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CAS: A Causal Attribution Score for Local and Global Explainable Artificial Intelligence
Michael Georgiades, Charalambia Varnava
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Summary
The paper introduces the Causal Attribution Score (CAS), a framework for causal explanation that distinguishes between predictive attribution (e.g., SHAP/LIME) and causal attribution. CAS uses an interventional coalition game and causal Shapley values to allocate joint intervention contrasts into Local CAS (magnitude and sign) and Global CAS summaries (instance-balanced and effect-mass). The method is validated against known-truth benchmarks and empirical datasets (401(k) and Pennsylvania reemployment bonus), demonstrating that Feature-CAS identifies different drivers of heterogeneity in treatment effects compared to predictive models.
Entities (11)
Relation Signals (9)
401(k) dataset → usedin → CAS
confidence 95% · On both empirical DoubleML datasets, 401(k) eligibility/net financial assets... Feature-CAS rankings
Pennsylvania reemployment bonus dataset → usedin → CAS
confidence 95% · Pennsylvania reemployment bonus/unemployment duration... Feature-CAS rankings
CAS → uses → Causal Shapley
confidence 95% · CAS starts from an identified interventional coalition game, allocates the joint intervention contrast with causal Shapley contributions
CAS → produces → Local CAS
confidence 92% · converts those raw outcome-scale effects into Local CAS
CAS → produces → Global CAS
confidence 92% · and two complementary Global CAS summaries
SHAP → comparedwith → Feature-CAS
confidence 90% · predictive SHAP/TreeSHAP rankings differed materially from Feature-CAS rankings
LIME → comparedwith → Feature-CAS
confidence 90% · LIME’s local surrogate differs from the Random-Forest prediction
Feature-CAS → isvariantof → CAS
confidence 88% · For feature-level comparison with predictive XAI, we use Feature-CAS, which explains the estimated treatment-effect surface
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Abstract
Abstract:Predictive explanation methods attribute a model output; they do not, by themselves, attribute an intervention effect on the real-world outcome. We introduce the Causal Attribution Score (CAS), a compact score architecture for causal explanation. CAS starts from an identified interventional coalition game, allocates the joint intervention contrast with causal Shapley contributions, and converts those raw outcome-scale effects into Local CAS, Signed Local CAS, and two complementary Global CAS summaries. The innovation is not a new Shapley formula, but a local-to-global causal reporting layer with an explicit intervention target. In the known-truth benchmark, eight repeated primary-interaction simulations (n = 2,200 each, three actions) gave mean Local CAS MAE of 0.107 for coalition-aware CAS, compared with 0.173 for one-at-a-time normalisation and 0.213 for a global normalised absolute ATE vector. The paired advantage over one-at-a-time normalisation increased from -0.003 under additivity to 0.091 under strong interactions. On both empirical DoubleML datasets, 401(k) eligibility/net financial assets (n = 9,915) and Pennsylvania reemployment bonus/unemployment duration (n = 5,099), predictive SHAP/TreeSHAP rankings differed materially from Feature-CAS rankings of treatment-effect modifiers. In Pennsylvania, dep1 (exactly one dependent) moved from predictive global rank 13 to Feature-CAS rank 2 and was the leading local Feature-CAS modifier. These results isolate the added value of separating what predicts the outcome from what explains heterogeneity in an estimated causal effect.
