CAS: A Causal Attribution Score for Local and Global Explainable Artificial Intelligence

# CAS: A Causal Attribution Score for Local and Global Explainable Artificial Intelligence
Source: [https://arxiv.org/html/2608.12555](https://arxiv.org/html/2608.12555)
Michael GeorgiadesAffiliation:Department of Computer ScienceAffiliation:Neapolis University PafosAffiliation:CyprusCharalambia Varnava

###### Abstract

Predictive explanation methods attribute a model output; they do not, by themselves, attribute an intervention effect on the real\-world outcome\. We introduce theCAS\(CAS\), 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 \(n=2,200n=2\{,\}200each, three actions\) gave mean Local CASMAE\(MAE\)0\.1070\.107for coalition\-aware CAS, compared with0\.1730\.173for one\-at\-a\-time normalisation and0\.2130\.213for a global normalised absoluteATE\(ATE\) vector\. The paired advantage over one\-at\-a\-time normalisation increased from−0\.003\-0\.003under additivity to0\.0910\.091under strong interactions\. On both empirical DoubleML datasets, 401\(k\) eligibility/net financial assets \(n=9,915n=9\{,\}915\) and Pennsylvania reemployment bonus/unemployment duration \(n=5,099n=5\{,\}099\), predictiveSHAP\(SHAP\)/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⋅\\cdotXAI⋅\\cdotSHAP⋅\\cdotLIME⋅\\cdotDML⋅\\cdottreatment effects

## 1Motivation and contribution

LIME\(LIME\) and SHAP explain why a fitted model produces a prediction\([8](https://arxiv.org/html/2608.12555#bib.bib1);[6](https://arxiv.org/html/2608.12555#bib.bib2)\)\. In both methods, the target is predictive:

f⁡\(x\)−fbase\.f\(x\)\-f\_\{\\mathrm\{base\}\}\.\(1\)An intervention\-oriented explanation has a different target:

𝔼⁡\[Y⁡\(a1\)−Y⁡\(a0\)∣X=x\]\.\\mathbb\{E\}\\\!\\left\[Y\(a\_\{1\}\)\-Y\(a\_\{0\}\)\\mid X=x\\right\]\.\(2\)Here,xxis the covariate profile at which the model is explained,fbasef\_\{\\mathrm\{base\}\}is a reference prediction \(e\.g\. the model’s average output\) andY⁡\(a1\)Y\(a\_\{1\}\)andY⁡\(a0\)Y\(a\_\{0\}\)are potential outcomes under target and baseline intervention profilesa1=a⁡\(Q\)a\_\{1\}=a\(Q\)anda0=a⁡\(∅\)a\_\{0\}=a\(\\varnothing\), formalised in Section[2](https://arxiv.org/html/2608.12555#S2)\. 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\([5](https://arxiv.org/html/2608.12555#bib.bib4)\)\. 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. 1\.define the causal quantity being explained;
2. 2\.allocate its joint intervention contrast in outcome units;
3. 3\.convert the allocation into comparable local scores;
4. 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\([7](https://arxiv.org/html/2608.12555#bib.bib7)\)\. Figure[1](https://arxiv.org/html/2608.12555#S1.F1)summarises the resulting progression from causal specification to local and global attribution\.

Causal specificationactionsQQ, baseline/targetInterventional gamevx​\(S\)v\_\{x\}\(S\)Raw causal allocationϕjC​\(x\)\\phi\_\{j\}^\{C\}\(x\)Local CASmagnitude \+ signGlobal CASinstance / effect\-masscausal estimand→\\rightarrowallocation→\\rightarrowcommunicationFigure 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\.
## 2CAS definitions

