{"id":"3a8b5770-98b4-4564-9e95-1726f1e6d8fd","arxiv_id":"2502.06749","paper_version":1,"verdict":"REJECT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"Rational strategic agents investing effort to pass a linear classifier favor desirable features only when those features have the best contribution-to-cost ratio, and uncertainty biases effort toward low-variance high-mean contributions.","lead":"This paper studies how agents invest effort to game or improve their treatment by a classifier, when changing one feature can change others through a causal graph. It gives conditions for when a designed classifier pushes agents toward desirable features, and shows uncertainty about the classifier or causal graph shifts effort toward high-expected, low-variance contributions.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The claim that C h stays Gaussian under Model 2 is false for DAGs with paths of length at least two, so the incomplete-information convexity and closed-form results (Lemmas 4-6, Theorem 3) are unsupported; results survive only for Model 1 or bipartite graphs.","rationale":"The paper's central contribution is a characterization of rational agent effort under complete and incomplete information. The complete-information results (Section 3) appear internally sound: Lemma 1 and Theorem 1 for l1, Lemma 2 and Theorem 2 for l_p, and the non-convexity/convexity analysis of classifier design (Lemma 3, Propositions 3-4) follow from direct convex/Karush-Kuhn-Tucker arguments. The incomplete-information analysis, however, is built on a single distributional assertion: that under Models 1 and 2, C h is Gaussian. Model 1 (deterministic C, Gaussian h) is fine. Model 2 (deterministic h, Gaussian edge weights) is not: C contains products of Gaussian random variables along multi-edge paths, and such products are not Gaussian. The assertion is used, not merely in passing: it transforms the chance constraint into a deterministic second-order cone constraint, which then yields convexity, feasibility thresholds, and the explicit effort formula e* = lambda*(k1 I + k2 Sigma_Ch)^{-1} mu_Ch. Without Gaussianity, none of Lemmas 4-6 or Theorem 3 is justified for general causal DAGs; the chance constraint may be non-convex or have a different solution structure. The paper's own experiments are run under Model 1, so they provide no validation of the Model 2 claims. Proposition 6 correctly identifies bipartite graphs as a diagonal-covariance special case, but the paper does not restrict its Model 2 theorems to that case; indeed, Section 2.2 defines the prior over all edge weights and Section 4.1 explicitly claims Gaussianity 'easy to see' for both information models. Because the abstract and Section 1 promise tractability under uncertainty over the causal graph, the central claim of the incomplete-information section is unsupported. The proposed test, a two-edge path with independent standard Gaussian weights, directly isolates the failure: the product w1 w2 is non-Gaussian and the SOC reformulation is not equivalent to the original chance constraint. If that test confirms the discrepancy, as the distributional facts indicate, the rejection verdict is warranted. If, contrary to expectation, the exact feasible set coincided with the SOC set, the concern would be refuted; but that would require the product distribution to be Gaussian, which is false. The secondary Monte-Carlo-based proof of Proposition 5 is not independently load-bearing given the primary gap, though it reinforces the impression of insufficient rigor in this section.","tokens_in":25104,"tokens_out":6739,"duration_ms":56860,"concrete_test":"Consider the 3-node DAG 1->2->3 with independent edge weights w1,w2 ~ N(0,1), known classifier h0=(0,0,1), and alpha=1, delta=0.1 (Model 2). Compute the exact distribution of g(e)=P(w1 w2 e >= 1). Check two things: (i) is w1 w2 Gaussian? It has variance 1 but kurtosis 9, so no; (ii) is the feasible set {e >= 0 : g(e) >= 0.9} equal to the SOC set {e >= 0 : 1 - p_delta |e| <= 0} used in Section 4? Evaluate g(e) numerically for a grid of e, and test convexity of the exact feasible set (e.g., whether e=1 and e=3 feasible implies e=2 feasible). If the feasible sets differ or non-convexity is exhibited, Lemma 4's reformulation and Theorem 3 are refuted for Model 2. An independent analytic check: compare P(w1 w2 >= t) with the Gaussian tail for t=1; the true tail is heavier, showing the Gaussian quantile is wrong.