What Is a Skill Worth? Structure-Aware Shapley Valuation of Agent Skills

# What Is a Skill Worth? Structure-Aware Shapley Valuation of Agent Skills
Source: [https://arxiv.org/html/2608.04562](https://arxiv.org/html/2608.04562)
Tao Li1Junfeng Liu2Qinghua Zhao3Yifan Li2Lei Wang2Bo Shao4Xuejun Liu1Linjun Shou4 1Nanjing University of Aeronautics and Astronautics2Pengcheng Laboratory 3Hefei University4Microsoft

###### Abstract

Agent skills are increasingly optimized by automated feedback loops, producing long structured artifacts whose internal value remains unclear\. We study*skill valuation*: assigning credit to the internal units of a fixed skill, such as rules, examples, scripts, and heuristics, under a fixed agent and held\-out task distribution\. Skill valuation differs from data or prompt\-span valuation because skill units are structured: they may depend on other units, belong to a document hierarchy, trigger agent behavior, and consume limited prompt context\. We introduceSkillSV, a structure\-aware Shapley\-style framework for skill valuation\.SkillSVcompiles a skill into units, dependencies, and hierarchy, so that only valid counterfactual skills are evaluated\. It uses paired deletion and length\-neutral padding to separate content value from context cost, and estimates the resulting values with a rollout\-budgeted estimator for noisy agent evaluations\. On four agentic benchmarks, we assess the*faithfulness*,*actionability*, and*explanation*ofSkillSV: it recovers unit interactions, preserves aggregate skill lift, and guides safe pruning and compression\.

*Keywords*Agent skills⋅\\cdotShapley value⋅\\cdotCredit assignment⋅\\cdotLLM agents⋅\\cdotPrompt compression

## 1Introduction

Agent skills are reusable artifacts that turn LLM reasoning into concrete proceduresDinget al\.\([2026](https://arxiv.org/html/2608.04562#bib.bib41)\)\. Automated optimizers increasingly write them, proposing and accepting edits under execution feedbackAgrawalet al\.\([2026](https://arxiv.org/html/2608.04562#bib.bib26)\); Yanget al\.\([2026a](https://arxiv.org/html/2608.04562#bib.bib28)\); Niet al\.\([2026](https://arxiv.org/html/2608.04562#bib.bib27)\); Yuksekgonulet al\.\([2024](https://arxiv.org/html/2608.04562#bib.bib25)\); Yanget al\.\([2026b](https://arxiv.org/html/2608.04562#bib.bib29)\); Alzubiet al\.\([2026](https://arxiv.org/html/2608.04562#bib.bib30)\)\. These loops raise aggregate scores, but return one long skill whose credit assignment is a black box\. What is each piece of a skill worth?

We focus on intra\-skill valuation, which attributes a skill’s performance to its internal unit \(e\.g\., a rule or a script\)\. This fine\-grained perspective is crucial because skills are inherently optimized, debugged, and evolved at the granularity of editable unitsRenet al\.\([2026](https://arxiv.org/html/2608.04562#bib.bib40)\)\. To make skill value well\-defined, we fix three ingredients: a skill artifact under evaluation, a target agent, and a held\-out task distribution with a metric\.

![Refer to caption](https://arxiv.org/html/2608.04562v1/x1.png)Figure 1:Why flat valuation misvalues agent skills, and howSkillSVfixes it\.While existing literature successfully assigns credit to training dataGhorbani and Zou \([2019](https://arxiv.org/html/2608.04562#bib.bib6)\); Wang and Jia \([2023](https://arxiv.org/html/2608.04562#bib.bib8)\); Malekiet al\.\([2013](https://arxiv.org/html/2608.04562#bib.bib9)\); Linet al\.\([2025](https://arxiv.org/html/2608.04562#bib.bib17)\)or attributes contributions to specific model parametersGhorbani and Zou \([2020](https://arxiv.org/html/2608.04562#bib.bib18)\); Liuet al\.\([2023](https://arxiv.org/html/2608.04562#bib.bib10)\); Mohammadi \([2024](https://arxiv.org/html/2608.04562#bib.bib11)\); Xuet al\.\([2026](https://arxiv.org/html/2608.04562#bib.bib20)\); Huanget al\.\([2026](https://arxiv.org/html/2608.04562#bib.bib19)\), these paradigms do not seamlessly transfer to agent skills\. A skill is not a singular data point or prompt, but a structured artifact that may combine Markdown hierarchy, triggers, rules, examples, scripts, and auxiliary files\.

This mismatch creates two challenges \(Fig\.[1](https://arxiv.org/html/2608.04562#S1.F1)\)\.

Challenge I: structural counterfactuals\.A coalition of skill units is not automatically a valid skill\. Removing one unit may leave a dangling reference, an undefined symbol, a missing script, or a section fragment detached from its lead\-in\. Scoring such broken artifacts would measure the agent’s robustness to malformed skills, not the value of the unit removed\. Skill valuation therefore needs a way to compile the artifact into units, dependencies, and hierarchy, and to evaluate only counterfactuals that still read like real skills\.

Challenge II: budgeted rollout estimation\.Even valid counterfactuals are expensive to score\. Each coalition must be rendered as a skill, given to the agent, run on held\-out tasks, and verified\. The estimates are noisy because task difficulty dominates many one\-unit differences\. In addition, deleting a unit changes both the information available to the agent and the prompt length, so a score change may reflect content, context occupancy, or both\.

*Why existing recipes fail\.*These two challenges explain why flat ablation\-style attribution is misleading for skills\. The same issue also affects flat Shapley\-style valuation: averaging over arbitrary subsets gives credit for contexts that are not valid skill artifactsGhorbani and Zou \([2019](https://arxiv.org/html/2608.04562#bib.bib6)\); Kwon and Zou \([2022](https://arxiv.org/html/2608.04562#bib.bib7)\)\. Leave\-one\-out \(LOO\) is the simplest failure caseCook \([1977](https://arxiv.org/html/2608.04562#bib.bib38)\)\. If a surviving unit depends on a removed one, LOO measures breakage rather than contribution\.

We proposeSkillSV, a structure\-aware valuation framework for agent skills\. It compiles a skill into units, dependencies, and hierarchy; defines Shapley\-style unit values over feasible insertion orders; and estimates them with paired deletion/padding rollouts under a fixed budget\. Its output supports editing decisions: keep, compress, or delete\.

Contributions\.\(i\) We formulate intra\-skill valuation as a structure\-constrained cooperative game, identifying structural counterfactuals and budgeted rollout estimation as the two challenges that distinguish skills from flat data valuation\. \(ii\) We introduce a deterministic compiler, paired deletion/padding counterfactuals, and a chain\-coupled estimator for noisy agent rollouts\. \(iii\) We show thatSkillSVrecovers interaction\-sensitive unit values, satisfies value closure, and guides safe pruning and compression with negligible performance loss across four benchmarks\.

## 2Related Work

We first introduce key concepts foundational to our skill valuation\. Appendix[A](https://arxiv.org/html/2608.04562#A1)provides an extended discussion and Appendix[B](https://arxiv.org/html/2608.04562#A2)summarizes notation\.

### 2\.1Agent Skills & Automated Skill Optimization

Agent skills are reusable artifacts that store task\-solving behavior, e\.g\., instructions, tool\-use procedures, examples, scripts, and domain heuristics\. They let language agents reuse prior experience across tasks instead of solving each instance from scratchDinget al\.\([2026](https://arxiv.org/html/2608.04562#bib.bib41)\); Renet al\.\([2026](https://arxiv.org/html/2608.04562#bib.bib40)\)\.

Recent work has moved from manually authored skills toward automated skill generation and refinement\. These methods optimize skills through execution feedbackYuksekgonulet al\.\([2024](https://arxiv.org/html/2608.04562#bib.bib25)\); Agrawalet al\.\([2026](https://arxiv.org/html/2608.04562#bib.bib26)\); Yanget al\.\([2026a](https://arxiv.org/html/2608.04562#bib.bib28)\), trajectory distillationNiet al\.\([2026](https://arxiv.org/html/2608.04562#bib.bib27)\); Yanget al\.\([2026b](https://arxiv.org/html/2608.04562#bib.bib29)\); Zhouet al\.\([2026](https://arxiv.org/html/2608.04562#bib.bib36)\), reinforcement learningShiet al\.\([2026](https://arxiv.org/html/2608.04562#bib.bib33)\); Visheet al\.\([2026](https://arxiv.org/html/2608.04562#bib.bib34)\), and evolutionary searchAlzubiet al\.\([2026](https://arxiv.org/html/2608.04562#bib.bib30)\); Zhanget al\.\([2026](https://arxiv.org/html/2608.04562#bib.bib32)\); Maet al\.\([2026b](https://arxiv.org/html/2608.04562#bib.bib31),[a](https://arxiv.org/html/2608.04562#bib.bib37)\)\. Across these lines, the common goal is to improve the performance of the resulting skill as a whole\. While existing methods improve overall skill performance, they rarely attribute these gains to specific internal components\.

### 2\.2Valuation via Cooperative Games

The Shapley value offers a principled way to assign credit when several components jointly determine a payoff\. A \(transferable\-utility\) cooperative game is a pair\(N,V\)\(N,V\)with player setN=\{1,…,n\}N=\\\{1,\\dots,n\\\}and characteristic functionV:2N→ℝV:2^\{N\}\\\!\\to\\\!\\mathbb\{R\}, whereV​\(S\)V\(S\)is the payoff a coalitionS⊆NS\\subseteq Ncan jointly secure\. The*Shapley value*\(Shapley,[1953](https://arxiv.org/html/2608.04562#bib.bib1)\)is the unique allocation satisfying efficiency, symmetry, linearity, and the null\-player axiom; in permutation form,

ϕi=1\|Π​\(N\)\|​∑π∈Π​\(N\)\[V​\(Siπ∪\{i\}\)−V​\(Siπ\)\],\\phi\_\{i\}\\;=\\;\\frac\{1\}\{\|\\Pi\(N\)\|\}\\sum\_\{\\pi\\in\\Pi\(N\)\}\\big\[V\(S\_\{i\}^\{\\pi\}\\cup\\\{i\\\}\)\-V\(S\_\{i\}^\{\\pi\}\)\\big\],\(1\)whereΠ​\(N\)\\Pi\(N\)is the set ofn\!n\!orderings ofNNandSiπ=\{j∈N:π​\(j\)<π​\(i\)\}S\_\{i\}^\{\\pi\}=\\\{\\,j\\in N:\\pi\(j\)<\\pi\(i\)\\,\\\}collects players precedingiiinπ\\pi\. Each summand is the*marginal contribution*ofiito the coalition it joins, so averaging over all orderings allocates credit fairly under interaction: redundant players split the value they duplicate, while complementary players are rewarded for the synergy they unlock\. Replacing the uniform average overΠ​\(N\)\\Pi\(N\)with an arbitrary distribution over orderings yields Weber’s*probabilistic values*\(Weber,[1988](https://arxiv.org/html/2608.04562#bib.bib2)\), which we later specialize to structure\-respecting skill orderings\.

### 2\.3Shapley\-Based Valuation in ML

Shapley\-based valuation has been widely applied in machine learning, either to internal components or to training data\. Component\-level methods score neuronsGhorbani and Zou \([2020](https://arxiv.org/html/2608.04562#bib.bib18)\), promptsLiuet al\.\([2023](https://arxiv.org/html/2608.04562#bib.bib10)\), model decisionsMohammadi \([2024](https://arxiv.org/html/2608.04562#bib.bib11)\), parametersXuet al\.\([2026](https://arxiv.org/html/2608.04562#bib.bib20)\), and mixture\-of\-experts expertsHuanget al\.\([2026](https://arxiv.org/html/2608.04562#bib.bib19)\), while data\-level methods estimate the worth of individual training examplesGhorbani and Zou \([2019](https://arxiv.org/html/2608.04562#bib.bib6)\); Kwon and Zou \([2022](https://arxiv.org/html/2608.04562#bib.bib7)\); Wang and Jia \([2023](https://arxiv.org/html/2608.04562#bib.bib8)\); Malekiet al\.\([2013](https://arxiv.org/html/2608.04562#bib.bib9)\)\. Across both lines, the units being valued are flat and unstructured, so uniform sampling over coalitions suffices\. In contrast, skill components are interdependent, which calls for structure\-respecting orderings rather than uniform ones\.