LetQ=\{1,…,q\}Q=\\\{1,\\ldots,q\\\}denote the declared actionable concepts\. Each actionjjhas a baseline levelaj0a\_\{j\}^\{0\}and a target levelaj1a\_\{j\}^\{1\}\. For coalitionS⊆QS\\subseteq Q, leta⁡\(S\)a\(S\)activate the target level for actions inSSand the baseline level otherwise\. For profilexx, define

vx​\(S\)=𝔼⁡\[Y⁡\(a⁡\(S\)\)∣X=x\]v\_\{x\}\(S\)=\\mathbb\{E\}\[Y\(a\(S\)\)\\mid X=x\]\(3\)and the joint intervention contrast

ΔQ​\(x\)=vx​\(Q\)−vx​\(∅\)\.\\Delta\_\{Q\}\(x\)=v\_\{x\}\(Q\)\-v\_\{x\}\(\\varnothing\)\.\(4\)
### 2\.1Raw causal contribution

CAS uses the classical Shapley value\([9](https://arxiv.org/html/2608.12555#bib.bib3)\)applied to the*interventional*game:

ϕjC​\(x\)=∑S⊆Q∖\{j\}\|S\|\!​\(q−\|S\|−1\)\!q\!​\[vx​\(S∪\{j\}\)−vx​\(S\)\]\.\\phi\_\{j\}^\{C\}\(x\)=\\sum\_\{S\\subseteq Q\\setminus\\\{j\\\}\}\\frac\{\|S\|\!\(q\-\|S\|\-1\)\!\}\{q\!\}\\left\[v\_\{x\}\(S\\cup\\\{j\\\}\)\-v\_\{x\}\(S\)\\right\]\.\(5\)The raw contributions are in outcome units and satisfy

∑j∈QϕjC​\(x\)=ΔQ​\(x\)\.\\sum\_\{j\\in Q\}\\phi\_\{j\}^\{C\}\(x\)=\\Delta\_\{Q\}\(x\)\.\(6\)Equation \([5](https://arxiv.org/html/2608.12555#S2.E5)\) is an allocation mechanism; CAS is the score layer defined next\.

### 2\.2Local CAS

Let

A\+​\(x\)=∑k∈Q\|ϕkC​\(x\)\|A\_\{\+\}\(x\)=\\sum\_\{k\\in Q\}\|\\phi\_\{k\}^\{C\}\(x\)\|\(7\)be local causal mass\. WhenA\+​\(x\)=0A\_\{\+\}\(x\)=0, the joint intervention effect is exactly zero atxx\(ΔQ​\(x\)=0\\Delta\_\{Q\}\(x\)=0by Equation \([6](https://arxiv.org/html/2608.12555#S2.E6)\)\)\. Local CAS is undefined at such instances and they are excluded from the population averages in Equations \([11](https://arxiv.org/html/2608.12555#S2.E11)\)–\([12](https://arxiv.org/html/2608.12555#S2.E12)\)\. ForA\+​\(x\)\>0A\_\{\+\}\(x\)\>0,

CASL\(j\)⁡\(x\)=\|ϕjC​\(x\)\|∑k\|ϕkC​\(x\)\|\\boxed\{\\operatorname\{CAS\}\_\{L\}^\{\(j\)\}\(x\)=\\frac\{\|\\phi\_\{j\}^\{C\}\(x\)\|\}\{\\sum\_\{k\}\|\\phi\_\{k\}^\{C\}\(x\)\|\}\}\(8\)is the fraction of absolute causal mass assigned to actionjj\. It is non\-negative and sums to one\. Direction is retained by

CAS~L\(j\)​\(x\)=ϕjC​\(x\)∑k\|ϕkC​\(x\)\|\\boxed\{\\widetilde\{\\operatorname\{CAS\}\}\_\{L\}^\{\(j\)\}\(x\)=\\frac\{\\phi\_\{j\}^\{C\}\(x\)\}\{\\sum\_\{k\}\|\\phi\_\{k\}^\{C\}\(x\)\|\}\}\(9\)with