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- Source: https://arxiv.org/abs/2608.12555v1
- Canonical: https://arxiv.org/abs/2608.12555v1
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CAS: A Causal Attribution Score for Local and Global Explainable Artificial Intelligence Michael Georgiades Affiliation: Department of Computer Science Affiliation: Neapolis University Pafos Affiliation: Cyprus Charalambia Varnava Affiliation: CaSToRC Affiliation: The Cyprus Institute Affiliation: Nicosia, Cyprus Abstract Predictive explanation methods attribute a model output; they do not, by themselves, attribute an intervention effect on the real-world outcome. We introduce the CAS (CAS), a compact score architecture for causal explanation. CAS starts from an identified interventional coalition game, allocates the joint intervention contrast with causal Shapley contributions and converts those raw outcome-scale effects into Local CAS, Signed Local CAS and two complementary Global CAS summaries. The innovation is not a new Shapley formula, but a local-to-global causal reporting layer with an explicit intervention target. In the known-truth benchmark, eight repeated primary-interaction simulations (n=2,200n=2,200 each, three actions) gave mean Local CAS MAE (MAE) 0.1070.107 for coalition-aware CAS, compared with 0.1730.173 for one-at-a-time normalisation and 0.2130.213 for a global normalised absolute ATE (ATE) vector. The paired advantage over one-at-a-time normalisation increased from −0.003-0.003 under additivity to 0.0910.091 under strong interactions. On both empirical DoubleML datasets, 401(k) eligibility/net financial assets (n=9,915n=9,915) and Pennsylvania reemployment bonus/unemployment duration (n=5,099n=5,099), predictive SHAP (SHAP)/TreeSHAP rankings differed materially from Feature-CAS rankings of treatment-effect modifiers. In Pennsylvania, dep1 (exactly one dependent) moved from predictive global rank 13 to Feature-CAS rank 2 and was the leading local Feature-CAS modifier. These results isolate the added value of separating what predicts the outcome from what explains heterogeneity in an estimated causal effect. Keywords causal attribution ⋅· XAI ⋅· SHAP ⋅· LIME ⋅· DML ⋅· treatment effects 1 Motivation and contribution LIME (LIME) and SHAP explain why a fitted model produces a prediction (8; 6). In both methods, the target is predictive: f(x)−fbase.f(x)-f_base. (1) An intervention-oriented explanation has a different target: [Y(a1)−Y(a0)∣X=x].E\! [Y(a_1)-Y(a_0) X=x ]. (2) Here, x is the covariate profile at which the model is explained, fbasef_base is a reference prediction (e.g. the model’s average output) and Y(a1)Y(a_1) and Y(a0)Y(a_0) are potential outcomes under target and baseline intervention profiles a1=a(Q)a_1=a(Q) and a0=a(∅)a_0=a( ), formalised in Section 2. A feature can be highly predictive because it is a proxy, confounder, descendant or marker of treatment selection without being an actionable cause. Conversely, an intervention can have a real effect, while receiving little predictive importance. Prior causal Shapley work incorporates causal structure into Shapley feature attribution (5). In this work, we propose CAS. CAS does not claim a new Shapley value. Its novelty is the score layer built around a declared intervention game: 1. define the causal quantity being explained; 2. allocate its joint intervention contrast in outcome units; 3. convert the allocation into comparable local scores; 4. aggregate local scores into distinct global summaries. This separation is the central contribution. The score is meaningful only under a declared intervention set and standard causal identification assumptions: well-defined interventions, consistency, conditional exchangeability, positivity/joint support and an interference specification (7). Figure 1 summarises the resulting progression from causal specification to local and global attribution. Causal specificationactions Q, baseline/targetInterventional gamevx(S)v_x(S)Raw causal allocationϕjC(x) _j^C(x)Local CASmagnitude + signGlobal CASinstance / effect-mass causal estimand → allocation → communication Figure 1: CAS is a score architecture built on an identified interventional target. The raw causal allocation remains in outcome units; Local and Global CAS provide relative explanatory summaries. 