## 3Method

### 3\.1Skill Compilation

Addressing*structural counterfactuals*, we build counterfactual skills by compiling the skill artifact into a structured game\. The compiler𝖢\\mathsf\{C\}maps a skill to a tripleG=\(N,D,H\)G=\(N,D,H\)with no model in the loop, whereNNare valuation units,DDrecords dependency constraints among units, andHHrecords the document hierarchy used to constrain insertion orders\.

#### UnitsNN\.

We cut valuation units at the granularity at which skill optimizers edit the artifact\(Yanget al\.,[2026a](https://arxiv.org/html/2608.04562#bib.bib28); Agrawalet al\.,[2026](https://arxiv.org/html/2608.04562#bib.bib26)\)\. In the Markdown body, each top\-level list item opens a unit \(e\.g\.,n1n\_\{1\}in Fig\.[3](https://arxiv.org/html/2608.04562#S3.F3)\), while indented continuations stay with their parent\. Headings are structural scaffolding rather than players: a heading is rendered whenever any unit under it survives\. Auxiliary files, such as scripts and references, are compiled as resource units \(shown in Fig\.[3](https://arxiv.org/html/2608.04562#S3.F3)\)\.

#### DependenciesDD\.

Dependencies implement the structural constraint introduced in Sec\.[1](https://arxiv.org/html/2608.04562#S1): a coalition is evaluated only if the units it keeps still form a skill that reads like a valid artifact\. An edge ofDDpoints from a unit to a unit it needs; feasible coalitions must therefore be downward closed under these edges\. We extract nine families of edges\. Three representative ones are as follows\.*Reference edges*cover the syntactically visible cases of markdown links, file\-path tokens, and verbatim heading mentions e\.g\.,n5→rn\_\{5\}\\rightarrow rin Fig\.[3](https://arxiv.org/html/2608.04562#S3.F3)\.*Semantic edges*bind a symbol’s definition to its use\. They carry no syntactic marker and are recovered by analysing the code blocks themselves e\.g\.,n4→sn\_\{4\}\\rightarrow s\.*Continuity edges*bind fragments that a patch\-editing history left adjacent, such as an orphaned table body and the unit carrying its header\. e\.g\.,\{n1→m,n4→m\}\\\{n\_\{1\}\\rightarrow m,n\_\{4\}\\rightarrow m\\\}\.

![Refer to caption](https://arxiv.org/html/2608.04562v1/x2.png)Figure 2:Compiled skill graphG=\(N,D,H\)G=\(N,D,H\)\.
![Refer to caption](https://arxiv.org/html/2608.04562v1/x3.png)Figure 3:Feasible prefixes and removal operators\.

#### HierarchyHH\.

The hierarchyHHrepresents the document tree \(from directories down to valuation units\)\. While dependenciesDDdictate*which*unit coalitions are valid,HHconstrains the*order*in which they are evaluated\. Relying solely onDDallows unnatural, interleaved evaluation orders \(e\.g\.,m,n1,n4,n2,n3m,n\_\{1\},n\_\{4\},n\_\{2\},n\_\{3\}\), where a unit might be judged against fragmented, partially opened sections\. To prevent this,HHenforces*contiguity*: sibling units must be grouped and evaluated together as a block \(e\.g\.,m,\[n1,n2,n3\],n4m,\[n\_\{1\},n\_\{2\},n\_\{3\}\],n\_\{4\}\)\. This guarantees that units are evaluated against complete, intact sections\. Because dependencies cannot enforce contiguity,HHrestricts evaluation permutations\(Winter,[1989](https://arxiv.org/html/2608.04562#bib.bib3)\), while𝖢\\mathsf\{C\}repairs structural conflicts betweenHHandDDduring sampling\.

#### Feasibilityℱ\\mathcal\{F\}\.

GivenNNandDD,ℱ\\mathcal\{F\}is the family of coalitions we allow the renderer to evaluate\. A subsetS⊆NS\\subseteq Nis feasible if it is downward closed under dependencies: wheneveru∈Su\\in Sandu→v∈Du\\to v\\in D, thenv∈Sv\\in S\(in Fig[3](https://arxiv.org/html/2608.04562#S3.F3)\(a\)\)\. Thus a kept unit must also keep the units it needs\. The purpose is distributional, not only syntactic\. Feasible coalitions should still read like real skills: no removed scripts referenced by surviving text, no undefined symbols, and no fragments detached from their lead\-in\. We do not fix infeasible coalitions by rewriting the remaining text, because that would measure deletion plus rewriting rather than unit value \.

Appendix[C](https://arxiv.org/html/2608.04562#A3)lists the full rule table and a walk\-through\.

### 3\.2From Skill Structure to Unit Value

The compiled structure tells us which pieces of a skill may coexist, but not how valuable any piece is\. We assign value by an insertion thought experiment: place the units in a valid order, add them one at a time, and measure how much the agent’s score changes when a given unit appears\. The value of a unit is the average of this*marginal gain*over valid partial\-skill contexts\.

Two choices are specific to skills\. First, not every subset is a valid partial skill: removing a target, a definition, or part of a section can leave a broken document\. This yields feasible insertion orders and their distributionμ\\mu\. Second, a valid subset is still only a set of units, not text\. Rendering it requires a removal operatorρ\\rho, whose choice determines whether removed content also shortens the prompt\.

#### Feasible orders andμ\\mu\.

A unit is valued by the score change when it is inserted into a partial skill\. We therefore only use insertion orders whose every prefix is itself a valid partial skill\. Formally, a permutation ofNNis feasible if each prefix lies inℱ\\mathcal\{F\}and is contiguous underHH\. Fig\.[3](https://arxiv.org/html/2608.04562#S3.F3)\(a\) shows the coalition\-level condition behind this definition: each prefix must be downward\-closed inDD, so a unit is never evaluated without the units it needs\.

Many feasible orders may remain\. Different samplers weight different partial skills, so we make this choice by indexing the value by a distributionμ\\muover feasible orders\. We do not claim a canonical uniform value: the layerwise sampler used in practice is generally not uniform over linear extensions, and uniform sampling is computationally hard\. Appendix[D](https://arxiv.org/html/2608.04562#A4)gives the sampler and its support properties\.

#### Rendering coalitions withρ\\rho\.

An insertion order tells us which coalitions to score, but a coalitionS⊆NS\\subseteq Nis not yet a prompt\. A removal operatorρ\\rhorendersSSas a skill by keeping the units inSSand handling the missing units inN∖SN\\setminus S\.

We requireρ\\rhoto be local: it may change only removed units, while surviving units remain byte\-for\-byte unchanged\. This makes the measured effect a deletion effect, not deletion plus rewriting\. Length is the second issue\. Removing a unit removes information and also shortens the prompt, so we use two local operators\.ρdel\\rho\_\{\\mathrm\{del\}\}deletes missing units outright;ρpad\\rho\_\{\\mathrm\{pad\}\}replaces them with length\-matched neutral placeholders\. Fig\.[3](https://arxiv.org/html/2608.04562#S3.F3)\(b\) illustrates the two renderings on the same feasible coalition\.

#### Unit value\.

Given a feasible\-order distributionμ\\muand a rendering operatorρ\\rho, the value of an execution unitiiis its expected marginal contribution\. This is the random\-order form of the Shapley marginal, with two changes: orders are restricted to feasible skill orders, and the uniform distribution is replaced by the declared distributionμ\\mu\(Ghorbani and Zou,[2019](https://arxiv.org/html/2608.04562#bib.bib6); Chiet al\.,[2025](https://arxiv.org/html/2608.04562#bib.bib5)\):

ϕi,ρ​\(μ\)=𝔼π∼μ​\[Vρ​\(Siπ∪\{i\}\)−Vρ​\(Siπ\)\],\\phi\_\{i,\\rho\}\(\\mu\)=\\mathbb\{E\}\_\{\\pi\\sim\\mu\}\\\!\\left\[V\_\{\\rho\}\\\!\\big\(S\_\{i\}^\{\\pi\}\\cup\\\{i\\\}\\big\)\-V\_\{\\rho\}\\\!\\big\(S\_\{i\}^\{\\pi\}\\big\)\\right\],\(2\)whereSiπS\_\{i\}^\{\\pi\}is the set of units precedingiiinπ\\pi\. The coalition value is the average verified rollout score after the agent is given the rendered skillρ​\(S\)\\rho\(S\):

Vρ​\(S\)=1M​∑t=1Mscore⁡\(Agent⁡\(ρ​\(S\),t\)\),V\_\{\\rho\}\(S\)=\\frac\{1\}\{M\}\\sum\_\{t=1\}^\{M\}\\operatorname\{score\}\\\!\\left\(\\operatorname\{Agent\}\(\\rho\(S\),t\)\\right\),\(3\)whereMMis the number of held\-out tasks,Agent⁡\(ρ​\(S\),t\)\\operatorname\{Agent\}\(\\rho\(S\),t\)is the agent’s rollout on taskttusing the rendered skillρ​\(S\)\\rho\(S\), andscore\\operatorname\{score\}is the programmatic task verifier\. For stochastic agents,score\\operatorname\{score\}denotes the rollout average used by the evaluator\.

With this definition, the two renderings give complementary views of the same unit\.ρdel\\rho\_\{\\mathrm\{del\}\}measures the net effect of removing the unit from the prompt, whileρpad\\rho\_\{\\mathrm\{pad\}\}measures its marginal contribution with the document footprint held fixed\. We therefore report:

content value=ϕi,ρpad,\\displaystyle=\\;\\phi\_\{i,\\rho\_\{\\mathrm\{pad\}\}\},context cost=ϕi,ρpad−ϕi,ρdel,\\displaystyle=\\;\\phi\_\{i,\\rho\_\{\\mathrm\{pad\}\}\}\-\\phi\_\{i,\\rho\_\{\\mathrm\{del\}\}\},net effect=ϕi,ρdel\.\\displaystyle=\\;\\phi\_\{i,\\rho\_\{\\mathrm\{del\}\}\}\.A unit with positive content value but large context cost is a compression target rather than a deletion target\. At the grand coalition the two renderings agree,Vρdel​\(N\)=Vρpad​\(N\)V\_\{\\rho\_\{\\mathrm\{del\}\}\}\(N\)=V\_\{\\rho\_\{\\mathrm\{pad\}\}\}\(N\), which gives a runtime check on the renderer\. That is, the context\-cost term is interpreted under the placeholder\-neutrality assumption\.