∑jCAS~L\(j\)​\(x\)=ΔQ​\(x\)A\+​\(x\)∈\[−1,1\],\\sum\_\{j\}\\widetilde\{\\operatorname\{CAS\}\}\_\{L\}^\{\(j\)\}\(x\)=\\frac\{\\Delta\_\{Q\}\(x\)\}\{A\_\{\+\}\(x\)\}\\in\[\-1,1\],\(10\)which follows from the triangle inequality,\|ΔQ​\(x\)\|=\|∑jϕjC​\(x\)\|≤∑j\|ϕjC​\(x\)\|=A\+​\(x\)\|\\Delta\_\{Q\}\(x\)\|=\|\\sum\_\{j\}\\phi\_\{j\}^\{C\}\(x\)\|\\leq\\sum\_\{j\}\|\\phi\_\{j\}^\{C\}\(x\)\|=A\_\{\+\}\(x\)\. The signed sum therefore exposes reinforcement or cancellation rather than behaving as a probability\.

### 2\.3Global CAS

Local and global explanation answer different questions\. The first Global CAS gives every profile equal weight:

CASG,inst\(j\)=𝔼X​\[CASL\(j\)⁡\(X\)\]\\boxed\{\\operatorname\{CAS\}\_\{G,\\mathrm\{inst\}\}^\{\(j\)\}=\\mathbb\{E\}\_\{X\}\[\\operatorname\{CAS\}\_\{L\}^\{\(j\)\}\(X\)\]\}\(11\)and answers:*which action is typically prominent across individuals?*

The second weights by causal\-effect mass:

CASG,mass\(j\)=𝔼X​\[\|ϕjC​\(X\)\|\]∑k𝔼X​\[\|ϕkC​\(X\)\|\]\\boxed\{\\operatorname\{CAS\}\_\{G,\\mathrm\{mass\}\}^\{\(j\)\}=\\frac\{\\mathbb\{E\}\_\{X\}\[\|\\phi\_\{j\}^\{C\}\(X\)\|\]\}\{\\sum\_\{k\}\\mathbb\{E\}\_\{X\}\[\|\\phi\_\{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\.4Why coalition\-aware CAS is more than normalisedCATE\(CATE\)

If the intervention game is additive,

vx​\(S\)=vx​\(∅\)\+∑j∈Sτj​\(x\),v\_\{x\}\(S\)=v\_\{x\}\(\\varnothing\)\+\\sum\_\{j\\in S\}\\tau\_\{j\}\(x\),\(13\)then the marginal contribution of actionjjis independent of coalition context and

ϕjC​\(x\)=τj​\(x\),CASL\(j\)⁡\(x\)=\|τj​\(x\)\|∑k\|τk​\(x\)\|\.\\phi\_\{j\}^\{C\}\(x\)=\\tau\_\{j\}\(x\),\\qquad\\operatorname\{CAS\}\_\{L\}^\{\(j\)\}\(x\)=\\frac\{\|\\tau\_\{j\}\(x\)\|\}\{\\sum\_\{k\}\|\\tau\_\{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\.

## 3Estimation and Feature\-CAS

For the empirical illustrations we use two datasets distributed with the PythonDoubleMLpackage\([2](https://arxiv.org/html/2608.12555#bib.bib6)\)\. The 401\(k\) sample containsn=9,915n=9\{,\}915observations from the 1991 Survey of Income and Program Participation and follows the 401\(k\) treatment\-response application of[1](https://arxiv.org/html/2608.12555#bib.bib8); we use eligibility \(e401\) as treatment and net financial assets \(net\_tfa\) as outcome\. The Pennsylvania sample containsn=5,099n=5\{,\}099observations from the Reemployment Bonus experiment analysed by[3](https://arxiv.org/html/2608.12555#bib.bib9); 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](https://arxiv.org/html/2608.12555#A1); no optional polynomial feature expansion is used\.