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Under Model 2 (Gaussian prior over edge weights, known classifier), Section 4.1 asserts that C h is Gaussian, calling it 'easy to see.' This is incorrect: C is a polynomial in the Gaussian edge weights, and for any DAG containing a directed path of length two or more, entries of C include products of independent Gaussians (e.g., for 1->2->3, C_{1,3}=w_{12}w_{23}). Products of non-degenerate Gaussians are not Gaussian. Hence C h0 is generally non-Gaussian, so (C h0)^T e is not Gaussian, and the chance constraint P((C h0)^T e >= alpha) >= 1-delta cannot be rewritten as the SOC constraint alpha - mu_Ch^T e - p_delta ||Sigma_Ch^{1/2} e||_2 <= 0. Lemma 4 (convexity), Lemma 5 (feasibility), Lemma 6 (l1 multi-feature effort), Theorem 3 (l2 closed form), and Corollary 1 (variance penalization) all depend on this Gaussian reduction. They are therefore unproven for Model 2 on general DAGs. The paper does not scope these claims to Model 1 or to bipartite graphs (Proposition 6), so the advertised tractability under 'uncertainty over the causal graph' is not established. A secondary weakness is that Proposition 5's non-convexity proof is a Monte Carlo plot rather than a rigorous mathematical counterexample; but the Gaussianity gap is the primary defect.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies strategic classification in a binary decision problem where a principal deploys a linear classifier and a cost-minimizing agent can exert effort on features embedded in a known DAG. Effort propagates along causal edges through a contribution matrix C, and the agent's objective is to pass the classifier with probability at least 1−δ. The authors introduce a notion of β-desirability to measure whether effort is concentrated on features the principal favors. Under complete information they characterize optimal effort profiles for ℓp costs (single-feature effort for p=1, proportional allocation for p>1), give sufficient conditions for β-desirable best responses, and analyze convexity of the classifier-design space. Under incomplete information, they model Gaussian priors over the classifier and/or edge weights and claim that partial uncertainty leads to a convex program with a closed-form optimal effort profile, while full uncertainty is non-convex. Numerical experiments on a cardiovascular disease dataset illustrate the behavior of β-desirability under classifier uncertainty.","tokens_in":25227,"tokens_out":11675,"duration_ms":109289,"significance":"The complete-information part is clean and potentially useful: the characterization of optimal effort under weighted ℓp costs, the β-desirability conditions, and the convexity results for single-desirable-feature design are coherent and the proofs check out. The paper also makes a good-faith effort to include proofs and a realistic experimental case study. However, the advertised core novelty is the incomplete-information analysis with uncertainty over the causal graph, and that part rests on a false Gaussianity assertion for Model 2. The results survive only when uncertainty is over the classifier (Model 1) or when the graph is bipartite, a much narrower scope than the paper claims. The feasibility characterization (Lemma 5) is also incorrect as stated. These are load-bearing issues, not presentation defects.","major_comments":[{"comment":"The claim that Ch is Gaussian under Model 2 is false for general DAGs. For example, take a chain 1→2→3 with independent Gaussian edge weights w12 and w23 and a deterministic classifier h0 = e3. Then (Ch0)_1 = w12·w23, a product of independent non-degenerate Gaussians, which is not Gaussian. Consequently the chance constraint cannot be rewritten as α − μ_Ch^T e − pδ·||Σ_Ch^{1/2} e||_2 ≤ 0, because (Ch)^T e is not Gaussian. This invalidates the reduction used in Lemma 4 (convexity), Lemma 5 (feasibility), Lemma 6 (ℓ1 multi-feature effort), Theorem 3 (ℓ2 closed form), and Corollary 1 (variance penalization) for Model 2 on general DAGs. Proposition 6 only handles bipartite graphs, and the paper does not scope the Section 4 results to Model 1 or to bipartite graphs. The paper's own discussion in §4.3 notes that non-diagonal Σ_Ch arises for Model 2 on non-bipartite graphs, but that observation is about covariance structure and does not repair the non-Gaussianity.","section":"§4.1, Eq. (10) and Eq. (11)"},{"comment":"The feasibility condition in Lemma 5 is wrong as stated. Take α > 0 and μ_Ch = 0 with Σ_Ch positive definite. For the regime δ < 1/2 used in Lemma 4, pδ = Φ^{-1}(δ) is negative, so the constraint becomes α − pδ·||Σ_Ch^{1/2} e||_2 ≤ 0, i.e., α + |pδ|·||Σ_Ch^{1/2} e||_2 ≤ 0, which is impossible for every e. Hence the problem is infeasible for all δ < 1/2. The lemma's formula Φ^{-1}(−||Σ_Ch^{-1/2} μ_Ch||_2) evaluates