The frontmatter unitmmis treated separately because it triggers the skill rather than executing one part of it\. We define its trigger value as

θm=Vρdel​\(\{m\}\)−Vρdel​\(∅\)\.\\theta\_\{m\}=V\_\{\\rho\_\{\\mathrm\{del\}\}\}\(\\\{m\\\}\)\-V\_\{\\rho\_\{\\mathrm\{del\}\}\}\(\\varnothing\)\.The deletion operator is used here becauseρpad​\(∅\)\\rho\_\{\\mathrm\{pad\}\}\(\\varnothing\)would be a document of placeholders rather than a bare agent\. Since every execution unit depends onmm, every feasible order begins withmm; all other units are therefore valued conditional on the skill being triggered

Algorithm 1Chain\-coupled task\-window estimation\.0:compiled game

G=\(N,D,H\)G=\(N,D,H\), stratified task panel, number of orders

KK, window size

bb, tolerance

τ\\tau
0:trigger value

θ^m\\hat\{\\theta\}\_\{m\}and unit values

ϕ^i,ρdel,ϕ^i,ρpad\\hat\{\\phi\}\_\{i,\\rho\_\{\\mathrm\{del\}\}\},\\hat\{\\phi\}\_\{i,\\rho\_\{\\mathrm\{pad\}\}\}for execution units

i≠mi\\neq m
1:initialize running sums for

θ^m\\hat\{\\theta\}\_\{m\}and all

ϕ^i,ρ\\hat\{\\phi\}\_\{i,\\rho\}to zero

2:for

k=1k=1to

KKdo

3:draw a feasible order

πk\\pi\_\{k\}respecting

DDand

HH
4:draw a stratified task window

BkB\_\{k\}of size

bb, independent of

πk\\pi\_\{k\}
5:evaluate

v¯ρdel​\(∅,Bk\)\\bar\{v\}\_\{\\rho\_\{\\mathrm\{del\}\}\}\(\\varnothing,B\_\{k\}\)and

v¯ρdel​\(\{m\},Bk\)\\bar\{v\}\_\{\\rho\_\{\\mathrm\{del\}\}\}\(\\\{m\\\},B\_\{k\}\)
6:add

v¯ρdel​\(\{m\},Bk\)−v¯ρdel​\(∅,Bk\)\\bar\{v\}\_\{\\rho\_\{\\mathrm\{del\}\}\}\(\\\{m\\\},B\_\{k\}\)\-\\bar\{v\}\_\{\\rho\_\{\\mathrm\{del\}\}\}\(\\varnothing,B\_\{k\}\)to the running sum for

θ^m\\hat\{\\theta\}\_\{m\}
7:evaluate the full\-skill anchor

v¯​\(N,Bk\)\\bar\{v\}\(N,B\_\{k\}\)\{same rendered skill for both operators\}

8:for

ρ∈\{ρdel,ρpad\}\\rho\\in\\\{\\rho\_\{\\mathrm\{del\}\},\\rho\_\{\\mathrm\{pad\}\}\\\}do

9:

S←\{m\}S\\leftarrow\\\{m\\\}
10:evaluate

prev←v¯ρ​\(S,Bk\)\\mathrm\{prev\}\\leftarrow\\bar\{v\}\_\{\\rho\}\(S,B\_\{k\}\)\{reuse the value above when

ρ=ρdel\\rho=\\rho\_\{\\mathrm\{del\}\}\}

11:foreach execution unit

iiafter

mmin

πk\\pi\_\{k\}do

12:

S←S∪\{i\}S\\leftarrow S\\cup\\\{i\\\}
13:evaluate

cur←v¯ρ​\(S,Bk\)\\mathrm\{cur\}\\leftarrow\\bar\{v\}\_\{\\rho\}\(S,B\_\{k\}\)
14:add

cur−prev\\mathrm\{cur\}\-\\mathrm\{prev\}to the running sum for

ϕ^i,ρ\\hat\{\\phi\}\_\{i,\\rho\}
15:

prev←cur\\mathrm\{prev\}\\leftarrow\\mathrm\{cur\}\{truncate only when the tolerance holds for both operators\}

16:if

\|cur−v¯​\(N,Bk\)\|≤τ\|\\mathrm\{cur\}\-\\bar\{v\}\(N,B\_\{k\}\)\|\\leq\\tauthen

17:stop this chain for operator

ρ\\rho
18:break

19:endif

20:endfor

21:endfor

22:endfor

23:divide every running sum by

KK
24:return

θ^m\\hat\{\\theta\}\_\{m\},

ϕ^i,ρdel\\hat\{\\phi\}\_\{i,\\rho\_\{\\mathrm\{del\}\}\},

ϕ^i,ρpad\\hat\{\\phi\}\_\{i,\\rho\_\{\\mathrm\{pad\}\}\}

### 3\.3Estimate Unit Values under a Rollout Budget

Addressing*budgeted rollout estimation*, this section estimates the value in Eq\.[2](https://arxiv.org/html/2608.04562#S3.E2)under a fixed rollout budget\. The difficulty is that no coalition value is a lookup: obtaining one requires re\-rendering the skill, running the frozen agent on held\-out tasks, and verifying every outcome\. A full two\-operator audit would require

Rfull=2⏟operators×K⏟orders×\(n\+1\)⏟prefix chain×M⏟tasksR\_\{\\mathrm\{full\}\}=\\underbrace\{2\}\_\{\\text\{operators\}\}\\times\\underbrace\{K\}\_\{\\text\{orders\}\}\\times\\underbrace\{\(n\+1\)\}\_\{\\text\{prefix chain\}\}\\times\\underbrace\{M\}\_\{\\text\{tasks\}\}agent rollouts, which is prohibitive for nontrivial skills\. Forn=50n=50units,M=40M=40tasks and onlyK=10K=10orders, already2×10×51×40=40,8002\\times 10\\times 51\\times 40=40\{,\}800agent rollouts\. This makes naive estimation impractical\.

Two observations make this reduction possible\. First, a sampled feasible order gives a whole prefix chain\. For an orderπk\\pi\_\{k\}, letSjS\_\{j\}be the set of the firstjjunits in that order, so

∅=S0⊂S1⊂⋯⊂Sn=N\.\\varnothing=S\_\{0\}\\subset S\_\{1\}\\subset\\cdots\\subset S\_\{n\}=N\.Each adjacent pair gives the marginal contribution of the unit added at that step\. Thus then\+1n\+1coalition evaluations along one order yield one marginal sample for every unit, rather than only for a single unit\.

Second, the task panel can be windowed only if the window is shared along the chain\. We draw a stratified task windowBkB\_\{k\}of sizeb≪Mb\\ll Mfor each sampled order and evaluate every prefix of that order on the sameBkB\_\{k\}\. Letv¯ρ​\(S,Bk\)\\bar\{v\}\_\{\\rho\}\(S,B\_\{k\}\)denote the average verified rollout score of coalitionSSon the task windowBkB\_\{k\}under rendering operatorρ\\rho:

v¯ρ​\(S,Bk\)=1b​∑t∈Bkscore⁡\(Agent⁡\(ρ​\(S\),t\)\)\.\\bar\{v\}\_\{\\rho\}\(S,B\_\{k\}\)=\\frac\{1\}\{b\}\\sum\_\{t\\in B\_\{k\}\}\\operatorname\{score\}\\\!\\left\(\\operatorname\{Agent\}\(\\rho\(S\),t\)\\right\)\.The marginal at stepjjis then the paired difference

v¯ρ​\(Sj,Bk\)−v¯ρ​\(Sj−1,Bk\)\.\\bar\{v\}\_\{\\rho\}\(S\_\{j\},B\_\{k\}\)\-\\bar\{v\}\_\{\\rho\}\(S\_\{j\-1\},B\_\{k\}\)\.Using the same tasks on both sides cancels much of the task\-level difficulty variation; using different windows would mix the unit effect with differences between tasks\. This pairing makes small windows usable, so the saved budget can be spent on sampling more feasible orders\.

We call this mechanism*chain\-coupled task windows*\. The resulting estimator overKKsampled feasible orders is

ϕ^i,ρ=1K​∑k=1K\[v¯ρ​\(Siπk∪\{i\},Bk\)−v¯ρ​\(Siπk,Bk\)\]\.\\hat\{\\phi\}\_\{i,\\rho\}=\\frac\{1\}\{K\}\\sum\_\{k=1\}^\{K\}\\left\[\\bar\{v\}\_\{\\rho\}\\big\(S\_\{i\}^\{\\pi\_\{k\}\}\\cup\\\{i\\\},B\_\{k\}\\big\)\-\\bar\{v\}\_\{\\rho\}\\big\(S\_\{i\}^\{\\pi\_\{k\}\},B\_\{k\}\\big\)\\right\]\.\(4\)Because each window is drawn independently of the sampled order and gives each task equal inclusion probability, Eq\.[4](https://arxiv.org/html/2608.04562#S3.E4)targets the panel\-level value in Eq\.[2](https://arxiv.org/html/2608.04562#S3.E2)\. The implementation enforces pairing by requiring the two sides of every marginal difference to use the identical task list\.

Windowing reduces the task factor, but a long chain can still spend rollouts on prefixes whose score has already matched the full skill\. We therefore use noise\-gated truncationGhorbani and Zou \([2019](https://arxiv.org/html/2608.04562#bib.bib6)\); Kwon and Zou \([2022](https://arxiv.org/html/2608.04562#bib.bib7)\)\. Each chain first evaluates the anchorv¯ρ​\(N,Bk\)\\bar\{v\}\_\{\\rho\}\(N,B\_\{k\}\)\. While scanning prefixes, if

\|v¯ρ​\(Sj,Bk\)−v¯ρ​\(N,Bk\)\|≤τ,\\left\|\\bar\{v\}\_\{\\rho\}\(S\_\{j\},B\_\{k\}\)\-\\bar\{v\}\_\{\\rho\}\(N,B\_\{k\}\)\\right\|\\leq\\tau,the remaining units on that chain receive zero marginal contribution\. The thresholdτ\\taucontrols truncation bias by bounding the total residual contribution left in the truncated suffix on that window\. Since the test uses a noisy window mean,bbandτ\\tauare calibrated jointly\. The trigger unitmmis minimal in every feasible order, so its trigger value is evaluated before truncation can occur\.

Algorithm[1](https://arxiv.org/html/2608.04562#alg1)also gives the actual cost\. Per sampled order, we evaluate four anchors:v¯ρdel​\(∅,Bk\)\\bar\{v\}\_\{\\rho\_\{\\mathrm\{del\}\}\}\(\\varnothing,B\_\{k\}\),v¯ρdel​\(\{m\},Bk\)\\bar\{v\}\_\{\\rho\_\{\\mathrm\{del\}\}\}\(\\\{m\\\},B\_\{k\}\),v¯ρpad​\(\{m\},Bk\)\\bar\{v\}\_\{\\rho\_\{\\mathrm\{pad\}\}\}\(\\\{m\\\},B\_\{k\}\), and the full\-skill anchorv¯​\(N,Bk\)\\bar\{v\}\(N,B\_\{k\}\)shared by both operators\. The remaining cost is two truncated prefix walks, one forρdel\\rho\_\{\\mathrm\{del\}\}and one forρpad\\rho\_\{\\mathrm\{pad\}\}\. Since the full\-skill value is already cached as the anchor, the last prefix need not be evaluated again\. Ifγ∈\(0,1\]\\gamma\\in\(0,1\]is the average fraction of intermediate prefixes that survive the truncation gate, the rollout cost is

R=K​b​\[4\+2​γ​\(n−2\)\],RRfull≈γ​bM\.R=K\\,b\\,\\big\[4\+2\\gamma\(n\-2\)\\big\],\\qquad\\frac\{R\}\{R\_\{\\mathrm\{full\}\}\}\\approx\\gamma\\,\\frac\{b\}\{M\}\.Forn=50n=50,M=40M=40,K=10K=10,b=8b=8, and a measuredγ≈0\.2\\gamma\\approx 0\.2, this gives a more than tenfold reduction\.

Appendix[E](https://arxiv.org/html/2608.04562#A5)gives the unbiasedness and variance analysis\.

## 4Experiments

We evaluate whetherSkillSVproduces values that are faithful to the defined game, useful for editing skills, and explanatory of optimized skill structure\. The experiments answer three questions:*\(i\) Faithfulness:*doesSkillSVrecover known unit values and preserve the aggregate value of real skills?*\(ii\) Actionability:*can the values guide pruning and compression without meaningful held\-out performance loss?*\(iii\) Explanation:*what attribution patterns in optimized skills make such compression possible?

### 4\.1Experimental Setup

Dataset\.We evaluateSkillSVon four agentic benchmarks covering different task types and interaction budgets: LiveMath\(Heet al\.,[2026](https://arxiv.org/html/2608.04562#bib.bib24)\), OfficeQA\(Opsahl\-Onget al\.,[2026](https://arxiv.org/html/2608.04562#bib.bib23)\), SpreadsheetBench\(SSB, in short\)\(Maet al\.,[2024](https://arxiv.org/html/2608.04562#bib.bib21)\), and ALFWorld\(Shridharet al\.,[2021](https://arxiv.org/html/2608.04562#bib.bib22)\)\. The details of the four benchmarks are summarized in Table[1](https://arxiv.org/html/2608.04562#S4.T1)\. For each benchmark, we evaluate the best\-performingskill\.mdgenerated by four automatic skill optimizers: Trace2Skill\(Niet al\.,[2026](https://arxiv.org/html/2608.04562#bib.bib27)\), TextGrad\(Yuksekgonulet al\.,[2024](https://arxiv.org/html/2608.04562#bib.bib25)\), GEPA\(Agrawalet al\.,[2026](https://arxiv.org/html/2608.04562#bib.bib26)\), and SkillOpt\(Yanget al\.,[2026a](https://arxiv.org/html/2608.04562#bib.bib28)\)\. Note that,SkillSVis optimizer\-agnostic: it values only the compiled skill document, independent of how the skill was produced\.