Treatment effects are estimated with cross\-fitting and an orthogonalAIPW\(AIPW\) score, followingDML\(DML\)\([4](https://arxiv.org/html/2608.12555#bib.bib5)\)\. Cross\-fitted AIPW estimation relies on the identification assumptions stated in Section[1](https://arxiv.org/html/2608.12555#S1): consistency, conditional exchangeability givenXXand positivity/overlap in treatment assignment\. Writingμ^d​i=μ^d​\(Xi\)\\widehat\{\\mu\}\_\{di\}=\\widehat\{\\mu\}\_\{d\}\(X\_\{i\}\)ande^i=e^​\(Xi\)\\widehat\{e\}\_\{i\}=\\widehat\{e\}\(X\_\{i\}\), the binary\-treatment score is

ψ^i=μ^1​i−μ^0​i\+Di​\(Yi−μ^1​i\)e^i−\(1−Di\)​\(Yi−μ^0​i\)1−e^i\.\\widehat\{\\psi\}\_\{i\}=\\widehat\{\\mu\}\_\{1i\}\-\\widehat\{\\mu\}\_\{0i\}\+\\frac\{D\_\{i\}\(Y\_\{i\}\-\\widehat\{\\mu\}\_\{1i\}\)\}\{\\widehat\{e\}\_\{i\}\}\-\\frac\{\(1\-D\_\{i\}\)\(Y\_\{i\}\-\\widehat\{\\mu\}\_\{0i\}\)\}\{1\-\\widehat\{e\}\_\{i\}\}\.\(15\)The ATE isτ^ATE=n−1​∑iψ^i\\widehat\{\\tau\}\_\{\\mathrm\{ATE\}\}=n^\{\-1\}\\sum\_\{i\}\\widehat\{\\psi\}\_\{i\}\. A second cross\-fitted learner estimates the heterogeneous effect

τ^i=g\(−k⁡\(i\)\)​\(Xi\),\\widehat\{\\tau\}\_\{i\}=g^\{\(\-k\(i\)\)\}\(X\_\{i\}\),\(16\)where observationiiis predicted by a model trained without its fold\.

For feature\-level comparison with predictiveXAI\(XAI\), we use*Feature\-CAS*, which explains the*estimated treatment\-effect surface*, not the outcome prediction:

τ^i=bi\+∑j=1pγi​j\.\\widehat\{\\tau\}\_\{i\}=b\_\{i\}\+\\sum\_\{j=1\}^\{p\}\\gamma\_\{ij\}\.\(17\)Here,ppis the number of pre\-treatment covariates andbib\_\{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,γi​j\\gamma\_\{ij\}is computed with TreeSHAP\([6](https://arxiv.org/html/2608.12555#bib.bib2)\)on the exact fold\-specific learner that generatedτ^i\\widehat\{\\tau\}\_\{i\}\. The implementation checks additivity and reconstructs the stored cross\-fitted CATE to numerical tolerance\. It does not explicitly enumerate all2p2^\{p\}covariate subsets\.

The corresponding local and global feature scores are

F​CASL,j⁡\(xi\)=\|γi​j\|∑k\|γi​k\|,F​CASG,j=𝔼​\|γj​\(X\)\|∑k𝔼​\|γk​\(X\)\|\.\\operatorname\{F\\\!CAS\}\_\{L,j\}\(x\_\{i\}\)=\\frac\{\|\\gamma\_\{ij\}\|\}\{\\sum\_\{k\}\|\\gamma\_\{ik\}\|\},\\qquad\\operatorname\{F\\\!CAS\}\_\{G,j\}=\\frac\{\\mathbb\{E\}\|\\gamma\_\{j\}\(X\)\|\}\{\\sum\_\{k\}\\mathbb\{E\}\|\\gamma\_\{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 producesγi​j\\gamma\_\{ij\}\.