2 CAS definitions Let Q=1,…,qQ=\1,…,q\ denote the declared actionable concepts. Each action j has a baseline level aj0a_j^0 and a target level aj1a_j^1. For coalition S⊆QS Q, let a(S)a(S) activate the target level for actions in S and the baseline level otherwise. For profile x, define vx(S)=[Y(a(S))∣X=x]v_x(S)=E[Y(a(S)) X=x] (3) and the joint intervention contrast ΔQ(x)=vx(Q)−vx(∅). _Q(x)=v_x(Q)-v_x( ). (4) 2.1 Raw causal contribution CAS uses the classical Shapley value (9) applied to the interventional game: ϕjC(x)=∑S⊆Q∖j|S|!(q−|S|−1)!q![vx(S∪j)−vx(S)]. _j^C(x)= _S Q \j\ |S|!(q-|S|-1)!q! [v_x(S∪\j\)-v_x(S) ]. (5) The raw contributions are in outcome units and satisfy ∑j∈QϕjC(x)=ΔQ(x). _j∈ Q _j^C(x)= _Q(x). (6) Equation (5) is an allocation mechanism; CAS is the score layer defined next. 2.2 Local CAS Let A+(x)=∑k∈Q|ϕkC(x)|A_+(x)= _k∈ Q| _k^C(x)| (7) be local causal mass. When A+(x)=0A_+(x)=0, the joint intervention effect is exactly zero at x (ΔQ(x)=0 _Q(x)=0 by Equation (6)). Local CAS is undefined at such instances and they are excluded from the population averages in Equations (11)–(12). For A+(x)>0A_+(x)>0, CASL(j)(x)=|ϕjC(x)|∑k|ϕkC(x)| CAS_L^(j)(x)= | _j^C(x)| _k| _k^C(x)| (8) is the fraction of absolute causal mass assigned to action j. It is non-negative and sums to one. Direction is retained by CAS~L(j)(x)=ϕjC(x)∑k|ϕkC(x)| CAS_L^(j)(x)= _j^C(x) _k| _k^C(x)| (9) with ∑jCAS~L(j)(x)=ΔQ(x)A+(x)∈[−1,1], _j CAS_L^(j)(x)= _Q(x)A_+(x)∈[-1,1], (10) which follows from the triangle inequality, |ΔQ(x)|=|∑jϕjC(x)|≤∑j|ϕjC(x)|=A+(x)| _Q(x)|=| _j _j^C(x)|≤ _j| _j^C(x)|=A_+(x). The signed sum therefore exposes reinforcement or cancellation rather than behaving as a probability. 2.3 Global CAS Local and global explanation answer different questions. The first Global CAS gives every profile equal weight: CASG,inst(j)=X[CASL(j)(X)] CAS_G,inst^(j)=E_X[CAS_L^(j)(X)] (11) and answers: which action is typically prominent across individuals? The second weights by causal-effect mass: CASG,mass(j)=X[|ϕjC(X)|]∑kX[|ϕkC(X)|] CAS_G,mass^(j)= E_X[| _j^C(X)|] _kE_X[| _k^C(X)|] (12) and answers: which action accounts for the greatest absolute causal effect in the population? This distinction is necessary because an action may be modest but frequently important, or rare but very large. Both global scores sum to one, but they represent different population summaries. 2.4 Why coalition-aware CAS is more than normalised CATE (CATE) If the intervention game is additive, vx(S)=vx(∅)+∑j∈Sτj(x),v_x(S)=v_x( )+ _j∈ S _j(x), (13) then the marginal contribution of action j is independent of coalition context and ϕjC(x)=τj(x),CASL(j)(x)=|τj(x)|∑k|τk(x)|. _j^C(x)= _j(x), _L^(j)(x)= | _j(x)| _k| _k(x)|. (14) Thus CAS deliberately reduces to normalised conditional treatment effects when there are no interactions. Its coalition-aware added value appears when the effect of one intervention depends on which other interventions are active. 3 Estimation and Feature-CAS For the empirical illustrations we use two datasets distributed with the Python DoubleML package (2). The 401(k) sample contains n=9,915n=9,915 observations from the 1991 Survey of Income and Program Participation and follows the 401(k) treatment-response application of 1; we use eligibility (e401) as treatment and net financial assets (net_tfa) as outcome. The Pennsylvania sample contains n=5,099n=5,099 observations from the Reemployment Bonus experiment analysed by 3; we use the treatment-group indicator (tg) and log unemployment duration (inuidur1). The exact variable definitions used in the analysis are listed in Appendix A; no optional polynomial feature expansion is used. Treatment effects are estimated with cross-fitting and an orthogonal AIPW (AIPW) score, following DML (DML) (4). Cross-fitted AIPW estimation relies on the identification assumptions stated in Section 1: consistency, conditional exchangeability given X and positivity/overlap in treatment assignment. Writing μ^di=μ^d(Xi) μ_di= μ_d(X_i) and e^i=e^(Xi) e_i= e(X_i), the binary-treatment score is ψ^i=μ^1i−μ^0i+Di(Yi−μ^1i)e^i−(1−Di)(Yi−μ^0i)1−e^i. ψ_i= μ_1i- μ_0i+ D_i(Y_i- μ_1i) e_i- (1-D_i)(Y_i- μ_0i)1- e_i. (15) The ATE is τ^ATE=n−1∑iψ^i