Table 1:Benchmark statistics and score anchors \(in %\)\.V​\(∅\)V\(\\varnothing\): target model with no skill;V​\(\{m\}\)V\(\\\{m\\\}\): metadata/trigger unit only;V​\(N\)V\(N\): full skill\. Total lift=V​\(N\)−V​\(∅\)=V\(N\)\-V\(\\varnothing\), brackets report the 95% bootstrap confidence interval \(CI\); content lift=V​\(N\)−V​\(\{m\}\)=V\(N\)\-V\(\\\{m\\\}\)\.BenchmarkMaxturnsV​\(∅\)V\(\\varnothing\)V​\(\{m\}\)V\(\\\{m\\\}\)V​\(N\)V\(N\)TotalliftContentliftLiveMath148\.348\.365\.0\+16\.7\+16\.7\[\+3\.3,\+30\.0\]\[\\,\+3\.3,\\ \+30\.0\\,\]\+16\.7\+16\.7OfficeQA2451\.253\.569\.8\+18\.6\+18\.6\[\+10\.5,\+27\.9\]\[\\,\+10\.5,\\ \+27\.9\\,\]\+16\.3\+16\.3Spreadsheet3027\.528\.778\.7\+51\.2\+51\.2\[\+40\.0,\+62\.5\]\[\\,\+40\.0,\\ \+62\.5\\,\]\+50\.0\+50\.0ALFWorld5088\.386\.796\.7\+8\.3\+8\.3\[\+1\.7,\+16\.7\]\[\\,\+1\.7,\\ \+16\.7\\,\]\+10\.0\+10\.0Baselines\.We compareSkillSVwith three baselines that rank the same compiled units in Sec\.[3](https://arxiv.org/html/2608.04562#S3): Closure\-LOOCook \([1977](https://arxiv.org/html/2608.04562#bib.bib38)\), an LLM judgeZhenget al\.\([2023](https://arxiv.org/html/2608.04562#bib.bib39)\), and random ranking\. Closure\-LOO ablates each unit with its dependency closure, while the LLM judge with GPT\-5\.5 scores units independently\. All methods use the same held\-out pruning protocol\.

Implementation details\.We compile skills into units and evaluate coalitions with a frozen target agent \(GPT\-5\.5\) under each benchmark’s native metric\. unit values are estimated with deletion \(ρdel\\rho\_\{\\mathrm\{del\}\}\) and length\-neutral padding \(ρpad\\rho\_\{\\mathrm\{pad\}\}\), which separate content contribution from context\-occupancy cost\. We use a budgeted estimator \(K=12K\{=\}12,b=8b\{=\}8,τ=0\.05\\tau\{=\}0\.05\) and a 1000\-replicate task\-level bootstrap\. Full experimental details are given in Appendix[F](https://arxiv.org/html/2608.04562#A6)\.

Table 2:Value\-closure check on four benchmarks\.Λ=V​\(N\)−V​\(\{m\}\)\\Lambda=V\(N\)\-V\(\\\{m\\\}\)is the content lift \(95% CIs in brackets\);✓/✗marks whether the estimator’s CI coversΛ\\Lambda\.SkillSVtotals recoverΛ\\Lambda, whereas LOO is often miscalibrated\. Note that, the inconsistency betweenV​\(N\)V\(N\)here and in Table[1](https://arxiv.org/html/2608.04562#S4.T1), because we evaluated it on the disjoint panel of held outs for fair comparison\.SizeCoalition valueLiftSkillSVClosure\-LOOBenchmarkUnitsTasksV​\(∅\)V\(\\varnothing\)V​\(\{m\}\)V\(\\\{m\\\}\)V​\(N\)V\(N\)Λ\\LambdaΣ​ϕ\\Sigma\\phiΣ​ϕΛ\\tfrac\{\\Sigma\\phi\}\{\\Lambda\}Σ​LOO\\Sigma\\mathrm\{LOO\}Σ​LOOΛ\\tfrac\{\\Sigma\\mathrm\{LOO\}\}\{\\Lambda\}LiveMath16600\.4830\.4830\.633\+0\.150\+0\.150\[\+0\.033,\+0\.283\]\[\\,\+0\.033,\\ \+0\.283\\,\]\+0\.146\+0\.146\[\+0\.073,\+0\.229\]\[\\,\+0\.073,\\ \+0\.229\\,\]0\.97×0\.97\\times✓−0\.125\-0\.125\[−1\.250,\+0\.875\]\[\\,\-1\.250,\\ \+0\.875\\,\]−0\.83×\-0\.83\\times✗OfficeQA25860\.4880\.5350\.733\+0\.198\+0\.198\[\+0\.116,\+0\.291\]\[\\,\+0\.116,\\ \+0\.291\\,\]\+0\.188\+0\.188\[\+0\.135,\+0\.240\]\[\\,\+0\.135,\\ \+0\.240\\,\]0\.95×0\.95\\times✓\+0\.875\+0\.875\[\+0\.250,\+1\.625\]\[\\,\+0\.250,\\ \+1\.625\\,\]4\.43×4\.43\\times✗Spreadsheet49800\.2750\.2750\.775\+0\.500\+0\.500\[\+0\.388,\+0\.613\]\[\\,\+0\.388,\\ \+0\.613\\,\]\+0\.479\+0\.479\[\+0\.396,\+0\.573\]\[\\,\+0\.396,\\ \+0\.573\\,\]0\.96×0\.96\\times✓−0\.500\-0\.500\[−1\.500,\+0\.500\]\[\\,\-1\.500,\\ \+0\.500\\,\]−1\.00×\-1\.00\\times✓ALFWorld48600\.8830\.9000\.950\+0\.050\+0\.050\[\+0\.000,\+0\.117\]\[\\,\+0\.000,\\ \+0\.117\\,\]\+0\.052\+0\.052\[\+0\.021,\+0\.083\]\[\\,\+0\.021,\\ \+0\.083\\,\]1\.04×1\.04\\times✓−0\.500\-0\.500\[−1\.125,\+0\.000\]\[\\,\-1\.125,\\ \+0\.000\\,\]−10\.00×\-10\.00\\times✗

### 4\.2SkillSVMeasures the Right Value

Recovering planted and real interactions\.We test faithfulness in two settings\. First, we use synthetic skills where the ground\-truth interaction pattern is known: a redundant OR pair, a complementary AND pair, a harmful unit, and a dead unit\. Second, we use real optimized skills of Spreadsheet and check whether the estimated values agree with direct interaction tests, such as removing one unit versus removing both units in a redundant pair\.

Fig\.[5](https://arxiv.org/html/2608.04562#S4.F5)\(a\) shows thatSkillSVrecovers the planted roles across all synthetic cases\. The baselines fail predictably: Closure\-LOO assigns zero to the redundant pair and double\-counts the complementary pair, the LLM judge is inconsistent, and random provides no meaningful signal\.

This effect extends to real skills \(Fig\.[5](https://arxiv.org/html/2608.04562#S4.F5)\(b\)\)\. For a matched redundant pair, removing either unit alone barely affects the score, whereas removing both causes a clear drop\.SkillSVcorrectly recover this\. The interaction value is real, yet entirely missed by single\-removal evaluations\.

![Refer to caption](https://arxiv.org/html/2608.04562v1/x4.png)Figure 4:SkillSVrecovers planted OR/AND unit values\.
![Refer to caption](https://arxiv.org/html/2608.04562v1/x5.png)Figure 5:AUC summary of the pruning curves\.

Recovering total value\.A second faithfulness check asks whether the unit scores add up to the observed gain from the skill content\. For each real skill, we first measure three anchor values: the bare agentV​\(∅\)V\(\\varnothing\), the trigger\-only skillV​\(\{m\}\)V\(\\\{m\\\}\), and the full skillV​\(N\)V\(N\)\. The content lift is

Λ=V​\(N\)−V​\(\{m\}\),\\Lambda=V\(N\)\-V\(\\\{m\\\}\),the improvement attributable to execution units beyond the frontmatter\. If an attribution method assigns calibrated unit values, then summing those unit values should recoverΛ\\Lambda\.

Table[2](https://arxiv.org/html/2608.04562#S4.T2)shows thatSkillSVpasses this closure check\. Across all four benchmarks,∑iϕi\\sum\_\{i\}\\phi\_\{i\}closely matches the measured content lift, with ratios from0\.95×0\.95\\timesto1\.04×1\.04\\times\. This is the*empirical counterpart of Shapley efficiency*: the budgeted estimator preserves the aggregate value of the skill content\. Closure\-LOO fails the same check, often with the wrong sign or totals several times larger thanΛ\\Lambda, because single\-deletion effects are not additive in the presence of redundancy and complementarity\.

### 4\.3SkillSV Enables Safe Compression

Efficacy of Safe Compression\.To directly evaluate the decision\-making utility of unit valuations, we conduct a pruning test: units are sequentially removed in ascending order of their estimated values, and performance is tracked on a held\-out panel\. An ideal valuation accurately isolates redundant content, preserving performance during early pruning steps and yielding a larger Area Under the Curve \(AUC\)\.

As shown in Fig\.[5](https://arxiv.org/html/2608.04562#S4.F5), results indicate thatSkillSVmost robustly guides “safe compression\.” In the pooled evaluation across all four benchmarks,SkillSVachieves a significantly higher AUC than all baselines \(\+0\.026 vs\. LOO, \+0\.049 vs\. LLM judge, \+0\.082 vs\. random, with no 95% CIs containing zero\), showing that the low\-value content it identifies can indeed be safely discarded\.

![Refer to caption](https://arxiv.org/html/2608.04562v1/x6.png)Figure 6:Held\-out pruning curves comparingSkillSVwith Closure\-LOO, an LLM judge, and random unit deletion\.Robustness at High Pruning Ratios\.Moreover, Fig\.[6](https://arxiv.org/html/2608.04562#S4.F6)reports the full held\-out pruning curves, where content units are removed from lowest to highest estimated value and each intermediate skill is re\-scored\.

An ideal valuation should maintain the full\-skill scoreV​\(N\)V\(N\)during early pruning, dropping toward the minimal\-skill floorV​\(\{m\}\)V\(\\\{m\\\}\)only after all expendable content is removed\. Across all four benchmarks,SkillSVsustains peak performance at significantly higher pruning ratios than Closure\-LOO, the LLM judge, or random ranking \(which degrades rapidly and even collapses belowV​\(\{m\}\)V\(\\\{m\\\}\)on OfficeQA\)\. This indicates thatSkillSVranks units by their true removable contribution rather than by isolated leave\-one\-out effects, enabling more aggressive compression before performance is affected\.

Lossless Skill Compression\.Hard deletion tests whether a valuation can rank units for removal, but editing a skill in practice is less binary: a low\-value unit may be deleted, merged with a redundant unit, or rewritten more compactly\. We therefore run a single attribution\-guided refinement step\. The editor receives the original skill and theSkillSVreport computed on panel A, including each unit’s content value, context cost, and net effect\. It is instructed to preserve high\-value content, remove harmful or near\-zero units, and compress units whose content value is positive but whose context cost is large\.

We then evaluate the revised skill once on the disjoint panel B and compare it with the original skill on the same tasks\. Table[3](https://arxiv.org/html/2608.04562#S4.T3)shows that the revised skills retain only 69% of the original tokens on average, yet no significant performance change on all 4 benchmarks\. Thus,SkillSVeffectively facilitates the lossless compression of converged skills\.