Feature\-CAS is deliberately not a direct instance of the interventional gamevx​\(S\)v\_\{x\}\(S\)in Equation \([3](https://arxiv.org/html/2608.12555#S2.E3)\): pre\-treatment covariates are effect modifiers, not declared actions with baseline/target levelsaj0,aj1a\_\{j\}^\{0\},a\_\{j\}^\{1\}\. Equation \([17](https://arxiv.org/html/2608.12555#S3.E17)\) applies the same Shapley\-type allocation logic to a different object, the already\-estimated scalarτ^i\\widehat\{\\tau\}\_\{i\}\. Predictive SHAP/TreeSHAP and LIME are evaluated on a Random\-Forest outcome model and explainf⁡\(x\)f\(x\); Feature\-CAS explainsτ^​\(x\)\\widehat\{\\tau\}\(x\)\. 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 fromXXalone\. Letmj=mediani⁡Xi​jm\_\{j\}=\\operatorname\{median\}\_\{i\}X\_\{ij\}and letsj=IQRi⁡Xi​js\_\{j\}=\\operatorname\{IQR\}\_\{i\}X\_\{ij\}\(withMAD\(MAD\)/standard\-deviation fallbacks if theIQR\(IQR\) is zero\)\. We select

ityp=arg⁡mini⁡\{1p​∑j=1p\(Xi​j−mjsj\)2\}1/2\.i\_\{\\mathrm\{typ\}\}=\\arg\\min\_\{i\}\\left\\\{\\frac\{1\}\{p\}\\sum\_\{j=1\}^\{p\}\\left\(\\frac\{X\_\{ij\}\-m\_\{j\}\}\{s\_\{j\}\}\\right\)^\{2\}\\right\\\}^\{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 hasq=3q=3declared interventions and therefore exactly enumerates all2q=82^\{q\}=8intervention coalitions before applying Equation \([5](https://arxiv.org/html/2608.12555#S2.E5)\)\. Empirical Feature\-CAS instead uses the tree\-specific TreeSHAP algorithm onp=9p=9orp=15p=15effect 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 exhaustive2p2^\{p\}enumeration, nor does it replace the causal identification assumptions forτ^\\widehat\{\\tau\}\.

## 4Results

### 4\.1Known\-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 \(Appendix[B](https://arxiv.org/html/2608.12555#A2)\)\. Table[1](https://arxiv.org/html/2608.12555#S4.T1)reports repeated\-seed results for the primary interaction scenario\. Raw contribution MAE evaluatesϕjC\\phi\_\{j\}^\{C\}directly; Local CAS MAE evaluates the normalised score; top\-action accuracy is

Pr\[argmaxj\|ϕ^jC\(x\)\|=argmaxj\|ϕjC,⋆\(x\)\|\]\.\\Pr\\\!\\left\[\\arg\\max\_\{j\}\|\\widehat\{\\phi\}\_\{j\}^\{C\}\(x\)\|=\\arg\\max\_\{j\}\|\\phi\_\{j\}^\{C,\\star\}\(x\)\|\\right\]\.\(20\)Each row in Table[1](https://arxiv.org/html/2608.12555#S4.T1)is the mean overR=8R=8independent seeds withn=2,200n=2\{,\}200instances 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 \[95% Monte Carlo interval\]\.The interaction\-strength stress test is a*separate repeated experiment*, not the subtraction of two entries in Table[1](https://arxiv.org/html/2608.12555#S4.T1)\. The paired Local CAS MAE reduction \(one\-at\-a\-time minus coalition\-aware CAS\) was0\.0660\.066\[0\.057, 0\.075\] in the primary\-interaction scenario and0\.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 \(Figure[2](https://arxiv.org/html/2608.12555#S4.F2)\)\. This directly matches Equation \([14](https://arxiv.org/html/2608.12555#S2.E14)\): coalition context adds information when interactions matter and should not create an artificial advantage in an additive game\.

![Refer to caption](https://arxiv.org/html/2608.12555v1/img/fig5_known_truth_advantage.png)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 overR=8R=8seeds; bars are descriptive 95% Monte Carlo intervals\.
### 4\.2Real\-data comparison with predictive explanations

The population figures use 301 explained profiles per dataset, while the underlying samples contain9,9159\{,\}915401\(k\) observations and5,0995\{,\}099Pennsylvania observations\. Table[2](https://arxiv.org/html/2608.12555#S4.T2)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](https://arxiv.org/html/2608.12555#S4.T2)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→\\rightarrowFeature\-CAS global rank\.Figures[3](https://arxiv.org/html/2608.12555#S4.F3)and[4](https://arxiv.org/html/2608.12555#S4.F4)show the full population\-trajectory comparison with enlarged manuscript\-scale axis labels\. In 401\(k\), SHAP and TreeSHAP emphasiseinc,piraandage, whereas Feature\-CAS shifts mass towardinc,ageandeduc\. In Pennsylvania, predictive explanations emphasiseagelt35,blackandagegt54, whereas Feature\-CAS placesfemale,dep1andlusdat the top of the causal\-effect\-modifier ordering\.

