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REVIEW 3 major objections 5 minor 39 references

Relative Explanations for Contextual Problems with Endogenous Uncertainty: An Application to Competitive Facility Location

T0 review · 3 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read A Wasserstein-regularized mixed-integer bilinear program computes relative counterfactual explanations for binary-decision contextual stochastic programs with endogenous uncertainty, demonstrated on competitive facility location.

desk verdict Solid formulation contribution, but an impossible runtime row and a missing advertised case study undercut the empirical headline; worth a serious referee after rework. read the letter →

arxiv 2506.19155 v4 pith:X2QOQCJ6 submitted 2025-06-23 math.OC

Jasone Ramírez-Ayerbe , Emma Frejinger This is my paper · ORCID
classification math.OC MSC 90B8090C1190C1590C26
keywords counterfactualexplanationscontextualstochasticoptimizationendogenousuncertaintycompetitivefacilitylocationmultinomiallogitmodelWassersteindistancemixed-integerbilinearprogrammingexplainable
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

Relative counterfactual explanations answer the question: what is the smallest change in the context that would make a desired decision acceptable? The paper claims these explanations can be computed for contextual stochastic optimization problems whose uncertainty is endogenous—the decision itself changes the distribution of the random outcome—even when the decision variables are binary, a setting it says earlier relative-explanation work did not cover. It makes the computation practical for a choice-based competitive facility location problem by minimizing an $\ell_1$ cost on transformed attractiveness covariates plus a 2-Wasserstein penalty that keeps the counterfactual demand distribution close to the factual one, and by reformulating the result as a mixed-integer bilinear program. If the claims hold, planners can learn, for example, which attractiveness improvements or environmental changes would justify opening a charging station that the current solution excludes, while limiting expected demand loss to a chosen factor $\alpha$.

What carries the argument

The load-bearing object is the relative counterfactual explanation: a new context $x$ for which some feasible decision $z\in\mathcal{D}$ earns a factor $\alpha$ of the factual expected reward. The argument runs through the standard mixed-integer linear reformulation of multinomial logit choice probabilities, extended so that the transformed attractiveness values $\phi_d(x)=\exp(\bar{v}_d(x_d))$ are decision variables rather than fixed parameters; this produces bilinear constraints linking the opening decisions, the attractiveness values, and the choice probabilities. The Wasserstein regularization enters as a finite optimal transport problem: with $\pi^n$ coupling the factual and counterfactual choice distributions, the objective contains $\lambda\sum_n\sum_{c,c'}\pi^n_{cc'}\delta(c,c')^2$. A model-free version that optimizes directly over probabilities gives a lower bound, and a greedy feasibility heuristic supplies the warm start that makes the bilinear program solvable within the hour limit on the test instances.

What would settle it

Run the instance families of Table 1 with identical warm starts and a one-hour limit for $\lambda=0$ and $\lambda=0.1$: if the unregularized model reaches optimality in less time on a majority of the larger instances, the claimed computational benefit of Wasserstein regularization is not general. A sharper test is to take instances where the factual choice distribution is nearly uniform, where the Wasserstein penalty should force the largest distortions and the unregularized objective should look relatively cheap; if $\lambda=0.1$ still dominates there, the speedup is robust to distribution shape.

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Extended reading notes

Core claim

The paper's central claim is that including the squared 2-Wasserstein distance between the factual and counterfactual choice distributions in the objective is not just a way to make counterfactuals realistic; it also accelerates solving the explanation problem and produces sparser solutions. In the multinomial logit facility location model, the explanation problem with a desired facility set is formulated as a single mixed-integer bilinear program over attractiveness variables $\phi_d(x)$, opening variables $z_d$, and optimal transport plans $\pi^n$, where the Wasserstein term is the transport cost between the factual and counterfactual choice distributions. A model-free relaxation over probability distributions provides a lower bound, and a constructive greedy heuristic provides a feasible warm start. On randomly generated instances, the regularized version with $\lambda=0.1$ reaches the one-hour time limit less often and yields sparser solutions than the unregularized version, while keeping the counterfactual choice probabilities close to the factual ones.