Table 3:Safe compression usingSkillSV\-only refinement\.BenchmarkMMTok\.VoldV\_\{\\text\{old\}\}VnewV\_\{\\text\{new\}\}Δ\\Delta\[95% CI\]LiveMath6296%61\.356\.5−4\.8\-4\.8\[−14\.5,\+4\.8\]\[\-14\.5,\+4\.8\]OfficeQA8680%64\.065\.1\+1\.2\+1\.2\[−7\.0,\+9\.3\]\[\-7\.0,\+9\.3\]Spreadsheet14057%73\.673\.6\+0\.0\+0\.0\[−5\.0,\+5\.0\]\[\-5\.0,\+5\.0\]ALFWorld6442%89\.187\.5−1\.6\-1\.6\[−9\.4,\+4\.7\]\[\-9\.4,\+4\.7\]*mean**69%**−1\.3\-1\.3*
### 4\.4Why Compression Works

The compression results above are possible because optimized skill files contain many units whose measured contribution is small\. We next give a deep analysis\.

First, within that content, value is far from uniform\. From Fig\.[7](https://arxiv.org/html/2608.04562#S4.F7)\(a\), the top 10% of units account for 21% of the mass on OfficeQA, 35% on LiveMath, 60% on Spreadsheet, and 100% on ALFWorld\. This concentration explains why low\-value pruning can remove substantial text before performance changes\.

Second, Fig\.[7](https://arxiv.org/html/2608.04562#S4.F7)\(b\) illustrates the same point at unit level on Spreadsheet\.SkillSVseparates a small number of influential units from values near the estimator’s noise floor\. By contrast, Closure\-LOO collapses estimates to zero because redundant units mask each other in the full context\. True sparsity emerges only through multi\-context evaluation\.

![Refer to caption](https://arxiv.org/html/2608.04562v1/x7.png)Figure 7:Concentration and resolution of skill values\.

## 5Conclusion

We introducedSkillSV, a framework for attributing the performance of an agent skill to the units inside the skill artifact\. The central point is that skill valuation is not flat ablation: counterfactual skills must respect dependencies and hierarchy, deletion must be separated from prompt\-length effects, and estimates must be obtained under a tight rollout budget\.SkillSVaddresses these requirements by compiling skills into structured games, valuing units over feasible insertion orders, and estimating paired deletion/padding counterfactuals with chain\-coupled task windows\.

Empirically,SkillSVrecovers interaction\-sensitive unit values, preserves aggregate skill lift, and supports safe pruning and compression across four agentic benchmarks\. These results suggest that optimized skills often contain redundant or low\-value content that can be identified without treating the skill as an opaque prompt\. More broadly, structure\-aware valuation provides a practical diagnostic layer for future skill optimization systems: not only whether a skill works, but which parts work, which parts are costly, and which parts can be safely changed\.

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## Appendix AExtended Related Work

The main related\-work section introduces the concepts needed to read the method\. Here we expand on two points: howSkillSVrelates to cooperative games with restricted cooperation, and how it differs from prior Shapley\-based valuation in machine learning\.

### A\.1Cooperative Games with Restricted Cooperation

Classical Shapley valuation averages marginal contributions uniformly over all player permutations, as in Eq\.[1](https://arxiv.org/html/2608.04562#S2.E1)\. Games with restricted cooperation retain this marginal\-contribution perspective but constrain which coalitions or joining orders are admissible, or change how admissible orders are weighted\.

Precedence\-constrained games restrict admissible orders to linear extensions of a partial order\[Faigle and Kern,[1992](https://arxiv.org/html/2608.04562#bib.bib4)\]\. Consequently, every prefix is downward closed: a player can enter only after its prerequisites\. This corresponds to the role of the dependency DAGDDinSkillSV\. Winter’s levels\-structure value instead imposes a nested organization on players and requires members of the same hierarchical block to remain contiguous in an admissible order\[Winter,[1989](https://arxiv.org/html/2608.04562#bib.bib3)\]\. This corresponds to the role of the document hierarchyHH\.

#### Relation to PC\-Winter\.

PC\-Winter\[Chiet al\.,[2025](https://arxiv.org/html/2608.04562#bib.bib5)\]combines precedence and hierarchy constraints derived from a graph computation structure, and values nodes over the resulting admissible orders\.SkillSVfollows the same broad perspective but differs in three ways\. First,DDandHHare independently compiled from skill dependencies and document structure, so their compatibility is not guaranteed\. Second, a coalition must be rendered into an executable skill byρ\\rho, making its payoffVρV\_\{\\rho\}renderer\-dependent\. Third, because our sampler need not be uniform over feasible orders, we explicitly index the value by its induced distributionμ\\mu, following the probabilistic\-value perspective\[Weber,[1988](https://arxiv.org/html/2608.04562#bib.bib2)\]\.

Thus,SkillSVis not simply PC\-Winter applied to a different data modality\. It adopts the general idea of restricting marginal\-contribution orders, while adding independently compiled dependency and hierarchy structures, renderer\-dependent counterfactuals, and an explicit feasible\-order distribution\. In the unconstrained special case with uniformμ\\mu, the conditional execution\-unit game reduces to classical Shapley valuation\.

### A\.2Shapley\-Based Valuation in Machine Learning

Shapley\-based and related cooperative\-game methods have been used to value training data, model components, and prompt or context fragments\. Many of these methods use a flat player set\. Grouped and hierarchical variants also exist, but their structure typically changes the attribution granularity or aggregation rule rather than declaring a text coalition structurally invalid\.SkillSVuses structure for the latter purpose: dependencies determine admissible coalitions, and hierarchy restricts admissible joining orders\.

#### Data and model\-component valuation\.

Data Shapley assigns credit to training examples through their marginal effects on model utility\[Ghorbani and Zou,[2019](https://arxiv.org/html/2608.04562#bib.bib6)\]\. Beta Shapley modifies the coalition weighting to improve robustness and estimation\[Kwon and Zou,[2022](https://arxiv.org/html/2608.04562#bib.bib7)\], while Data Banzhaf provides a related cooperative\-game alternative\[Wang and Jia,[2023](https://arxiv.org/html/2608.04562#bib.bib8)\]\. The same perspective has been applied to model components, including neurons\[Ghorbani and Zou,[2020](https://arxiv.org/html/2608.04562#bib.bib18)\], parameters\[Xuet al\.,[2026](https://arxiv.org/html/2608.04562#bib.bib20)\], and mixture\-of\-experts experts\[Huanget al\.,[2026](https://arxiv.org/html/2608.04562#bib.bib19)\]\. These methods capture redundancy and complementarity, but their interventions remain defined without introducing dependency\-closed coalitions or hierarchy\-constrained orders\.

#### Prompt and context attribution\.

Text\-based attribution is closer to skill valuation\. Prior work values prompt segments or input attributes\[Liuet al\.,[2023](https://arxiv.org/html/2608.04562#bib.bib10), Mohammadi,[2024](https://arxiv.org/html/2608.04562#bib.bib11)\], tokens\[Horovicz and Goldshmidt,[2024](https://arxiv.org/html/2608.04562#bib.bib15), Xiaoet al\.,[2025](https://arxiv.org/html/2608.04562#bib.bib14)\], context sentences\[Cohen\-Wanget al\.,[2024](https://arxiv.org/html/2608.04562#bib.bib16)\], in\-context demonstrations\[Xieet al\.,[2025](https://arxiv.org/html/2608.04562#bib.bib12)\], and source documents\[Ye and Yoganarasimhan,[2025](https://arxiv.org/html/2608.04562#bib.bib13)\]\. These methods generally construct coalitions by deleting, masking, or selecting text fragments\. However, they do not explicitly model a dependency relation under which removing one fragment can invalidate another, or a document hierarchy that constrains the contexts in which a fragment should be evaluated\.

Skills make these constraints consequential\. A surviving unit may reference a removed script, use a deleted definition, or become detached from its structural lead\-in\.SkillSVtherefore evaluates only dependency\-closed coalitions and hierarchy\-compatible orders\. Its local renderer additionally leaves surviving units unchanged, avoiding counterfactuals that combine deletion with uncontrolled rewriting\.

#### Removal effects and agentic evaluation\.

Deleting prompt text can change both its information content and its context footprint\. Prior attribution methods typically include both effects in a single intervention\.SkillSVinstead pairs deletion with length\-matched padding\. Under the placeholder\-neutrality assumption stated in Sec\.[3\.2](https://arxiv.org/html/2608.04562#S3.SS2), the two interventions support separate estimates of content value, context cost, and net effect\. This interpretation is therefore conditional on the placeholder acting only through its context footprint\.

The utility oracle also differs\. Context\-attribution methods typically explain a prediction, output probability, or generated response, whereas skill valuation measures verified performance over a held\-out task distribution after multi\-step agent execution\. Each coalition may therefore require multiple stochastic, tool\-using rollouts\. Our estimator builds on permutation sampling and truncation developed for Data Shapley\[Ghorbani and Zou,[2019](https://arxiv.org/html/2608.04562#bib.bib6), Malekiet al\.,[2013](https://arxiv.org/html/2608.04562#bib.bib9)\], but couples adjacent marginals through shared task windows to reduce variance from the agentic oracle\.

In summary, prompt attribution provides the closest textual intervention setting, while restricted\-cooperation games provide the closest structural formalism\.SkillSVconnects the two by valuing editable text units only through counterfactuals that remain valid, executable skills\.

## Appendix BNotation

Table[4](https://arxiv.org/html/2608.04562#A2.T4)summarizes the notation in order of its use: skill compilation, structural counterfactuals, unit valuation, and budgeted estimation\.

Table 4:Summary of notation\.SymbolMeaning*Skill structure and compilation*𝖢\\mathsf\{C\}Deterministic compiler mapping a skill artifact toGGG=\(N,D,H\)G=\(N,D,H\)Compiled skill structureN,nN,\\ nSet of valuation units, including the trigger unit;n=\|N\|n=\|N\|DDDependency DAG;u→vu\\to vmeans that unituurequires unitvvHHDocument hierarchy over files, sections, and valuation unitsmmFrontmatter/trigger unit required by every execution unitiiAn execution unit,i∈N∖\{m\}i\\in N\\setminus\\\{m\\\}*Coalitions and counterfactuals*SSCoalition of retained units,S⊆NS\\subseteq Nℱ\\mathcal\{F\}Dependency\-closed coalitions:\{S⊆N:u∈S,u→v∈D⇒v∈S\}\\\{S\\subseteq N:u\\in S,\\ u\\to v\\in D\\Rightarrow v\\in S\\\}ρ\\rhoLocal counterfactual rendering operator mappingSSto a skill artifactρdel\\rho\_\{\\mathrm\{del\}\}Deletes all absent unitsN∖SN\\setminus Sρpad\\rho\_\{\\mathrm\{pad\}\}Replaces absent units with length\-matched neutral placeholdersπ\\piFeasible permutation whose prefixes lie inℱ\\mathcal\{F\}and respectHH\-contiguityΠ​\(N\)\\Pi\(N\)Set of all permutations ofNNΠfeas​\(D,H\)\\Pi\_\{\\mathrm\{feas\}\}\(D,H\)Set of feasible permutations underDDandHHμ\\muDeclared distribution over feasible permutationsSiπS\_\{i\}^\{\\pi\}Units precedingiiin permutationπ\\pi*Tasks, value, and valuation*𝒯,t\\mathcal\{T\},\\ tTarget task distribution and a taskt∼𝒯t\\sim\\mathcal\{T\}TA,TBT\_\{A\},T\_\{B\}Disjoint attribution and held\-out confirmation panels, respectivelyMA,MBM\_\{A\},M\_\{B\}Panel sizes,MA=\|TA\|M\_\{A\}=\|T\_\{A\}\|andMB=\|TB\|M\_\{B\}=\|T\_\{B\}\|Agent\\operatorname\{Agent\}Frozen target agentscore\\operatorname\{score\}Programmatic benchmark\-native task verifiervρ​\(S,t\)v\_\{\\rho\}\(S,t\)Per\-task score:score⁡\(Agent⁡\(ρ​\(S\),t\)\)\\operatorname\{score\}\\\!\\left\(\\operatorname\{Agent\}\(\\rho\(S\),t\)\\right\)Vρ​\(S\)V\_\{\\rho\}\(S\)Coalition value underρ\\rho, obtained by averagingvρ​\(S,t\)v\_\{\\rho\}\(S,t\)over the attribution panelTAT\_\{A\}Vρdel​\(∅\)V\_\{\\rho\_\{\\mathrm\{del\}\}\}\(\\varnothing\)Bare\-agent value with no skillϕi,ρ​\(μ\)\\phi\_\{i,\\rho\}\(\\mu\)Expected marginal contribution of execution unitiiunder rendererρ\\rhoand feasible\-order distributionμ\\muθm\\theta\_\{m\}Trigger value:Vρdel​\(\{m\}\)−Vρdel​\(∅\)V\_\{\\rho\_\{\\mathrm\{del\}\}\}\(\\\{m\\\}\)\-V\_\{\\rho\_\{\\mathrm\{del\}\}\}\(\\varnothing\)Λ\\LambdaContent lift:Vρdel​\(N\)−Vρdel​\(\{m\}\)V\_\{\\rho\_\{\\mathrm\{del\}\}\}\(N\)\-V\_\{\\rho\_\{\\mathrm\{del\}\}\}\(\\\{m\\\}\)*Budgeted estimator*KKNumber of sampled feasible permutationsWk,bW\_\{k\},\\ bStratified task window for permutationkkand its size,Wk⊆TAW\_\{k\}\\subseteq T\_\{A\}and\|Wk\|=b\|W\_\{k\}\|=bv¯ρ​\(S,Wk\)\\bar\{v\}\_\{\\rho\}\(S,W\_\{k\}\)Mean score of coalitionSSon windowWkW\_\{k\}underρ\\rhoϕ^i,ρ\\hat\{\\phi\}\_\{i,\\rho\}Budgeted estimate ofϕi,ρ​\(μ\)\\phi\_\{i,\\rho\}\(\\mu\)θ^m\\hat\{\\theta\}\_\{m\}Estimated trigger valueτ\\tauTruncation tolerance relative to the full\-skill anchorγ\\gammaAverage fraction of intermediate prefixes evaluated before truncationR,RfullR,\\ R\_\{\\mathrm\{full\}\}Rollout cost of the budgeted and full estimators, respectively
## Appendix CCompiler Specification