![Refer to caption](https://arxiv.org/html/2608.12555v1/img/fig1_401k_population.png)Figure 3:401\(k\): population cumulative contribution trajectories \(fulln=9,915n=9\{,\}915; 301 profiles displayed\)\. SHAP and TreeSHAP explain the Random\-Forest outcome model; Feature\-CAS explains the exact cross\-fitted DML\-CATE surface\.![Refer to caption](https://arxiv.org/html/2608.12555v1/img/fig2_bonus_population.png)Figure 4:Pennsylvania bonus: population cumulative contribution trajectories \(fulln=5,099n=5\{,\}099; 301 profiles displayed\)\. 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 \([19](https://arxiv.org/html/2608.12555#S3.E19)\), which removes the earlier asymmetry of selecting a profile for an extreme CATE\. For 401\(k\), the selected row is 4124 \(robustXX\-distance 0\.0653\): the reference\-run DML CATE is approximately $6,848 and the ATE $7,872\. SHAP and LIME both rankpiraandincfirst, whereas Feature\-CAS ranksinc,piraanddb\. Feature\-CAS reconstructs the stored cross\-fitted CATE with absolute error3\.71×10−53\.71\\times 10^\{\-5\}outcome units, corresponding to relative error5\.4×10−95\.4\\times 10^\{\-9\}\. LIME’s local surrogate differs from the Random\-Forest prediction by 9\.4%, with local weightedR2=0\.417R^\{2\}=0\.417\(Figure[5](https://arxiv.org/html/2608.12555#S4.F5)\)\. Because this row is selected as the most central observed profile under the robustXX\-distance rule, the remaining LIME discrepancy should be interpreted as local surrogate fidelity rather than as deliberate selection of a causal outlier\.

![Refer to caption](https://arxiv.org/html/2608.12555v1/img/fig3_401k_local.png)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 fromXXalone\.For Pennsylvania, the selected row is 158 \(robustXX\-distance 0\.258\): the reference\-run DML CATE is−0\.0749\-0\.0749and the ATE−0\.07423\-0\.07423\. Predictive SHAP ranksagelt35,lusdandq2highest; LIME ranksblack,agegt54andagelt35; Feature\-CAS ranksdep1,femaleandq2\. Feature\-CAS reconstruction error is1\.10×10−91\.10\\times 10^\{\-9\}outcome units, corresponding to relative error1\.46×10−81\.46\\times 10^\{\-8\}\. LIME is locally close to the Random\-Forest prediction, with a 0\.31% surrogate discrepancy and localR2=0\.808R^\{2\}=0\.808\(Figure[6](https://arxiv.org/html/2608.12555#S4.F6)\)\. The contrast with 401\(k\) illustrates that LIME fidelity is profile\- and surface\-dependent and should be reported rather than assumed\.

![Refer to caption](https://arxiv.org/html/2608.12555v1/img/fig4_bonus_local.png)Figure 6:Pennsylvania method\-neutral typical profile\. The leading Feature\-CAS modifiers aredep1,femaleandq2, 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 \(Table[3](https://arxiv.org/html/2608.12555#S4.T3)\)\. The ATE and CATE values printed in Figures[5](https://arxiv.org/html/2608.12555#S4.F5)and[6](https://arxiv.org/html/2608.12555#S4.F6)are the single reference\-run estimates used to construct those exact decompositions; Table[3](https://arxiv.org/html/2608.12555#S4.T3)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 overR=5R=5repetitions\. Values are mean±\\pmstandard deviation \[descriptive 95% Monte Carlo interval\]\.