to Φ^{-1}(0) = −∞ in this case, so the lemma incorrectly declares feasibility for every real δ. The proof's derivation around the case d* = Σ_Ch^{-1} μ_Ch also breaks down precisely when μ_Ch = 0, since the Cauchy-Schwarz equality direction is vacuous.","section":"Appendix C.3, Lemma 5"},{"comment":"The proof of Proposition 5 is not a rigorous mathematical proof. It presents a Monte Carlo plot of f(e) = P[(ωh)e ≥ α] for independent standard normals ω and h and asserts that the function is 'clearly' not concave. An empirical plot does not establish non-convexity of the feasible set or of the optimization problem; it does not rule out, for instance, that the feasible set is convex in some other representation, and it provides no certificate for the claimed counterexample. A rigorous counterexample, or an analytic argument showing a specific non-convex feasible set, is needed to support the negative tractability claim for Model 3.","section":"Appendix C.2, Proposition 5"}],"minor_comments":[{"comment":"The text refers to 'Lemma ??' in the introduction to Section 4; the cross-reference should be to Lemma 4.","section":"§4, first paragraph"},{"comment":"The displayed condition in Theorem 2 is typeset in a way that is hard to read: the expression 'βp 1 − β2' does not clearly indicate whether the intended factor is β·p, β_p, or β/√(1−β²). The surrounding discussion and the p=2 reduction should be checked for consistency.","section":"Theorem 2"},{"comment":"Figure 5 has no axis labels and is used as the sole evidence for Proposition 5; beyond the mathematical issue raised above, the figure should at least specify what quantity is plotted and how the Monte Carlo estimate was obtained.","section":"Figure 5"},{"comment":"The discussion of what it means for Σ_Ch to be non-diagonal is helpful, but it should explicitly acknowledge that when G is non-bipartite and uncertainty is over edge weights, the random vector Ch itself is generally not jointly Gaussian, so the tractability claims of §4.1 do not apply in that case.","section":"§4.3"}],"recommendation":"reject","confidential_remarks":"The complete-information material is solid and could form the basis of a useful paper, but the incomplete-information half, which is the advertised novelty, contains a false distributional claim and an incorrect feasibility theorem. These are not local presentation issues; they require either substantially restricting the model (to Model 1 or bipartite graphs) or developing a genuinely different tractability argument. Given the current scope of the claims, I do not see a path to acceptance without major new work."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The complete-information half of this paper is solid and genuinely useful. The authors characterize optimal effort profiles for ℓ1 and ℓp costs, give clean β-desirability conditions, and correctly identify the non-convexity of classifier design plus the single-desirable-feature convex special case. Those proofs check out, and the model itself—causal graphs plus Gaussian priors in strategic classification—is a reasonable and timely extension of Kleinberg-Raghavan. Credit is due for that part.\n\nThe problem is the incomplete-information section, and your stress-test note lands. Under Model 2 (Gaussian priors over edge weights), the paper asserts that C h is Gaussian because it is 'easy to see.' That is false for any DAG containing a path of length two or more: entries of C are products of Gaussian edge weights, and products of non-degenerate Gaussians are not Gaussian. So the chance constraint cannot be rewritten as a second-order cone constraint in the way the paper does. Lemma 4 (convexity), Lemma 5 (feasibility), Lemma 6 (ℓ1 multi-feature effort), Theorem 3 (ℓ2 closed form), and Corollary 1 (variance penalization) all inherit this flaw. The claims are only safe for Model 1 or for bipartite graphs under Model 2, which the paper does not clearly scope. That is a load-bearing gap, not a minor technicality, because the abstract and introduction advertise tractability under 'uncertainty about the causal graph.'\n\nA secondary weakness: Proposition 5's non-convexity proof for Model 3 is a Monte Carlo plot, not a mathematical counterexample. It is probably true, but as written it is not a proof.