The compiler𝖢\\mathsf\{C\}maps a skill artifact toG=\(N,D,H\)G=\(N,D,H\)through four deterministic passes: unitization, dependency extraction, graph normalization, and dependency–hierarchy reconciliation\. Each output unit records its source span, and each dependency edge records its rule identifier, source and target units, and triggering evidence\. The compiled structure can therefore be audited against the original artifact\.

Table 5:Dependency rules enabled in the compiler\.RuleNameConditionEmitted edge and evidenceR1linkUnituucontains a Markdown link resolving to unit or sectionxx\.u→primary⁡\(x\)u\\to\\operatorname\{primary\}\(x\); link and resolved target\.R2pathUnituucontains a path naming resource unitrr\.u→ru\\to r; matched path\.R3heading\-ref†Unituucontains a verbatim heading mention of at least two words and eight characters\.u→primary⁡\(x\)u\\to\\operatorname\{primary\}\(x\); matched heading\.R4def\-useA supported code block inuuuses a symbol defined in unitvv\.u→vu\\to v; symbol and definition/use locations\.R5example\-refA workflow unituuexplicitly invokes an example resourcerrexposed by the skill manifest\.u→ru\\to r; matched example identifier and resource path\.R6triggeriiis an execution unit\.i→mi\\to m; structural convention\.R7table\-contA unituucontains headerless table rows continuing a table introduced by unitvv\.u→vu\\to v; row and header locations\.R8list\-cont†An unordered top\-level itemuuimmediately refines the nearest preceding ordered itemvvin the same section\.u→vu\\to v; the two list markers\.R9lead\-in†Unitℓ\\ellends in a colon and immediately introduces an uninterrupted run of unitsu1,…,uku\_\{1\},\\ldots,u\_\{k\}\.uj→ℓu\_\{j\}\\to\\ellfor eachjj; lead\-in and run spans\.
†Deterministic surface heuristic\. We ablate R3, R8, and R9 separately because their evidence is weaker than explicit links, paths, and code\-level def–use relations\. R5 is enabled but emits no edge in the reported experiments because none of the evaluated artifacts exposes a separate example resource\.

### C\.1Unitization and Scaffolding

The frontmatter is compiled as the trigger unitmm\. In the Markdown body, each list marker at indentation zero opens a body unit\. Its indented bullets, code blocks, tables, and continuation lines remain attached until the next indentation\-zero item, heading, protected\-region boundary, or end of file; blank lines alone do not terminate a unit\. Each auxiliary file exposed by the skill manifest is compiled as a resource unit\.

A region marked by the producing optimizer as atomically rewritten is compiled as one protected unit\. Any top\-level content not covered by the preceding rules is also compiled as one protected unit and flagged in the compiler report\. This fallback ensures that every non\-scaffold source span belongs to exactly one unit\.

Headings and horizontal separators are scaffold rather than players\. Underρdel\\rho\_\{\\mathrm\{del\}\}, a scaffold span is rendered exactly when at least one unit in its subtree survives\. Underρpad\\rho\_\{\\mathrm\{pad\}\}, an otherwise absent scaffold span is replaced by a length\-matched neutral span\. Thus scaffold rendering is a deterministic function of the coalition rather than an additional player\.

### C\.2Dependency Extraction

For every emitted edgeu→vu\\to v, unituurequires unitvv\. Table[5](https://arxiv.org/html/2608.04562#A3.T5)lists the rules enabled in our experiments\.

#### Worked example\.

Fig\.[3](https://arxiv.org/html/2608.04562#S3.F3)illustrates one compiler trace\. The instruction unitn4n\_\{4\}usesOUTPUT\_PATH, whose definition occurs in the resource unitsscorresponding totemplate\.py\. Rule R4 therefore emits the edgen4→sn\_\{4\}\\to s\. Its audit record stores the rule identifier, symbol name, and definition and use locations\. The remaining edges are recorded analogously under the rules in Table[5](https://arxiv.org/html/2608.04562#A3.T5)\.

### C\.3Graph Normalization

Three conventions are applied after edge extraction\. First, content\-derived rules do not emit edges frommm, preservingmmas the unique minimal trigger unit\. Second, a reference to a section is resolved to its primary unit: the section’s body unit when present, and otherwise its first descendant unit\. This convention applies to R1 and R3\.

Third, every strongly connected component ofDDis contracted into an atomic composite unit\. Its member spans remain in their original document positions but are retained or removed jointly\. The compiler records all members of each contraction because contraction reduces valuation resolution\. The hierarchy is then recomputed before compatibility checking\.

### C\.4Dependency–Hierarchy Compatibility

After cycle contraction,DDis a DAG, but it may still conflict with hierarchical contiguity\. For each internal hierarchy nodehh, let its children form candidate blocks\. Construct a quotient graphQhQ\_\{h\}over these blocks: if a unit under childccdepends on a unit under childc′c^\{\\prime\}, add the precedence arcc′→cc^\{\\prime\}\\to c\. Self\-arcs are ignored\. Assuming each child is internally feasible, its children can be ordered as contiguous blocks if and only ifQhQ\_\{h\}is acyclic\.

The compiler checks hierarchy nodes bottom\-up\. At nodehh, it initially treats each child subtree as one local sampling block\. WheneverQhQ\_\{h\}is cyclic, every child block on a deterministic witnessing cycle is expanded into its repaired child blocks, and the quotient graph is rebuilt\. Expansion relaxes contiguity only at the conflicting boundary: active descendant blocks remain intact, and ancestor constraints are unchanged\. The process produces a set of repaired sampling blocksCh∗C\_\{h\}^\{\*\}whose quotient graph is acyclic\. Every expansion and its witnessing cycle are recorded in the compiler report\.

This procedure terminates because each expansion replaces a non\-leaf block by blocks strictly lower in the finite hierarchy\. In the limiting case,Ch∗C\_\{h\}^\{\*\}consists of individual units; its quotient graph is then a subgraph of the acyclic dependency graph\. The repaired hierarchy therefore admits at least one feasible order while retaining every contiguity constraint that was not implicated in a conflict\.

## Appendix DThe Feasible\-Order Sampler

Section[3\.2](https://arxiv.org/html/2608.04562#S3.SS2)defines unit value with respect to a distributionμ\\muover feasible insertion orders\. This appendix specifies the sampler inducing the experimentalμ\\muand establishes its soundness, full support, and nonuniformity\. Throughout this section,HHdenotes the repaired hierarchy returned by the compatibility pass in Appendix[C\.4](https://arxiv.org/html/2608.04562#A3.SS4)\.

### D\.1Feasible Orders

Write an order asπ=\(π1,…,πn\)\\pi=\(\\pi\_\{1\},\\ldots,\\pi\_\{n\}\)and let

Sjπ=\{π1,…,πj\}S\_\{j\}^\{\\pi\}=\\\{\\pi\_\{1\},\\ldots,\\pi\_\{j\}\\\}be itsjjth prefix\. We say thatπ\\pi*respectsHH*if the units in every active hierarchy block appear consecutively\. A feasible order must both respectHHand have only dependency\-closed prefixes\. Thus,

Πfeas​\(D,H\)≔\{π∈Π​\(N\)\|Sjπ∈ℱ\(j=1,…,n\),π​respects​H\}\.\\Pi\_\{\\mathrm\{feas\}\}\(D,H\)\\coloneqq\\left\\\{\\pi\\in\\Pi\(N\)\\ \\middle\|\\ \\begin\{subarray\}\{c\}S\_\{j\}^\{\\pi\}\\in\\mathcal\{F\}\\quad\(j=1,\\ldots,n\),\\\\ \\pi\\text\{ respects \}H\\end\{subarray\}\\right\\\}\.\(5\)
The two conditions have distinct roles\. Prefix feasibility ensures that a unit never appears before a unit it requires, while hierarchy compatibility prevents units from different document blocks from being arbitrarily interleaved\. Because every execution unit has the dependencyi→mi\\to m, every feasible order begins with the trigger unitmm, consistent with the trigger/execution decomposition in Sec\.[3\.2](https://arxiv.org/html/2608.04562#S3.SS2)\.

### D\.2Recursive Block Sampler

At an internal hierarchy nodehh, letCh∗C\_\{h\}^\{\*\}be the repaired sampling blocks returned by the compatibility pass\. We construct a block graphQhQ\_\{h\}overCh∗C\_\{h\}^\{\*\}: if some unit below blockccdepends on a unit below blockc′c^\{\\prime\}, we add the precedence arcc′→cc^\{\\prime\}\\to c\. Thus every edge points from a prerequisite block to a block that requires it\. The compatibility pass guarantees that every such block graph is acyclic\.

Algorithm 2SampleOrder\(h\)\(h\)0:repaired hierarchy node

hh, dependency DAG

DD
0:an order of all units in

L​\(h\)L\(h\)
1:if

hhis a unitthen

2:return

\[h\]\[h\]
3:endif

4:

R←Ch∗R\\leftarrow C\_\{h\}^\{\*\},

π←\[\]\\pi\\leftarrow\[\\,\]
5:build the block graph

QhQ\_\{h\}over

Ch∗C\_\{h\}^\{\*\}
6:while

R≠∅R\\neq\\varnothingdo

7:

A←\{c∈R:∄​c′∈R​with​c′→c​in​Qh\}A\\leftarrow\\\{c\\in R:\\nexists\\,c^\{\\prime\}\\in R\\text\{ with \}c^\{\\prime\}\\to c\\text\{ in \}Q\_\{h\}\\\}
8:draw

ccuniformly from

AA
9:

π←π∥SampleOrder​\(c\)\\pi\\leftarrow\\pi\\mathbin\{\\\|\}\\textsc\{SampleOrder\}\(c\)
10:

R←R∖\{c\}R\\leftarrow R\\setminus\\\{c\\\}
11:endwhile

12:return

π\\pi

The eligible setAAcontains exactly the remaining sampling blocks whose prerequisite blocks have already been emitted\. SinceQhQ\_\{h\}is acyclic,AAis non\-empty at every iteration\. The order supplied to Algorithm[1](https://arxiv.org/html/2608.04562#alg1)isSampleOrder​\(root⁡\(H\)\)\\textsc\{SampleOrder\}\(\\operatorname\{root\}\(H\)\)\.

