] minus a divergence penalty rather than choosing an action directly.
    This variational form is assumed at Eq. (25) and Eq. (78), following Mattsson and Weibull, Ortega and Braun, and Genewein et al.
  • ad hoc to paper Deontology corresponds to the divergence regularizer R(p,q) and to the support of the prior q.
    The mapping is introduced around Eq. (78) and is the central conceptual claim; it is not derived from moral theory or from data.
  • ad hoc to paper A restriction of a constitutional right is justified iff the restricted action leads to a legal state at least as desirable and the net public utility exceeds the disutility cost.
    Section 7 sets the condition beta times expected utility greater than Dx(d(p, kappa_x)); this encodes legal proportionality as a utility/disutility comparison without independent derivation from legal texts.
  • domain assumption The inverse temperature beta is external and must be fixed by a competent authority.
    Stated in the Conclusions and in Section 7; therefore the model has no internal mechanism to set its central trade-off.
  • domain assumption The constraint-robust setting assumes a source probability P on world states and a fixed rate R.
    Section 6, Eq. (53); this is inherited from rational inattention and information bottleneck theory.
  • standard math Standard background theorems: disintegration theorem, convexity of KL divergence, Bauer maximum principle, information monotonicity.
    Used at Lemmas 3.1 through 4.5 and in Section 6 without proof.
invented entities (2)
  • Ought co-sheaf OX
    purpose: Formalizes the set of permitted actions per state or law as a set-valued co-sheaf on a partition topology.
    Introduced in Section 2.2; no independent empirical handle, it is a mathematical encoding of the legal notion of permitted acts.
  • Utility expansion path gamma+ and contraction path gamma-
    purpose: Define the curve of optimal policies as the allowed rate R increases or decreases.
    Geometric constructs in Section 6.3 defined by tangency to mutual-information level sets; no independent falsifiable prediction is attached.

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Pith. "Pith review of The geometry of moral decision making." pith.science (2026). https://pith.science/paper/LPXGOAAP

@misc{pith2026250108865,
  author       = {Pith},
  title        = {Pith review of: The geometry of moral decision making},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LPXGOAAP}},
  note         = {Machine review of arXiv:2501.08865}
}
read the original abstract

We show how (resource) bounded rationality can be understood as the interplay of two fundamental moral principles: deontology and utilitarianism. In particular, we interpret deontology as a regularisation function in an optimal control problem, coupled with a free parameter, the inverse temperature, to shield the individual from expected utility. We discuss the information geometry of bounded rationality and aspects of its relation to rate distortion theory. A central role is played by Markov kernels and regular conditional probability, which are also studied geometrically. A gradient equation is used to determine the utility expansion path. Finally, the framework is applied to the analysis of a disutility model of the restriction of constitutional rights that we derive from legal doctrine. The methods discussed here are also relevant to the theory of autonomous agents.

Figures

Figures reproduced from arXiv: 2501.08865 by the authors.

Figure 1
Figure 1. Left panel: Ought set Ox ⊂ A of allowed actions in situation x. A particular norm or law corresponds to a co-section σ : X → Y with y = σ(x). ΓX global co-section. Right panel: Utility of all actions. Maximum utility u ∗ x achieved at a ∗ , but not eligible because a ∗ ∈/ Ox. for a left action, and refer to a.x as the consequence of action a on state x. If (A, A ) is a measurable monoid, we require the action to be … view at source ↗
Figure 3
Figure 3. Conditional Markov kernel Let us briefly consider the impact of the operations introduced on information. The sequence of random variables X ∼ p, Y ∼ K∗p and Z = g ◦ Y ∼ (g∗K)∗p form a Markov chain X → Y → g ◦ Y . The ‘Data Processing Inequality’ implies that I(P, K∗P) ≥ I(P,(g∗K)∗P), (15) or, in a more conventional form, I(X, Y ) ≥ I(X, g(Y )). Therefore, it is impossible to increase the amount of information conta… view at source ↗
Figure 4
Figure 4. Left panel 4a: Submanifolds MP, Er and Er ′, r < r′ , of ∆(X ×Y ). Black contour lines surround the coloured areas corresponding to the different values of the free energy difference Fβ := Eπ[U] − 1 β I(π). The blue dashed lines show contours of constant mutual information. Right panel 4b: Submanifolds MP (fibre over P), E⊥⊥ (exponential family of product measures) and E0(P) of ∆(X × Y ). E0(P) = MP ∩ E⊥⊥. Section σ… view at source ↗
Figures from the paper (5 more)
Figure 5
Figure 5. Figure 5: Set of product measures E0(P) (red diagonal) in MP and E0(ν) (dark green diagonal) in Mν. Left panel 5a: m-convex set of couplings of P and ν in ∆(X × Y ), given by the intersection Γ(P, ν) = Mν ∩ MP, and E0(P) ∩ E0(ν) = P ⊗ ν. Right panel 5b: Rear view of the probabil…
Figure 6
Figure 6. Figure 6: The probability simplex with priors q1, q2 and q3 and e geodesics γi through qi (blue curves), parameterised by β. The contours show different KL spheres in ∆2 with different radii, all centred on point q3. The geodesic has the following asymptotic bounds: lim t→+∞ γ (…
Figure 7
Figure 7. Figure 7: Red diagonal line E0(P): Locus of product measures in MP. The dashed contours show mutual information values with I(P; K) = R for different values of R. The blue contour lines show values of EP⋊K[U] for U¯(R) ≥ R0. The red curved lines show the paths of utility expansi…
Figure 9
Figure 9. Figure 9: The [PITH_FULL_IMAGE:figures/full_fig_p028_9.png]
Figure 8
Figure 8. Figure 8: Disutility function D(d) For β → 0, Fβ[p] < 0 for all p ∈ ∆m, because D(d) ≥ dmax − dj ∗ > 0, i.e. it is bounded from below, and thus (77) has no solution. Between the two extremes there exists β0 > 0 such that Fβ0+ε[p] > 0 for some p ∈ ∆m and for all ε > 0. Furthermor…

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.