τ_ATE=n^-1 _i ψ_i. A second cross-fitted learner estimates the heterogeneous effect τ^i=g(−k(i))(Xi), τ_i=g^(-k(i))(X_i), (16) where observation i is predicted by a model trained without its fold. For feature-level comparison with predictive XAI (XAI), we use Feature-CAS, which explains the estimated treatment-effect surface, not the outcome prediction: τ^i=bi+∑j=1pγij. τ_i=b_i+ _j=1^p _ij. (17) Here, p is the number of pre-treatment covariates and bib_i is the fold-specific TreeSHAP base value determined from the held-out fold’s CATE learner and its training-fold background. Because the second-stage CATE learner is a Random Forest, γij _ij is computed with TreeSHAP (6) on the exact fold-specific learner that generated τ^i τ_i. The implementation checks additivity and reconstructs the stored cross-fitted CATE to numerical tolerance. It does not explicitly enumerate all 2p2^p covariate subsets. The corresponding local and global feature scores are FCASL,j(xi)=|γij|∑k|γik|,FCASG,j=|γj(X)|∑k|γk(X)|.F\!CAS_L,j(x_i)= | _ij| _k| _ik|, \!CAS_G,j= E| _j(X)| _kE| _k(X)|. (18) These are effect-modifier attributions: they identify which pre-treatment covariates explain heterogeneity in the estimated causal effect. They do not claim that directly manipulating a covariate such as age produces γij _ij. Feature-CAS is deliberately not a direct instance of the interventional game vx(S)v_x(S) in Equation (3): pre-treatment covariates are effect modifiers, not declared actions with baseline/target levels aj0,aj1a_j^0,a_j^1. Equation (17) applies the same Shapley-type allocation logic to a different object, the already-estimated scalar τ^i τ_i. Predictive SHAP/TreeSHAP and LIME are evaluated on a Random-Forest outcome model and explain f(x)f(x); Feature-CAS explains τ^(x) τ(x). The comparison is therefore intentionally between different explanation targets. Fair local-profile selection. To avoid selecting a profile because one explainer makes it extreme, all local comparisons use the same observed row chosen from X alone. Let mj=medianiXijm_j=median_iX_ij and let sj=IQRiXijs_j=IQR_iX_ij (with MAD (MAD)/standard-deviation fallbacks if the IQR (IQR) is zero). We select ityp=argmini1p∑j=1p(Xij−mjsj)21/2.i_typ= _i \ 1p _j=1^p ( X_ij-m_js_j )^2 \^1/2. (19) No outcome, treatment, model prediction, SHAP/LIME value, ATE, CATE or Feature-CAS quantity enters this selection rule. The same row is then explained by all three local methods. Exact enumeration versus exact reconstruction. The core known-truth CAS benchmark below has q=3q=3 declared interventions and therefore exactly enumerates all 2q=82^q=8 intervention coalitions before applying Equation (5). Empirical Feature-CAS instead uses the tree-specific TreeSHAP algorithm on p=9p=9 or p=15p=15 effect modifiers. Accordingly, “exact Feature-CAS reconstruction” means exact additive reconstruction of the fitted fold-specific CATE-model output up to numerical tolerance; it does not mean exhaustive 2p2^p enumeration, nor does it replace the causal identification assumptions for τ τ. 4 Results 4.1 Known-truth validation of the CAS core The strongest test of the core score is a structural benchmark in which every interventional coalition value is known (Appendix B). Table 1 reports repeated-seed results for the primary interaction scenario. Raw contribution MAE evaluates ϕjC _j^C directly; Local CAS MAE evaluates the normalised score; top-action accuracy is Pr[argmaxj|ϕ^jC(x)|=argmaxj|ϕjC,⋆(x)|]. \! [ _j| φ_j^C(x)|= _j| _j^C, (x)| ]. (20) Each row in Table 1 is the mean over R=8R=8 independent seeds with n=2,200n=2,200 instances per seed. Brackets are descriptive 95% Monte Carlo intervals across seeds. Table 1: Repeated known-truth primary-interaction benchmark (R=8R=8, n=2,200n=2,200, q=3q=3, eight exactly enumerated intervention coalitions per profile). Lower is better for MAE; higher is better for top-action accuracy. Values are mean [95% Monte Carlo interval]. Method Raw ϕφ MAE Local CAS MAE Top-action accuracy Coalition-aware CAS 0.321 [0.312, 0.330] 0.107 [0.100, 0.114] 0.803 [0.787, 0.819] One-at-a-time normalisation 0.500 [0.480, 0.520] 0.173 [0.165, 0.182] 0.775 [0.752, 0.798] Global normalised |ATE||ATE| 0.617 [0.606, 0.627] 0.213 [0.196, 0.230] 0.762 [0.751, 