\n\nI would not desk-reject this. The complete-information results and the overall framing deserve referee time, and the incomplete-information section is fixable—either by restricting the claims to bipartite graphs or Model 1, or by handling the non-Gaussian case with different tools. But as submitted, the central claim is not supported, so the paper needs major revision. If the authors resubmit after scoping or fixing, I would be willing to review again.","headline":"The complete-information results are clean and worth keeping; the incomplete-information section rests on a false Gaussianity claim that leaves the paper's main advertised tractability results unsupported.","tokens_in":25935,"tokens_out":1346,"would_cite":false,"duration_ms":13745,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["91A80","90C25"],"pacs":[],"model":"deepseek-v4-flash","headline":"A linear classifier steers strategic agents' effort toward desirable features exactly when those features win the contribution-per-cost ranking; uncertainty then turns optimal effort into a mean–variance tradeoff that avoids high-variance…","keywords":["strategic classification","causal graphs","beta-desirability","contribution matrix","incomplete information","Gaussian priors","effort profiles","convex optimization"],"falsifier":"Run the agent's true optimization on a three-node chain $X_1 \\to X_2 \\to X_3$ with independent Gaussian edge weights $w_1, w_2$, a known classifier, and $\\ell_2$ costs: the contribution of $X_3$ to the score contains the product $w_1 w_2$, so one can compare the true optimal effort (computed by sampling the product distribution) against the claimed formula $\\lambda^*(k_1 I + k_2 \\Sigma_{Ch})^{-1}\\mu_{Ch}$ evaluated with the Gaussian moments; a discrepancy — or, at small $\\delta$, non-convexity of the true feasible set — would show the closed form is an approximation, not an exact solution, for general causal graphs.","tokens_in":24695,"feed_emoji":"🎯","tokens_out":14638,"duration_ms":115628,"temperature":0.7,"pith_summary":"This paper asks when a decision-maker who publishes a linear classifier can get self-interested agents to spend their effort on features the decision-maker actually wants changed, rather than on cheaper or easier-to-game ones. It models feature interdependencies through a causal graph, so effort on one feature propagates to others; the paper's central object is the contribution matrix $C$, which aggregates every directed path's weight, making the score change from effort $e$ equal to $(Ch_0)^\\top e$ in the full-knowledge case. In that complete-information setting the paper pins down exactly when rational agents' optimal effort is $\\beta$-desirable: for $\\ell_1$ costs, when the best contribution-to-cost feature is desirable (Theorem 1), and for $p>1$, when desirable contributions dominate in the relevant norm comparison (Theorem 2). With Gaussian uncertainty over the classifier or the graph's edge weights, the agent's problem becomes convex and the optimal profile takes the semi-closed form $e^\\star = \\lambda^*(k_1 I + k_2 \\Sigma_{Ch})^{-1} \\mu_{Ch}$, meaning effort rises with expected contribution and falls with variance. The upshot for a designer is a concrete checklist for when incentives work, plus a warning: uncertainty can systematically push agents away from exactly the features the principal wants to encourage.","feed_headline":"Uncertainty pushes strategic agents away from risky features","feed_subtitle":"Causal-graph analysis gives exact conditions for steering effort to desirable features — uncertainty redirects it.","key_machinery":"The load-bearing object is the contribution matrix $C = \\sum_{k=0}^{|F|} A^k$ of the acyclic causal graph, whose entry $C_{ij}$ sums the products of edge weights over all directed paths from feature $i$ to feature $j$; it converts an exogenous effort profile $e$ into net feature change $\\Delta x = C^\\top e$, so the classifier sees score change $(Ch)^\\top e$. All of the paper's tractability results hinge on the distribution of $Ch$: in the complete-information case $Ch_0$ is a fixed vector and the ratios $(Ch_0)_f/c_f$ are the 'bang per buck' that determine the optimal profile; under partial uncertainty the assumption that $Ch$ is Gaussian rewrites the probability constraint as $\\alpha - \\mu_{Ch}^\\top e - \\Phi^{-1}(\\delta)\\,\\|\\Sigma_{Ch}^{1/2} e\\|_2 \\le 0$, a convex constraint whose KKT analysis yields the closed form $e^\\star = \\lambda^*(k_1 I + k_2 \\Sigma_{Ch})^{-1}\\mu_{Ch}$. The same Gaussian object supplies the feasibility boundary in Lemma 5, the diagonal-covariance closed form of Corollary 1, and the experimental behavior of $\\beta$-desirability as uncertainty grows.","core_discovery":"The paper's central discovery is that causality turns strategic best-response into linear algebra: because net feature change is $C^\\top e$, the effective incentive signal sent by a classifier is the vector $Ch_0$, the total