### D\.3Sampler Correctness

###### Proposition 1\(Validity and full support\)\.

Assume that the repaired block graph at every hierarchy node is acyclic after the compatibility pass\. Ifμ\\muis induced by Algorithm[2](https://arxiv.org/html/2608.04562#alg2), then

supp⁡\(μ\)=Πfeas​\(D,H\)\.\\operatorname\{supp\}\(\\mu\)=\\Pi\_\{\\mathrm\{feas\}\}\(D,H\)\.

###### Proof\.

We first show that every sampled order is feasible\. The recursive call emits all units under a selected sampling block before selecting another block at the same repaired boundary, so the resulting order respectsHH\. For any dependencyu→vu\\to v, consider the lowest hierarchy node placinguuandvvin different sampling blocks\. Its block graph contains an arc from the block ofvvto the block ofuu\. The sampler therefore emits the block containingvvbefore the block containinguu\. Dependencies internal to a block are handled recursively\. Hence every prerequisite precedes the unit requiring it, and every prefix lies inℱ\\mathcal\{F\}\.

Conversely, letπ∈Πfeas​\(D,H\)\\pi\\in\\Pi\_\{\\mathrm\{feas\}\}\(D,H\)\. Becauseπ\\pirespectsHH, it induces an ordering of the repaired sampling blocks at every internal node\. Whenever the next blockccin this ordering is considered,cccannot depend on an unplaced blockc′c^\{\\prime\}; otherwiseπ\\piwould place a dependent block before one of its prerequisites\. Thusccbelongs to the sampler’s eligible set\. Every eligible block is selected with positive probability, so the finite sequence of choices producingπ\\pialso has positive probability\. ∎

The first inclusion guarantees that Algorithm[1](https://arxiv.org/html/2608.04562#alg1)evaluates only valid partial skills\. The reverse inclusion shows that the sampler does not structurally exclude any feasible order\. It does not imply that feasible orders receive similar probabilities or will all be observed under a finite budget; the induced nonuniform distribution is characterized next\.

### D\.4Whyμ\\muIs Not Uniform

Algorithm[2](https://arxiv.org/html/2608.04562#alg2)samples uniformly from the locally eligible blocks, but this does not imply a uniform distribution over complete feasible orders\. Consider a flat hierarchy over\{x,y,z\}\\\{x,y,z\\\}with the dependencyy→xy\\to x, soxxmust precedeyy\. The feasible orders are

x​y​z,x​z​y,z​x​y\.xyz,\\qquad xzy,\\qquad zxy\.Initially, the eligible units are\{x,z\}\\\{x,z\\\}\. Choosingzzforces the orderz​x​yzxy, whereas choosingxxleaves two possible continuations\. Therefore,

μ​\(z​x​y\)=12,μ​\(x​y​z\)=μ​\(x​z​y\)=14,\\mu\(zxy\)=\\frac\{1\}\{2\},\\qquad\\mu\(xyz\)=\\mu\(xzy\)=\\frac\{1\}\{4\},instead of1/31/3for every order\. A globally uniform sampler would choosexxwith probability2/32/3, proportional to its two feasible continuations, rather than uniformly from the current eligible set\.

Thus, the sampler induces a specific nonuniform distributionμ\\mu\. Changing the sampler would change the value in Eq\.[2](https://arxiv.org/html/2608.04562#S3.E2), soμ\\muis part of the estimand rather than merely an implementation detail\.

## Appendix EEstimator Analysis

Once the value definition is fixed, the estimator is the Monte Carlo average of marginal gains over feasible orders\. The design question is therefore not how to construct a new estimand, but how to estimate the declared value under an expensive rollout oracle without corrupting each marginal comparison\. We address this by binding each sampled order to one shared task window and evaluating every prefix of that order on the same tasks\.

Table 6:Benchmark statistics and evaluation protocols\. Split sizes follow the released SkillOpt manifests\. “Max turns” denotes the per\-episode interaction limit specified by each environment\.BenchmarkTask typeTrainValTestMax turnsReported metricsLiveMathmath theorem MCQ35181241accuracy \(EM\)OfficeQAdoc\-grounded agentic QA502417224EM / F1SpreadsheetBenchspreadsheet manipulation804028030cell pass \(soft/hard\)ALFWorldembodied household tasks391813450task successTable 7:Disjoint task panels allocated by one seeded shuffle without replacement\. Panel A is used for attribution and ranking; panel B is used only for held\-out confirmation\. The listed sizes satisfy\|TA\|\+\|TB\|≤\|test\|\|T\_\{A\}\|\+\|T\_\{B\}\|\\leq\|\\mathrm\{test\}\|\.BenchmarkTestPanel APanel BLiveMath1246062OfficeQA1728686SpreadsheetBench28080140ALFWorld1346064### E\.1One Order Gives a Chain of Execution Marginals

Every feasible order begins with the trigger unitmm\. Write its execution\-unit suffix as\(i1,…,in−1\)\(i\_\{1\},\\ldots,i\_\{n\-1\}\)and define

S0=\{m\},Sj=\{m,i1,…,ij\},Sn−1=N\.S\_\{0\}=\\\{m\\\},\\qquad S\_\{j\}=\\\{m,i\_\{1\},\\ldots,i\_\{j\}\\\},\\qquad S\_\{n\-1\}=N\.The marginal ofiji\_\{j\}on the full attribution panel is

Vρ​\(Sj\)−Vρ​\(Sj−1\)\.V\_\{\\rho\}\(S\_\{j\}\)\-V\_\{\\rho\}\(S\_\{j\-1\}\)\.Averaging this quantity over feasible orders givesϕij,ρ​\(μ\)\\phi\_\{i\_\{j\},\\rho\}\(\\mu\)\. Thus one sampled order yields one marginal sample for every execution unit\. The trigger is not included in this chain: its value is estimated separately asθm=Vρdel​\(\{m\}\)−Vρdel​\(∅\)\\theta\_\{m\}=V\_\{\\rho\_\{\\mathrm\{del\}\}\}\(\\\{m\\\}\)\-V\_\{\\rho\_\{\\mathrm\{del\}\}\}\(\\varnothing\), avoiding the semantically ambiguous padded empty artifact\.

### E\.2Replacing the Attribution Panel by a Window

Evaluating every prefix on allMAM\_\{A\}tasks inTAT\_\{A\}is expensive\. For each sampled orderπk\\pi\_\{k\}, we independently draw a stratified task windowWk⊆TAW\_\{k\}\\subseteq T\_\{A\}of sizeb≪MAb\\ll M\_\{A\}\. The window design is independent ofπk\\pi\_\{k\}and gives every task equal marginal inclusion probability\. We estimate

v¯ρ​\(S,Wk\)=1b​∑t∈Wkvρ​\(S,t\)\.\\bar\{v\}\_\{\\rho\}\(S,W\_\{k\}\)=\\frac\{1\}\{b\}\\sum\_\{t\\in W\_\{k\}\}v\_\{\\rho\}\(S,t\)\.Consequently, for every fixed coalitionSS,

𝔼Wk​\[v¯ρ​\(S,Wk\)\]=1MA​∑t∈TAvρ​\(S,t\)=Vρ​\(S\)\.\\mathbb\{E\}\_\{W\_\{k\}\}\\\!\\left\[\\bar\{v\}\_\{\\rho\}\(S,W\_\{k\}\)\\right\]=\\frac\{1\}\{M\_\{A\}\}\\sum\_\{t\\in T\_\{A\}\}v\_\{\\rho\}\(S,t\)=V\_\{\\rho\}\(S\)\.Applying this identity to both sides of each marginal gives

ϕ^i,ρ=1K​∑k=1K\[v¯ρ​\(Siπk∪\{i\},Wk\)−v¯ρ​\(Siπk,Wk\)\],\\hat\{\\phi\}\_\{i,\\rho\}=\\frac\{1\}\{K\}\\sum\_\{k=1\}^\{K\}\\left\[\\bar\{v\}\_\{\\rho\}\(S\_\{i\}^\{\\pi\_\{k\}\}\\cup\\\{i\\\},W\_\{k\}\)\-\\bar\{v\}\_\{\\rho\}\(S\_\{i\}^\{\\pi\_\{k\}\},W\_\{k\}\)\\right\],and, when truncation is disabled,

𝔼​\[ϕ^i,ρ\]=ϕi,ρ​\(μ\)\.\\mathbb\{E\}\[\\hat\{\\phi\}\_\{i,\\rho\}\]=\\phi\_\{i,\\rho\}\(\\mu\)\.The same windows give the trigger estimator

θ^m=1K​∑k=1K\[v¯ρdel​\(\{m\},Wk\)−v¯ρdel​\(∅,Wk\)\],\\hat\{\\theta\}\_\{m\}=\\frac\{1\}\{K\}\\sum\_\{k=1\}^\{K\}\\left\[\\bar\{v\}\_\{\\rho\_\{\\mathrm\{del\}\}\}\(\\\{m\\\},W\_\{k\}\)\-\\bar\{v\}\_\{\\rho\_\{\\mathrm\{del\}\}\}\(\\varnothing,W\_\{k\}\)\\right\],which is unbiased for the panel\-level trigger value\.

### E\.3Why Each Chain Uses One Shared Window

The two coalitions in a marginal must be evaluated on the same tasks\. Otherwise their difference mixes the unit effect with differences in task difficulty\. With a shared windowWW,

v¯ρ​\(S∪\{i\},W\)−v¯ρ​\(S,W\)=1b​∑t∈W\[vρ​\(S∪\{i\},t\)−vρ​\(S,t\)\]\.\\bar\{v\}\_\{\\rho\}\(S\\cup\\\{i\\\},W\)\-\\bar\{v\}\_\{\\rho\}\(S,W\)=\\frac\{1\}\{b\}\\sum\_\{t\\in W\}\\left\[v\_\{\\rho\}\(S\\cup\\\{i\\\},t\)\-v\_\{\\rho\}\(S,t\)\\right\]\.Easy and difficult tasks therefore cancel inside the paired difference\. Ifsd2s\_\{d\}^\{2\}denotes the finite\-panel variance of the task\-level marginal in brackets, uniform sampling without replacement gives

VarW⁡\[v¯ρ​\(S∪\{i\},W\)−v¯ρ​\(S,W\)\]=\(1−bMA\)​sd2b\.\\operatorname\{Var\}\_\{W\}\\\!\\left\[\\bar\{v\}\_\{\\rho\}\(S\\cup\\\{i\\\},W\)\-\\bar\{v\}\_\{\\rho\}\(S,W\)\\right\]=\\left\(1\-\\frac\{b\}\{M\_\{A\}\}\\right\)\\frac\{s\_\{d\}^\{2\}\}\{b\}\.Independent windows would instead retain the two level\-score variances and lose their positive task\-difficulty covariance\.

SharingWkW\_\{k\}across the entire chain also preserves exact telescoping:

∑j=1n−1\[v¯ρ​\(Sj,Wk\)−v¯ρ​\(Sj−1,Wk\)\]=v¯ρ​\(N,Wk\)−v¯ρ​\(\{m\},Wk\)\.\\sum\_\{j=1\}^\{n\-1\}\\left\[\\bar\{v\}\_\{\\rho\}\(S\_\{j\},W\_\{k\}\)\-\\bar\{v\}\_\{\\rho\}\(S\_\{j\-1\},W\_\{k\}\)\\right\]=\\bar\{v\}\_\{\\rho\}\(N,W\_\{k\}\)\-\\bar\{v\}\_\{\\rho\}\(\\\{m\\\},W\_\{k\}\)\.For deletion, adding the separately measured trigger difference gives the full\-skill lift over the bare agent on every untruncated chain\.