0.773] The interaction-strength stress test is a separate repeated experiment, not the subtraction of two entries in Table 1. The paired Local CAS MAE reduction (one-at-a-time minus coalition-aware CAS) was 0.0660.066 [0.057, 0.075] in the primary-interaction scenario and 0.0910.091 [0.080, 0.103] under strong interactions. With interactions removed it was −0.003-0.003 [−0.010-0.010, 0.003], statistically indistinguishable from zero at this Monte Carlo resolution (Figure 2). This directly matches Equation (14): coalition context adds information when interactions matter and should not create an artificial advantage in an additive game. Figure 2: Paired Local CAS MAE advantage over one-at-a-time normalisation across additive, primary-interaction and strong-interaction known-truth scenarios. Points are mean paired reductions over R=8R=8 seeds; bars are descriptive 95% Monte Carlo intervals. 4.2 Real-data comparison with predictive explanations The population figures use 301 explained profiles per dataset, while the underlying samples contain 9,9159,915 401(k) observations and 5,0995,099 Pennsylvania observations. Table 2 summarises the global estimand change and adds conditional row-bootstrap intervals (B=500B=500) for the displayed population feature-mass scores. The numerical values in Table 2 are bootstrap means, whereas the feature ordering in the trajectory plots is determined from the fitted-sample global masses; the small numerical differences do not change the reported ranks. These intervals quantify finite-profile variability conditional on the fitted models; they do not propagate refitting uncertainty. TreeSHAP gives the same top-three predictive ordering as KernelSHAP in both datasets and nearly identical mass estimates. Table 2: Global feature-mass comparison on the 301 displayed profiles. Values are mean mass [95% conditional row-bootstrap interval], B=500B=500. The rank shift is KernelSHAP global rank → Feature-CAS global rank. Dataset Predictive SHAP top 3 Feature-CAS top 3 Illustrative rank shift 401(k) inc 0.315 [0.288,0.342]; pira 0.302 [0.290,0.313]; age 0.138 [0.125,0.149] inc 0.395 [0.365,0.426]; age 0.157 [0.141,0.173]; educ 0.104 [0.090,0.117] pira: 2 → 7 Pennsylvania bonus agelt35 0.207 [0.199,0.214]; black 0.157 [0.144,0.170]; agegt54 0.131 [0.121,0.141] female 0.123 [0.112,0.134]; dep1 0.121 [0.106,0.137]; lusd 0.108 [0.096,0.118] dep1: 13 → 2 Figures 3 and 4 show the full population-trajectory comparison with enlarged manuscript-scale axis labels. In 401(k), SHAP and TreeSHAP emphasise inc, pira and age, whereas Feature-CAS shifts mass toward inc, age and educ. In Pennsylvania, predictive explanations emphasise agelt35, black and agegt54, whereas Feature-CAS places female, dep1 and lusd at the top of the causal-effect-modifier ordering. Figure 3: 401(k): population cumulative contribution trajectories (full n=9,915n=9,915; 301 profiles displayed). SHAP and TreeSHAP explain the Random-Forest outcome model; Feature-CAS explains the exact cross-fitted DML-CATE surface. Figure 4: Pennsylvania bonus: population cumulative contribution trajectories (full n=5,099n=5,099; 301 profiles displayed). Predictive SHAP/TreeSHAP and Feature-CAS produce different feature orderings because they explain different estimands. The local comparison now uses the method-neutral typical profile in Equation (19), which removes the earlier asymmetry of selecting a profile for an extreme CATE. For 401(k), the selected row is 4124 (robust X-distance 0.0653): the reference-run DML CATE is approximately $6,848 and the ATE $7,872. SHAP and LIME both rank pira and inc first, whereas Feature-CAS ranks inc, pira and db. Feature-CAS reconstructs the stored cross-fitted CATE with absolute error 3.71×10−53.71× 10^-5 outcome units, corresponding to relative error 5.4×10−95.4× 10^-9. LIME’s local surrogate differs from the Random-Forest prediction by 9.4%, with local weighted R2=0.417R^2=0.417 (Figure 5). Because this row is selected as the most central observed profile under the robust X-distance rule, the remaining LIME discrepancy should be interpreted as local surrogate fidelity rather than as deliberate selection of a causal outlier. Figure 5: 401(k) method-neutral typical profile. SHAP and LIME explain the