contribution of each feature to the score along all causal paths. With this object the paper proves that under $\\ell_1$ costs a rational agent invests in a single feature, the one maximizing $(Ch_0)_f / c_f$, so guaranteeing desirable behavior reduces to making the best bang-per-buck feature desirable; under $\\ell_p$ costs with $p>1$ the optimal effort spreads across all contributing features with $e^\\star_f \\propto ((Ch_0)_f / c_f)^{1/(p-1)}$, and $\\beta$-desirability holds when the desirable features' contributions dominate in the corresponding norm, e.g. $\\|(Ch_0)_D\\|_2 \\ge (\\beta/\\sqrt{1-\\beta^2})\\|(Ch_0)_U\\|_2$ for unit costs at $p=2$. The paper further shows that the set of classifiers inducing desirable effort is generally non-convex when more than one feature is desirable, that it becomes convex when the principal targets a single desirable feature or only bounds undesirable contribution, and that under partial Gaussian uncertainty the agent's chance constraint reduces to a convex inequality and the optimal effort becomes $e^\\star = \\lambda^*(k_1 I + k_2 \\Sigma_{Ch})^{-1}\\mu_{Ch}$ — so uncertainty acts like a ridge penalty that diverts effort from high-variance features. Experiments based on a cardiovascular-disease risk graph show agents investing in desirable, unobserved lifestyle features purely through causal propagation, with desirability decreasing in the level of uncertainty.","pith_inferences":["An implication the authors leave implicit: the closed form $e^\\star = \\lambda^*(k_1 I + k_2 \\Sigma_{Ch})^{-1}\\mu_{Ch}$ is structurally a ridge-type shrinkage, which suggests a design lever the paper does not develop — a principal can redirect effort not only by changing classifier weights but by deliberately shaping agents' uncertainty, for instance by publishing a simpler or more stable classifie","Because $Ch$ is a product of Gaussian edge weights along paths, the closed forms for graph uncertainty are exact only when every path has length one (e.g., bipartite graphs); for deeper DAGs the paper's Gaussian assumption is an approximation, and a natural testable extension is whether that approximation degrades gracefully in the small-variance regime or fails qualitatively in the tails of the p","A concrete extension with fairness implications: if different population groups hold priors with different variances, the same published classifier will induce group-specific effort profiles, so the $\\beta$-desirability gap across groups is, in this model, a function of information asymmetry rather than of the classifier weights alone."],"forward_implications":["With $\\ell_1$ costs, if a desirable feature offers the unique best contribution-to-cost ratio, no further design effort is needed: every best response is $\\beta$-desirable for every $\\beta \\in (0,1]$.","With $\\ell_p$ costs, $p>1$, the principal can certify desirable behavior in closed form by comparing the $\\ell^{2/(p-1)}$-weighted norms of $(Ch_0)$ over desirable versus undesirable features; for unit costs and $p=2$ the condition is the $\\ell_2$-norm comparison $\\|(Ch_0)_D\\|_2 \\ge (\\beta/\\sqrt{1-\\beta^2})\\|(Ch_0)_U\\|_2$.","Designing classifiers to induce desirable effort is generally non-convex and therefore hard, but convex and efficiently solvable when the principal targets exactly one desirable feature (for $p \\in [1,3]$) or relaxes the goal to bounding undesirable contribution.","Under partial Gaussian uncertainty the agent's best response is a tractable convex program with solution $e^\\star_f = \\lambda^*(\\mu_{Ch})_f / (k_1 + k_2 (\\Sigma_{Ch})_f)$ in the diagonal case: effort increases with a feature's expected contribution and decreases with its variance, so variance concentrated on desirable features directly harms $\\beta$-desirability.","In the cardiovascular case study, agents invest substantial effort in desirable lifestyle features even when the classifier puts zero weight on them, because those features causally drive the observed risk factors the classifier does use; raising classifier uncertainty or lowering the required pass probability $\\delta$ degrades desirability, and beyond a threshold the agent's problem is infeasible"],"supporting_citations":[{"why":"Supplies the closest prior model of strategic effort on general causal graphs in scoring and regression settings; the paper extends it to classification, $\\ell_p$ costs, and incomplete information.","marker":"Kleinberg and Raghavan [2020]"},{"why":"Establishes the framing that strategic classification is causal modeling in disguise and that gaming versus improvement depends on the causal graph, motivating