## 5What 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\|A​T​E\|\|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\.

## 6Conclusion

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 ADataset variables

Table[4](https://arxiv.org/html/2608.12555#A1.T4)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](https://arxiv.org/html/2608.12555#bib.bib8), while the Pennsylvania Reemployment Bonus experiment is documented by[3](https://arxiv.org/html/2608.12555#bib.bib9)\. Variable names and preprocessing follow the DoubleML data interface\([2](https://arxiv.org/html/2608.12555#bib.bib6)\)\.

Table 4:Variables used in the empirical analysis\.Dataset / roleVariableDefinition401\(k\), outcomenet\_tfaNet financial assets, US dollars\.401\(k\), treatmente401Eligibility for a 401\(k\) plan\.401\(k\), covariateageAge\.401\(k\), covariateincIncome, US dollars\.401\(k\), covariateeducEducation in years\.401\(k\), covariatefsizeFamily size\.401\(k\), covariatemarrMarried indicator\.401\(k\), covariatetwoearnTwo\-earner household indicator\.401\(k\), covariatedbDefined\-benefit pension indicator\.401\(k\), covariatepiraIndividual retirement account participation indicator\.401\(k\), covariatehownHome\-owner indicator\.Pennsylvania, outcomeinuidur1DoubleML\-preprocessed log duration of the first unemployment spell; the raw measure is in weeks\.Pennsylvania, treatmenttgBinary treatment\-group indicator in the preprocessed bonus\-analysis sample\.Pennsylvania, covariatefemaleFemale indicator\.Pennsylvania, covariateblackBlack\-race indicator\.Pennsylvania, covariateothraceOther\-race indicator: non\-white, non\-black, non\-Hispanic\.Pennsylvania, covariatedep1Indicator for exactly one dependent\.Pennsylvania, covariatedep2Indicator for exactly two dependents\.Pennsylvania, covariateq2Enrollment\-quarter 2 indicator; quarter 1 is the reference\.Pennsylvania, covariateq3Enrollment\-quarter 3 indicator; quarter 1 is the reference\.Pennsylvania, covariateq4Enrollment\-quarter 4 indicator; quarter 1 is the reference\.Pennsylvania, covariateq5Enrollment\-quarter 5 indicator; quarter 1 is the reference\.Pennsylvania, covariateq6Enrollment\-quarter 6 indicator; quarter 1 is the reference\.Pennsylvania, covariateagelt35Indicator for claimant age below 35\.Pennsylvania, covariateagegt54Indicator for claimant age above 54\.Pennsylvania, covariatedurableIndicator for an occupation in durable manufacturing\.Pennsylvania, covariatelusdIndicator for filing at a low\-unemployment / short\-duration site \(Coatesville, Reading or Lancaster\)\.Pennsylvania, covariatehusdIndicator for filing at a high\-unemployment / short\-duration site \(Lewistown, Pittston or Scranton\)\.
## Appendix BKnown\-truth structural benchmark

The reproducible known\-truth benchmark uses six independent standard\-normal covariatesX1,…,X6X\_\{1\},\\ldots,X\_\{6\}, three binary interventionsA1,A2,A3A\_\{1\},A\_\{2\},A\_\{3\}and

μ⁡\(X\)\\displaystyle\\mu\(X\)=0\.6X1−0\.35X2\+0\.25X32\+0\.20sinX4\+0\.15X5X6,\\displaystyle=0\.6X\_\{1\}\-0\.35X\_\{2\}\+0\.25X\_\{3\}^\{2\}\+0\.20\\sin X\_\{4\}\+0\.15X\_\{5\}X\_\{6\},\(21\)τ1​\(X\)\\displaystyle\\tau\_\{1\}\(X\)=1\+0\.65​X1−0\.20​X2,\\displaystyle=1\+0\.65X\_\{1\}\-0\.20X\_\{2\},\(22\)τ2​\(X\)\\displaystyle\\tau\_\{2\}\(X\)=0\.35\+0\.70sinX2\+0\.25X3,\\displaystyle=0\.35\+0\.70\\sin X\_\{2\}\+0\.25X\_\{3\},\(23\)τ3​\(X\)\\displaystyle\\tau\_\{3\}\(X\)=0\.55−0\.75​𝟙​\(X1<0\)\+0\.35​X4\.\\displaystyle=0\.55\-0\.75\\mathbb\{1\}\(X\_\{1\}<0\)\+0\.35X\_\{4\}\.\(24\)Interaction functions are