predictive outcome model; Feature-CAS explains the exact fold-specific cross-fitted DML CATE. All three methods explain the same row selected from X alone. For Pennsylvania, the selected row is 158 (robust X-distance 0.258): the reference-run DML CATE is −0.0749-0.0749 and the ATE −0.07423-0.07423. Predictive SHAP ranks agelt35, lusd and q2 highest; LIME ranks black, agegt54 and agelt35; Feature-CAS ranks dep1, female and q2. Feature-CAS reconstruction error is 1.10×10−91.10× 10^-9 outcome units, corresponding to relative error 1.46×10−81.46× 10^-8. LIME is locally close to the Random-Forest prediction, with a 0.31% surrogate discrepancy and local R2=0.808R^2=0.808 (Figure 6). The contrast with 401(k) illustrates that LIME fidelity is profile- and surface-dependent and should be reported rather than assumed. Figure 6: Pennsylvania method-neutral typical profile. The leading Feature-CAS modifiers are dep1, female and q2, whereas SHAP and LIME emphasise different predictive features. Split-seed stability. To give a separate view of model/split variability, the DML pipeline was rerun over five independent split/model seeds, while holding the same method-neutral row fixed (Table 3). The ATE and CATE values printed in Figures 5 and 6 are the single reference-run estimates used to construct those exact decompositions; Table 3 instead reports means across five reruns, so the table values are not expected to equal the figure annotations exactly. These are descriptive algorithmic-stability intervals, not formal causal confidence intervals. Table 3: DML split-seed stability over R=5R=5 repetitions. Values are mean ± standard deviation [descriptive 95% Monte Carlo interval]. Dataset ATE Typical-row CATE 401(k) 7,915 ± 130 [7,753, 8,077] 7,218 ± 1,821 [4,957, 9,479] Pennsylvania −0.07427±0.00296-0.07427± 0.00296 [−0.07795,−0.07059-0.07795,-0.07059] −0.1548±0.1198-0.1548± 0.1198 [−0.3036,−0.0061-0.3036,-0.0061] 5 What CAS adds The results support three narrow claims. 1. CAS changes the explained object. Predictive SHAP/LIME answer “what contributed to this prediction?”. CAS answers “how is an identified intervention effect allocated?”. Feature-CAS correspondingly asks which pre-treatment variables explain heterogeneity in that causal effect. 2. Local and global causal importance are not interchangeable. Local CAS is a compositional explanation for one profile. Instance-Balanced Global CAS describes typical local prominence, whereas Effect-Mass Global CAS describes population causal mass. One cannot in general replace these quantities with a single global |ATE||ATE| ranking. 3. Coalition awareness matters exactly when interactions matter. Under additivity, CAS reduces to normalised CATEs. Under interaction, one-at-a-time effects omit coalition context. The repeated known-truth benchmark and interaction stress test show that the recovery advantage grows with interaction strength and vanishes under additivity. This is the principal evidence that coalition-aware CAS is more than a relabelling of one-at-a-time treatment effects. CAS remains an explanatory score, not an optimisation objective. Raw signed effects and the outcome direction must be retained when an action is to be chosen. 6 Conclusion CAS provides a minimal local-to-global language for causal attribution. It begins with a declared interventional target, preserves a raw outcome-scale causal allocation, then separates magnitude, sign, individual prominence and population effect mass. The framework deliberately collapses to normalised conditional treatment effects in additive settings and departs from them only when coalition context changes marginal effects. The evidence supports this narrow claim from two directions. In known truth, coalition-aware recovery improves as interaction strength increases and shows no artificial advantage under additivity. In the two empirical datasets, method-neutral local profiles and population summaries show that predictive SHAP/LIME importance can differ substantially from the covariates that explain heterogeneity in an estimated DML treatment effect. Together, these results establish the added value of CAS without treating predictive attribution and causal attribution as interchangeable tasks. Appendix A Dataset variables Table 4 lists the exact treatment, outcome and base covariates used in the empirical analysis. The 401(k) data source is the 1991 Survey of Income and Program Participation application used by 1, while the Pennsylvania Reemployment Bonus experiment is documented by 3. Variable names and preprocessing follow the DoubleML data interface (2). Table 4: Variables used in the empirical analysis. Dataset / role Variable Definition 401(k), outcome net_tfa Net financial assets, US dollars. 