the contribution-matrix model.","marker":"Miller et al. [2020]"},{"why":"Provides the causal strategic regression baseline on special linear causal graphs that the paper generalizes to arbitrary DAGs.","marker":"Shavit et al. [2020]"},{"why":"Defines the foundational strategic-classification agent best-response problem that the paper's principal-agent model builds on.","marker":"Hardt et al. [2016]"},{"why":"Supplies the linear-programming corner-point argument used to prove the single-feature characterization of optimal effort under $\\ell_1$ costs (Lemma 1).","marker":"Bertsimas and Tsitsiklis [1997]"},{"why":"Background on information discrepancy in strategic learning, the incomplete-information thread the paper brings to causal settings.","marker":"Bechavod et al. [2022]"},{"why":"Provides the expert-survey causal graph and feature set for the cardiovascular disease case study used in the numerical experiments.","marker":"Hasani et al. [2024]"}],"fun_headline_variants":["Causality and uncertainty steer strategic effort","Uncertainty drives agents to low-risk features","Causal structure decides desirable agent effort","Ridge-like uncertainty redirects agent investment"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The tractable incomplete-information results assume that the product $Ch$ of the classifier with the causal contribution matrix is Gaussian, which is guaranteed only when the classifier alone is uncertain or the graph is bipartite; for deeper causal graphs the contribution of a feature is a product of Gaussian edge weights and is not Gaussian, so the closed forms hold as approximations rather than exact characterizations.","fun_headline_variants_meta":{"raw":{"variants":["Causality and uncertainty steer strategic effort","Uncertainty drives agents to low-risk features","Causal structure decides desirable agent effort","Ridge-like uncertainty redirects agent investment"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00049,"raw_usage":{"total_tokens":2511,"prompt_tokens":1149,"completion_tokens":1362,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":765,"completion_tokens_details":{"reasoning_tokens":1307}},"tokens_in":765,"tokens_out":1362,"duration_ms":9326,"temperature":1.0,"reasoning_tokens":1307,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-08T14:28:03.485653+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the agent's true optimization on a three-node chain $X_1 \\to X_2 \\to X_3$ with independent Gaussian edge weights $w_1, w_2$, a known classifier, and $\\ell_2$ costs: the contribution of $X_3$ to the score contains the product $w_1 w_2$, so one can compare the true optimal effort (computed by sampling the product distribution) against the claimed formula $\\lambda^*(k_1 I + k_2 \\Sigma_{Ch})^{-1}\\mu_{Ch}$ evaluated with the Gaussian moments; a discrepancy — or, at small $\\delta$, non-convexity of the true feasible set — would show the closed form is an approximation, not an exact solution, for general causal graphs.","supporting_citations":[{"cited_title":"Strategic classification is causal modeling in disguise","cited_arxiv_id":null,"evidence_quote":"Establishes the framing that strategic classification is causal modeling in disguise and that gaming versus improvement depends on the causal graph, motivating the contribution-matrix model."},{"cited_title":"Causal strategic linear regression","cited_arxiv_id":null,"evidence_quote":"Provides the causal strategic regression baseline on special linear causal graphs that the paper generalizes to arbitrary DAGs."},{"cited_title":"Strategic classification","cited_arxiv_id":null,"evidence_quote":"Defines the foundational strategic-classification agent best-response problem that the paper's principal-agent model builds on."},{"cited_title":"Introduction to linear optimization","cited_arxiv_id":null,"evidence_quote":"Supplies the linear-programming corner-point argument used to prove the single-feature characterization of optimal effort under $\\ell_1$ costs (Lemma 1)."},{"cited_title":"Information discrepancy in strategic learning","cited_arxiv_id":null,"evidence_quote":"Background on information discrepancy in strategic learning, the incomplete-information thread the paper brings to causal settings."},{"cited_title":"Constructing causal pathways for premature cardiovascular disease mortality using directed acyclic graphs with integrating evidence synthesis and expert knowledge","cited_arxiv_id":null,"evidence_quote":"Provides the expert-survey causal graph and feature set for the cardiovascular disease case study used in the numerical experiments."}],"review_version":1}