γ12​\(X\)\\displaystyle\\gamma\_\{12\}\(X\)=s⁡\{0\.90\+0\.30​𝟙​\(X1\>0\)\},\\displaystyle=s\\\{0\.90\+0\.30\\mathbb\{1\}\(X\_\{1\}\>0\)\\\},\(25\)γ23​\(X\)\\displaystyle\\gamma\_\{23\}\(X\)=s⁡\(−0\.75\+0\.25​X2\),\\displaystyle=s\(\-0\.75\+0\.25X\_\{2\}\),\(26\)γ13​\(X\)\\displaystyle\\gamma\_\{13\}\(X\)=0\.45​s​tanh⁡\(X3\+X5\),\\displaystyle=0\.45s\\tanh\(X\_\{3\}\+X\_\{5\}\),\(27\)wheres∈\{0,1,1\.8\}s\\in\\\{0,1,1\.8\\\}gives the additive, primary\-interaction and strong\-interaction scenarios\. The structural outcome is

Y=μ⁡\(X\)\+∑j=13Aj​τj​\(X\)\+γ12​A1​A2\+γ23​A2​A3\+γ13​A1​A3\+ε,ε∼N⁡\(0,0\.82\)\.Y=\\mu\(X\)\+\\sum\_\{j=1\}^\{3\}A\_\{j\}\\tau\_\{j\}\(X\)\+\\gamma\_\{12\}A\_\{1\}A\_\{2\}\+\\gamma\_\{23\}A\_\{2\}A\_\{3\}\+\\gamma\_\{13\}A\_\{1\}A\_\{3\}\+\\varepsilon,\\qquad\\varepsilon\\sim N\(0,0\.8^\{2\}\)\.\(28\)The three treatment propensities are

e1​\(X\)\\displaystyle e\_\{1\}\(X\)=logit−1⁡\(−0\.15\+0\.60​X1−0\.25​X4\),\\displaystyle=\\operatorname\{logit\}^\{\-1\}\(\-0\.15\+0\.60X\_\{1\}\-0\.25X\_\{4\}\),\(29\)e2​\(X\)\\displaystyle e\_\{2\}\(X\)=logit−1⁡\(0\.10−0\.45​X2\+0\.30​X5\),\\displaystyle=\\operatorname\{logit\}^\{\-1\}\(0\.10\-0\.45X\_\{2\}\+0\.30X\_\{5\}\),\(30\)e3​\(X\)\\displaystyle e\_\{3\}\(X\)=logit−1⁡\(−0\.05\+0\.40​X3−0\.25​X6\),\\displaystyle=\\operatorname\{logit\}^\{\-1\}\(\-0\.05\+0\.40X\_\{3\}\-0\.25X\_\{6\}\),\(31\)withAj∼Bernoulli⁡\(ej​\(X\)\)A\_\{j\}\\sim\\operatorname\{Bernoulli\}\(e\_\{j\}\(X\)\)conditionally onXX\. For each profile, the benchmark evaluates all eight intervention coalitions exactly, computes the true causal Shapley allocation by Equation \([5](https://arxiv.org/html/2608.12555#S2.E5)\) and compares it with allocations obtained from an estimated Random\-Forest outcome surface\. The results in Section[4](https://arxiv.org/html/2608.12555#S4)usen=2,200n=2\{,\}200andR=8R=8independent seeds per scenario\.

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