401(k), treatment e401 Eligibility for a 401(k) plan. 401(k), covariate age Age. 401(k), covariate inc Income, US dollars. 401(k), covariate educ Education in years. 401(k), covariate fsize Family size. 401(k), covariate marr Married indicator. 401(k), covariate twoearn Two-earner household indicator. 401(k), covariate db Defined-benefit pension indicator. 401(k), covariate pira Individual retirement account participation indicator. 401(k), covariate hown Home-owner indicator. Pennsylvania, outcome inuidur1 DoubleML-preprocessed log duration of the first unemployment spell; the raw measure is in weeks. Pennsylvania, treatment tg Binary treatment-group indicator in the preprocessed bonus-analysis sample. Pennsylvania, covariate female Female indicator. Pennsylvania, covariate black Black-race indicator. Pennsylvania, covariate othrace Other-race indicator: non-white, non-black, non-Hispanic. Pennsylvania, covariate dep1 Indicator for exactly one dependent. Pennsylvania, covariate dep2 Indicator for exactly two dependents. Pennsylvania, covariate q2 Enrollment-quarter 2 indicator; quarter 1 is the reference. Pennsylvania, covariate q3 Enrollment-quarter 3 indicator; quarter 1 is the reference. Pennsylvania, covariate q4 Enrollment-quarter 4 indicator; quarter 1 is the reference. Pennsylvania, covariate q5 Enrollment-quarter 5 indicator; quarter 1 is the reference. Pennsylvania, covariate q6 Enrollment-quarter 6 indicator; quarter 1 is the reference. Pennsylvania, covariate agelt35 Indicator for claimant age below 35. Pennsylvania, covariate agegt54 Indicator for claimant age above 54. Pennsylvania, covariate durable Indicator for an occupation in durable manufacturing. Pennsylvania, covariate lusd Indicator for filing at a low-unemployment / short-duration site (Coatesville, Reading or Lancaster). Pennsylvania, covariate husd Indicator for filing at a high-unemployment / short-duration site (Lewistown, Pittston or Scranton). Appendix B Known-truth structural benchmark The reproducible known-truth benchmark uses six independent standard-normal covariates X1,…,X6X_1,…,X_6, three binary interventions A1,A2,A3A_1,A_2,A_3 and μ(X) μ(X) =0.6X1−0.35X2+0.25X32+0.20sinX4+0.15X5X6, =0.6X_1-0.35X_2+0.25X_3^2+0.20 X_4+0.15X_5X_6, (21) τ1(X) _1(X) =1+0.65X1−0.20X2, =1+0.65X_1-0.20X_2, (22) τ2(X) _2(X) =0.35+0.70sinX2+0.25X3, =0.35+0.70 X_2+0.25X_3, (23) τ3(X) _3(X) =0.55−0.75(X1<0)+0.35X4. =0.55-0.751(X_1<0)+0.35X_4. (24) Interaction functions are γ12(X) _12(X) =s0.90+0.30(X1>0), =s\0.90+0.301(X_1>0)\, (25) γ23(X) _23(X) =s(−0.75+0.25X2), =s(-0.75+0.25X_2), (26) γ13(X) _13(X) =0.45stanh(X3+X5), =0.45s (X_3+X_5), (27) where s∈0,1,1.8s∈\0,1,1.8\ gives the additive, primary-interaction and strong-interaction scenarios. The structural outcome is Y=μ(X)+∑j=13Ajτj(X)+γ12A1A2+γ23A2A3+γ13A1A3+ε,ε∼N(0,0.82).Y=μ(X)+ _j=1^3A_j _j(X)+ _12A_1A_2+ _23A_2A_3+ _13A_1A_3+ , N(0,0.8^2). (28) The three treatment propensities are e1(X) e_1(X) =logit−1(−0.15+0.60X1−0.25X4), =logit^-1(-0.15+0.60X_1-0.25X_4), (29) e2(X) e_2(X) =logit−1(0.10−0.45X2+0.30X5), =logit^-1(0.10-0.45X_2+0.30X_5), (30) e3(X) e_3(X) =logit−1(−0.05+0.40X3−0.25X6), =logit^-1(-0.05+0.40X_3-0.25X_6), (31) with Aj∼Bernoulli(ej(X))A_j (e_j(X)) conditionally on X. For each profile, the benchmark evaluates all eight intervention coalitions exactly, computes the true causal Shapley allocation by Equation (5) and compares it with allocations obtained from an estimated Random-Forest outcome surface. The results in Section 4 use n=2,200n=2,200 and R=8R=8 independent seeds per scenario. References Abadie (2003) A. Abadie Semiparametric instrumental variable estimation of treatment response models. Journal of Econometrics 113 (2), p. 231–263. 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