### E\.4Noise\-Gated Truncation

Before scanning a chain, we evaluate the full\-skill anchorv¯ρ​\(N,Wk\)\\bar\{v\}\_\{\\rho\}\(N,W\_\{k\}\)\. If a prefixSjS\_\{j\}satisfies

\|v¯ρ​\(Sj,Wk\)−v¯ρ​\(N,Wk\)\|≤τ,\\left\|\\bar\{v\}\_\{\\rho\}\(S\_\{j\},W\_\{k\}\)\-\\bar\{v\}\_\{\\rho\}\(N,W\_\{k\}\)\\right\|\\leq\\tau,we stop the chain and assign zero marginal contribution to its remaining execution units\. The omitted window\-level marginals telescope to

∑ℓ\>j\[v¯ρ​\(Sℓ,Wk\)−v¯ρ​\(Sℓ−1,Wk\)\]=v¯ρ​\(N,Wk\)−v¯ρ​\(Sj,Wk\),\\sum\_\{\\ell\>j\}\\left\[\\bar\{v\}\_\{\\rho\}\(S\_\{\\ell\},W\_\{k\}\)\-\\bar\{v\}\_\{\\rho\}\(S\_\{\\ell\-1\},W\_\{k\}\)\\right\]=\\bar\{v\}\_\{\\rho\}\(N,W\_\{k\}\)\-\\bar\{v\}\_\{\\rho\}\(S\_\{j\},W\_\{k\}\),whose absolute value is at mostτ\\tau\. Thusτ\\taubounds the aggregate suffix discarded by one operator on one sampled window\.

This is an aggregate, window\-level statement\. Individual omitted marginals can be larger and cancel within the suffix, and the bound does not directly apply to the full panel\. We therefore report truncation as a controlled approximation and compare it with an untruncated control run rather than claiming a per\-unit error bound\.

## Appendix FExperimental Details and Extended Analyses

This appendix documents the protocol underlying the main\-text experiments and provides additional analyses of estimator stability, decision support, and hierarchical attribution\. The main text asks whetherSkillSVis faithful and actionable\. The analyses below test whether those conclusions can instead be explained by task sampling, rollout budget, or a particular way of aggregating unit values\.

### F\.1Experimental Protocol

Benchmarks and splits\.We evaluate four agentic benchmarks spanning different verification regimes and interaction horizons\.LiveMath\[Heet al\.,[2026](https://arxiv.org/html/2608.04562#bib.bib24)\]consists of mathematician\-level multiple\-choice problems answered in a single turn\.OfficeQA\[Opsahl\-Onget al\.,[2026](https://arxiv.org/html/2608.04562#bib.bib23)\]evaluates document\-grounded enterprise question answering with up to2424tool turns\.SpreadsheetBench\[Maet al\.,[2024](https://arxiv.org/html/2608.04562#bib.bib21)\]requires spreadsheet manipulation over trajectories of up to3030turns\.ALFWorld\[Shridharet al\.,[2021](https://arxiv.org/html/2608.04562#bib.bib22)\]evaluates embodied household task completion over up to5050interaction steps\. This range lets us test whether block valuation remains useful as evaluation moves from a single verified answer to a long agent trajectory\. Table[6](https://arxiv.org/html/2608.04562#A5.T6)summarizes the released splits and evaluation protocols\.

Each benchmark exposes a programmatic verifier\. The coalition utility is the primary scalar field fixed in that benchmark’s run configuration; additional verifier outputs are reported as secondary metrics and are not combined post hoc\.

Skill selection\.For each benchmark, we run Trace2Skill\[Niet al\.,[2026](https://arxiv.org/html/2608.04562#bib.bib27)\], TextGrad\[Yuksekgonulet al\.,[2024](https://arxiv.org/html/2608.04562#bib.bib25)\], GEPA\[Agrawalet al\.,[2026](https://arxiv.org/html/2608.04562#bib.bib26)\], and SkillOpt\[Yanget al\.,[2026a](https://arxiv.org/html/2608.04562#bib.bib28)\]\. We select the skill with the highest*validation*score and freeze the selected artifact before constructing the test panels\. Consequently, neither attribution scores nor held\-out pruning results participate in skill selection\.SkillSVreceives only the resulting skill directory and does not inspect its optimizer or optimization history\.

Target agent and scoring\.Every coalition is executed by the same frozen GPT\-5\.5 model snapshot, with identical decoding, tool, and interaction settings\. Each rollout runs in an isolated sandbox and is scored by the benchmark’s programmatic verifier\. Therefore, differences between two coalition scores arise only from the rendered skill and rollout stochasticity, rather than model updates or cross\-rollout state\. The exact endpoint identifier, decoding parameters, retry policy, and environment configuration are fixed in the released run manifest\.

Disjoint screening and confirmation panels\.We apply one seeded shuffle \(seed4242\) to each test split and allocate panelsTAT\_\{A\}andTBT\_\{B\}without replacement; thus the panels are disjoint by construction\. Panel A supplies the task windowsWkW\_\{k\}used to estimate unit values and induce rankings\. Panel B is never used to select the skill, tune estimator hyperparameters, rank units, or propose edits\. It is reserved for reported held\-out anchors, pruning curves, refinement evaluation, and audits\. Panel sizes are fixed before inspecting coalition scores\. Panel B is evaluated in full at every held\-out checkpoint, making its size the main determinant of pruning\-curve uncertainty\. Table[7](https://arxiv.org/html/2608.04562#A5.T7)reports the pre\-specified panel sizes\.

![Refer to caption](https://arxiv.org/html/2608.04562v1/x8.png)Figure 8:Pairwise agreement among attribution rankings\. We report a tie\-aware Kendall coefficient and assess significance against permutation null distributions that preserve each method’s tie structure\. Held\-out pruning performance, rather than agreement itself, determines ranking utility\.SkillSVconfiguration\.We compile each selected skill at block granularity and drawK=12K=12feasible orders from the declared order distribution\. Each order respects the dependency DAG and the repaired hierarchy\. Independently for each order, we draw a stratified task windowWkW\_\{k\}ofb=8b=8tasks fromTAT\_\{A\}and reuse it for every adjacent marginal on that prefix chain\. The window draw is independent of the order and gives every task equal marginal inclusion probability\.

For a feasible coalitionSS,ρdel\\rho\_\{\\mathrm\{del\}\}deletes units outsideSS, whereasρpad\\rho\_\{\\mathrm\{pad\}\}replaces their spans with neutral placeholders matched to the target model’s token length\. Feasibility is enforced when coalitions and orders are constructed; the rendering operators do not independently expand a dependency closure\. In contrast, the held\-out pruning walks in the main text remove the reverse\-dependency closure of a selected unit whenever this is necessary to keep the remaining coalition feasible\.

We use noise\-gated truncation withτ=0\.05\\tau=0\.05\. A suffix is skipped only after the current prefix score is withinτ\\tauof the full\-skill anchor on the same task window\. For OfficeQA, where both operators are evaluated, the suffix is truncated only when the gate holds for both operators\. The tolerance bounds the aggregate omitted suffix on that sampled window; it does not bound every individual omitted marginal\.

We evaluate bothρdel\\rho\_\{\\mathrm\{del\}\}andρpad\\rho\_\{\\mathrm\{pad\}\}on OfficeQA, our two\-operator audit, and useρdel\\rho\_\{\\mathrm\{del\}\}on the other benchmarks\. Thus all four benchmarks support the net\-value and pruning results, while conclusions about content value and context\-occupancy cost are scoped to OfficeQA\.

Uncertainty estimation\.Unit\-value uncertainty is computed by resampling complete\(πk,Wk\)\(\\pi\_\{k\},W\_\{k\}\)chains, preserving the dependence among all marginals obtained from the same order and task window\. Held\-out anchors and pruning curves instead use paired task\-level bootstrap samples from panel B\. For comparisons between two pruning methods, the same bootstrap task sample is used to recompute both curves and their AUC difference\. We use10001000bootstrap replicates throughout\.

![Refer to caption](https://arxiv.org/html/2608.04562v1/x9.png)Figure 9:OfficeQA block\-level decision map\. The horizontal axis is content valueϕi,ρpad\\phi\_\{i,\\rho\_\{\\mathrm\{pad\}\}\}and the vertical axis is context\-occupancy costϕi,ρpad−ϕi,ρdel\\phi\_\{i,\\rho\_\{\\mathrm\{pad\}\}\}\-\\phi\_\{i,\\rho\_\{\\mathrm\{del\}\}\}\. The decomposition distinguishes content that should be retained from useful but overly costly content that should be compressed\.
### F\.2Estimator Calibration and Stability

Selecting the task\-window size\.Figure[10](https://arxiv.org/html/2608.04562#A6.F10)evaluates the variance–budget trade\-off underlying the choiceb=8b=8\. For each candidate window size, we compare chain\-coupled evaluation, in which both sides of every marginal use the same tasks, with an unpaired alternative using independently sampled tasks\. Pairing removes the shared task\-difficulty component and leaves only variation in task\-level marginal effects\. At the smallest windows, pairing reduces the measured variance to roughly one sixth of the unpaired alternative\. Atb=MAb=M\_\{A\}, the two procedures coincide, providing an implementation consistency check\. We chooseb=8b=8as the smallest window that covers all required task strata while providing a stable truncation comparison\.

![Refer to caption](https://arxiv.org/html/2608.04562v1/x10.png)Figure 10:Variance–budget trade\-off as task\-window sizebbvaries\. Chain\-coupled evaluation uses the same task window on both sides of each marginal; the unpaired alternative samples them independently\. The marked settingb=8b=8is used in all main experiments\.Calibrating truncation\.Window size and truncation tolerance cannot be selected independently: smaller windows make the saturation comparison noisier, whereas a larger tolerance can discard meaningful suffix effects\. We calibrate\(b,τ\)\(b,\\tau\)on planted\-value games and compare the selectedτ=0\.05\\tau=0\.05configuration with an untruncated control run\. In accordance with Appendix[E](https://arxiv.org/html/2608.04562#A5), this audit evaluates aggregate and per\-unit discrepancies separately; only the aggregate omitted suffix has a deterministic window\-level bound\.

### F\.3Additional Attribution Diagnostics

From values to keep–compress–delete decisions\.The pruning experiments in the main text use deletion values to test whether low\-valued content can be removed\. The two\-operator OfficeQA audit provides a more refined decision view\. Figure[9](https://arxiv.org/html/2608.04562#A6.F9)places each block according to its content valueϕi,ρpad\\phi\_\{i,\\rho\_\{\\mathrm\{pad\}\}\}and context\-occupancy costϕi,ρpad−ϕi,ρdel\\phi\_\{i,\\rho\_\{\\mathrm\{pad\}\}\}\-\\phi\_\{i,\\rho\_\{\\mathrm\{del\}\}\}\. Positive\-content blocks with small occupancy cost are natural retention targets, whereas positive\-content blocks with substantial occupancy cost are candidates for compression rather than deletion\. A block whose confidence interval includes zero on a decision axis, or whose magnitude lies below the estimator’s reporting resolution, is treated as unresolved rather than assigned a categorical action\. In this audit the occupancy effects are measurable but substantially narrower than the deletion values, so the decomposition is used as a targeted diagnostic rather than as a claim that context cost dominates these skills\.

Relationship to baseline rankings\.Figure[8](https://arxiv.org/html/2608.04562#A6.F8)compares the rankings induced bySkillSV, Closure\-LOO, the LLM judge, and random ordering, retaining ties rather than breaking them arbitrarily\. We compare each observed coefficient with a permutation null that preserves the two rankings’ tie structures and apply a Holm correction across benchmark–method comparisons\. No method pair exceeds this corrected chance baseline\. Low agreement does not by itself establish that one ranking is better; that evidence comes from the value\-closure and held\-out pruning experiments in the main text\. Instead, this analysis shows thatSkillSVis not merely a rescaled version of either learned or leave\-one\-out baselines\.

Hierarchical localization of value\.Unit values can be aggregated through the hierarchy produced by the compiler\. For an internal node, we sum the values of all descendant execution units; the node is not separately re\-estimated\. Figure[11](https://arxiv.org/html/2608.04562#A6.F11)applies this aggregation to LiveMath\. It complements the main\-text concentration analysis by showing where the concentrated value resides in the document structure, rather than only how much value is captured by the top\-ranked blocks\.

![Refer to caption](https://arxiv.org/html/2608.04562v1/x11.png)Figure 11:Hierarchical value tree for the LiveMath skill\. Leaf nodes are compiled valuation units, and each internal\-node value is the sum of its descendant unit values\. The visualization localizes the value concentration reported in the main text to specific sections of the skill\.