Load-bearing premise

The framework assumes the conditional distribution $P(y\mid z,x)$ is known, and every explanation is relative to that fixed model; if the model is misspecified, the computed counterfactual changes are not reliable evidence about what would happen in the world.

Editorial extensions

If this is right

  • For any desired facility configuration, planners get a concrete, minimal context shift that makes that configuration acceptable, expressed in the same attractiveness or environmental variables the demand model uses.
  • The same mixed-integer bilinear template transfers to other binary-decision problems with endogenous uncertainty—network design, assortment planning, school siting—as long as the outcome distribution is known and depends on covariates through a monotone transformation.
  • The model-free lower bound gives a certificate of how much structural realism costs: if the bound is far below the model-based value, no context can realize the cheap distributional shift the bound finds.
  • Regularized explanations are sparser and produce smoother demand transitions, so they are more likely to be perceived as plausible by stakeholders than explanations that change many small features.
  • Because the desired space $\mathcal{D}$ can be any subset of configurations, the same code can answer what would make any one of a family of alternatives acceptable, not just a single expert-chosen target.

Reading between the lines

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

  • I infer that the Wasserstein penalty will matter most when the factual choice probabilities are concentrated: with sharply peaked distributions the transport cost forces probability mass to move along short paths, whereas the unregularized $\ell_1$ objective can jump to an arbitrary distribution at the same covariate cost; the paper's experiments do not vary the concentration of the factual distri
  • A natural extension the paper leaves open is to estimate $P(y\mid z,x)$ from data and propagate estimation error into the explanations; I would expect the slack in the demand constraint to be a useful diagnostic for when a found counterfactual is an artifact of a misspecified model.
  • I infer from the reported instances that the speedup from regularization is most reliable for small budgets: several $r=8$ instances still reach the time limit even with $\lambda=0.1$, so the computational benefit may weaken as the budget grows.
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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 / 5 minor

Summary. The paper studies relative counterfactual explanations for contextual stochastic optimization problems in which the outcome distribution depends on both the context and the decisions. It formulates the explanation task as a non-convex optimization problem (2), introduces a cost that combines ℓ1 dissimilarity in a transformed feature space with a 2-Wasserstein regularization term (4), and proposes a model-free lower bound plus a greedy warm start. The method is specialized to a choice-based competitive facility location problem under a multinomial logit model; after a Haase-type reformulation, explanations are computed by solving the mixed-integer bilinear program (9). Section 5 reports experiments on a small illustrative instance and on synthetic instances of varying size, claiming that moderate Wasserstein regularization reduces computation time and yields sparser counterfactuals. The supplied abstract also claims validation on a real-world electric vehicle charging station planning case study in Montreal, but this case study is not present in the manuscript body.

Significance. If the computational claims held, this would be a useful contribution: it extends relative counterfactual explanations from continuous expert-provided target solutions to problems with binary decisions and endogenous uncertainty, and it allows a desired feasible set rather than a single target solution. The mathematical derivation from the MNL model to formulation (9) follows Haase (2009) and is internally consistent, and the authors make code and instances publicly available, which supports reproducibility. The lower-bound and warm-start scheme is a sensible practical addition. The significance of the empirical claims is, however, currently undermined by an arithmetically impossible row in Table 1 and by an abstract that advertises a real-world validation absent from the body; the modeling contribution itself appears sound.

major comments (3)
  1. [Section 5, Table 1] The row for N=100, D=10, r=8, λ=0.1 reports TL=5 and average time 1,094.136 s. Since five runs hit the 3,600 s time limit, any valid average over the ten instances must be at least 1,800 s, so the reported average is arithmetically impossible. This row directly supports the paper's headline claim that λ=0.1 reduces computation time, and the table as reported therefore does not support that claim. Please correct the data, recalculate the summary statistics, or rerun the affected instances before the efficiency claim can be accepted.
  2. [Abstract; Section 5] The abstract supplied with the paper states that the framework is validated on a real-world case study of electric vehicle charging station planning in Montreal. The full text contains no such case study: Section 5 contains only the illustrative example and the synthetic instances of Tables 1–2, and Section 6 does not mention Montreal. Either the case study and its results must be added to the manuscript, or the abstract's validation claim must be removed or corrected.
  3. [Section 5, Table 1] The statement that Wasserstein regularization yields sparser counterfactuals is stronger than what the table shows. For N=200, D=20, r=8, sparsity increases from 0.075 at λ=0 to 0.105 at λ=0.1 and 0.205 at λ=1, and several λ=1 rows, such as N=100, D=10, r=8, are less sparse than the corresponding λ=0 row. The qualitative claim in the text should be qualified as instance-dependent, or the analysis should explain when and why the sparsity improvement holds.
minor comments (5)
  1. [Section 4.1] The symbol D is used both for the set of candidate locations and for the desired feasible space; constraint (8g), written as "zd∈D", appears to be a typo for "z∈D" (or "z∈\mathcal{D}"), and the notation should be disambiguated.
  2. [Section 4.2] In the warm-start description, Step 2 says to calculate the factual captured demand "for each facility d," but the displayed quantity Q_factual = ∑_n q_n ∑_d P0_n(d) is a single total over all facilities; please clarify the wording.
  3. [Section 5] The phrase "the Wasserstein term is scaled by its bound" is not defined; please specify the exact scaling factor used in the objective before reporting the experiments.
  4. [Equations (9g)–(9i)] The constraint ranges "d∈D e∈E n∈N" appear to include e∈E unintentionally in (9g), which does not depend on e; this should be corrected for readability.
  5. [Figures 1 and 2] The captions contain typos: "solid dimonds" should be "solid diamonds" in Figure 1, and "a a solution" should be "a solution" in Figure 2.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the counterfactual formulation and bilinear reformulation are derived from explicit definitions and an external reformulation, with self-citations confined to background and notation.

full rationale

The derivation chain is self-contained. Definition 1.1 fixes the notion of a relative explanation; Problem (2) is the literal optimization formulation of that definition, and (3) merely restricts the distribution P to a family H. In the CFLP application, (8) is obtained by substituting the MNL choice probabilities (7) and the cost L = l1 + lambda*W2 into (2)-(3), and (9) is an algebraic reformulation of (8) using the external Haase (2009) linearization plus the added constraint (9i). None of these steps fits a parameter to the quantity being predicted. The Wasserstein term is an explicit design choice governed by the hyperparameter lambda, and the efficiency/sparsity claims are empirical comparisons of lambda = 0 versus lambda > 0 on the same instances; observing a lower Wasserstein distance under regularization is the intended effect of the objective, and the paper labels it 'as expected' rather than presenting it as an independent prediction. The self-citations (Bogetoft et al. 2024; Carrizosa et al. 2024; Legault and Frejinger 2025; Sadana et al. 2025) appear only in related-work discussion, as notation, or as a survey pointer, and no load-bearing uniqueness theorem or ansatz is imported from the authors' own prior work. The known-distribution assumption is explicitly scoped in Section 1 and is a modeling limitation, not a circular input. Separate empirical-validity concerns exist: Table 1's N=100, D=10, r=8, lambda=0.1 row reports TL=5 with a 1094-second average, which is arithmetically impossible under a 3600-second time limit, and the abstract's advertised Montreal case study does not appear in the body. These concerns affect the factual support for the runtime and validation claims, but they are not circularity in the derivation chain.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the assumption that the choice model is known and correctly specified, that covariates are continuous and unbounded so a feasible counterfactual exists, and that the chosen Wasserstein metric captures the right notion of plausibility. The free parameters are user choices (λ, α, ε, scaling) that are not derived or fit to external data. No new entities are invented.

free parameters (5)
  • λ (Wasserstein regularization weight) = 0, 0.1, 1 (experiments)
    Trade-off parameter between contextual change (ℓ1 in exp-space) and distributional shift; selected by the authors for the experiments, not tuned by a principled rule.
  • α (allowed performance loss factor) = 1 in all experiments
    User-specified factor in Definition 1.1; experiments fix it to 1, meaning captured demand must not decrease.
  • ϵ (positive lower bound in lower-bound formulation) = not reported
    Parameter in (10c) forcing a strictly positive probability for assigned customers; value is not stated.
  • Wasserstein scaling factor = not reported
    The term is 'scaled by its bound' (Section 5) to be comparable to the ℓ1 term; the actual scale is not specified.
  • greedy warm-start attractiveness constant c = increased until demand constraint met
    Used in the constructive greedy algorithm in Section 4.2 to build an initial feasible solution; value is instance-dependent.
assumptions (5)
  • domain assumption P(y|z,x) is known, with MNL parameters β given
    Section 1: 'We assume that this distribution is known, and we do not tackle the challenging task of learning this distribution from data.'
  • domain assumption Contextual covariates are continuous and unbounded above (X ⊆ R_+^{dx})
    Section 4.1: 'Provided that ϕ_d(x) is unbounded from above, a feasible solution to (8) always exists.'
  • domain assumption Utility \bar{v}_d is linear in features for the post-processing recovery of x
    Section 4.1: 'In many applications, the utility \bar{v}_d is a weighted sum of contextual features'; the recovery step is stated to be convex only in that case.
  • domain assumption MNL proportional substitution pattern is a valid model of customer choice
    Section 4: the CFLP application relies on the MNL model and its proportional substitution property.
  • domain assumption Squared 2-Wasserstein with Euclidean ground distance on facility locations is the appropriate dissimilarity for distributions
    Section 3.1: proposed and used to define the regularization; its desirability is demonstrated empirically, not derived.

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Pith. "Pith review of Relative Explanations for Contextual Problems with Endogenous Uncertainty: An Application to Competitive Facility Location." pith.science (2026). https://pith.science/paper/X2QOQCJ6

@misc{pith2026250619155,
  author       = {Pith},
  title        = {Pith review of: Relative Explanations for Contextual Problems with Endogenous Uncertainty: An Application to Competitive Facility Location},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/X2QOQCJ6}},
  note         = {Machine review of arXiv:2506.19155}
}
read the original abstract

In this paper, we consider a contextual stochastic optimization problem in which unknown parameters follow distributions that depend on contextual covariates and decisions. The problem is motivated by transportation infrastructure decisions such as facility location or network design. In such high-stakes settings, decisions must often be communicated, justified, and reconsidered under alternative stakeholder requirements. To this end, we propose a framework for computing relative counterfactual explanations. These explanations identify the smallest changes in the covariates required for a solution to satisfy prescribed constraints while limiting the performance loss to a controlled level. Whereas relative explanations have been introduced in prior literature, to the best of our knowledge, this is the first work focusing on problems with binary decision variables and endogenous uncertainty. We propose a methodology that uses the Wasserstein distance as a regularization term in the objective. Beyond improving tractability, this regularization yields explanations with desirable structural properties: it produces sparser counterfactuals, induces smoother transitions in the underlying choice distributions, and keeps the counterfactual behavior close to realistic demand patterns. We illustrate the method using a choice-based competitive facility location problem and present numerical experiments that demonstrate its ability to efficiently compute sparse, plausible, and interpretable explanations. We further validate the framework on a real-world case study of electric vehicle charging station planning in Montreal, where the explanations reveal the minimal capacity investments and environmental conditions required to justify including a candidate location in the charging network.

Figures

Figures reproduced from arXiv: 2506.19155 by the authors.

Figure 1
Figure 1. Illustration showing candidate (triangles), competitor (solid dimonds) and customer (solid [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Probability distributions for each customer. Each bar color represents the distribution [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗

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