A Mathematical Framework for AI-Human Integration in Work
Authors: L. Elisa Celis
Lingxiao Huang
Nisheeth K. Vishnoi
Venue: ICML 2025 poster (2025)
Keywords: AI and work
human-AI collaboration
worker-job fit modeling
phase transition
Primary area: social_aspects
TL;DR: We present a mathematical framework for modeling jobs, workers, and metrics to evaluate worker-job fit, and use the framework to highlight the benefit of AI-human integration.

Abstract: The rapid rise of Generative AI (GenAI) tools has sparked debate over their role in complementing or replacing human workers across job contexts. We present a mathematical framework that models jobs, workers, and worker-job fit, introducing a novel decomposition of skills into decision-level and action-level subskills to reflect the complementary strengths of humans and GenAI. We analyze how changes in subskill abilities affect job success, identifying conditions for sharp transitions in success probability. We also establish sufficient conditions under which combining workers with complementary subskills significantly outperforms relying on a single worker. This explains phenomena such as *productivity compression*, where GenAI assistance yields larger gains for lower-skilled workers. We demonstrate the framework's practicality using data from O*NET and Big-Bench Lite, aligning real-world data with our model via subskill-division methods. Our results highlight when and how GenAI complements human skills, rather than replacing them.

A Mathematical Framework for AI-Human Integration in Work
L. Elisa Celis! Lingxiao Huang? Nisheeth K. Vishnoi
Abstract
These advances have intensified focus on their implica- tions for work and workers. Institutions such the BBC (Annabelle Liang, 2024; Chris Vallance, 2023), the IMF (Cazzaniga et al.). and the World Economic Forum (Shine & Whiting, 2023), along with many re- searchers (Brynjolfsson et al., 2023; Felten et al., 2023; Noy & Zhang, 2023; Agarwal et al., 2023; Angelova et al., 2023; Cabrera et al., 2023; Vaccaro et al., 2024; Jaffe et al., 2024; Otis et al., 2024), have explored the evolving labor landscape. A growing view holds that GenAl may recom- pose, rather than eliminate, work. Autor (2024) argues that GenAl has the potential to enable middle-skill workers to take on tasks traditionally reserved for high-skill experts. Still, some see GenAl as a disruptive force: the IMF esti- mates that nearly 40% of jobs could be affected, raising con- cerns about large-scale displacement (Cazzaniga et al.;; Chris Vallance, 2023; Microsoft, 2023; Paradis, 2024; Kochhar, 2023). Others emphasize complementarity GenAl tools can enhance human capabilities rather than replace them (Ace- moglu & Johnson, 2023). Empirical studies show that such tools improve the performance of less experienced workers, narrowing the productivity gap with more skilled profes- sionals (Brynjolfsson al., 2023; Noy & Zhang, 2023). This raises central question: Do GenAl tools substitute for human workers-or or enable them to succeed in new ways?
The rapid rise of Generative AI (GenAI) tools has sparked debate over their role in complementing replacing human workers across job contexts. We present mathematical framework that mod- els jobs, workers, and worker-job fit, introducing a novel decomposition of skills into decision-level and action-level subskills to reflect the comple- mentary strengths of humans and GenAI. We an- alyze how changes in subskill abilities affect job success, identifying conditions for sharp transi- tions in success probability. We also establish suf- ficient conditions under which combining work- ers with complementary subskills significantly outperforms relying on a single worker. This ex- plains phenomena such as productivity compres- sion, where GenAl assistance yields larger gains for lower-skilled workers. We demonstrate the framework's ' practicality using data from OENET and Big-bench Lite, aligning real-world data with methods. Our
our model via results highlight when and how GenAl comple- ments human skills, rather than replacing them.
1. Introduction
Recent studies have empirically examined GenAl's impact on various aspects of work, including accuracy, productivity, and implementation cost (Agarwal et al., 2023; Brynjolf- sson et al., 2023; Vaccaro al., 2024; Jaffe et al., 2024; Klubnikin al., 2024; Brodsky, 2024; Anthropic, 2023; Guo et al., 2025). Vaccaro al. (2024), for example, ana- lyzed 106 experiments comparing human-Al collaboration to human- or Al-only performance on job tasks. They found that while collaboration improves outcomes in content cre- ation, it lags in decision-making underscoring the nuanced ways GenAl complements human skills. Brymjolfsson et al. (2023) and Jaffe et al. (2024) studied GenAl integration in real-world workflows, showing measurable productivity gains. In one case, Al-assisted customer service agents re- solved 14% more issues per hour, suggesting that GenAl can amplify human efficiency. Other work has highlighted implementation barriers: studies such as Klubnikin et al. (2024); Brodsky (2024) identify hidden costs and opera- tional challenges that hinder widespread GenAI adoption.
The rapid emergence of capabilities in Generative Artificial Intelligence (GenAI) has drawn global attention. Multi- modal models like OpenAl's GPT-4 and DeepMind's Gem- ini seamlessly interpret and generate text and images, trans- forming tasks such as content creation, summarization, and contextual understanding (OpenAI, 2023; DeepMind, 2023). Similarly, models like OpenAl's Codex and GitHub Copilot show strong performance in code generation and debugging (OpenAI, 2024; Jaffe et al., 2024). Notably, GPT-4 scores on standardized tests, including SAT, GRE, and AP, are comparable to those of human test-takers (OpenAI, 2023).
'Yale University. USA. "State Key Laboratory of Novel Soft- ware Technology. New Cornerstone Science Laboratory, Nan- jing University, China. Correspondence to: Nisheeth K. Vishnoi <nisheeth. vishnoi @ gmai..com>.
Proceedings of the 42nd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

---

A Mathematical Framework for Al-Human Integration in Work
This paper focuses on understanding the impact of GenAl on job accuracy. Addressing this requires modeling jobs, worker abilities (whether human or AI), and how these abil- ities relate to job performance. A key resource we draw on is the Occupational Information Network (O*NET) (U.S. Department of Labor, Employment and Training Admin- istration, 2023), a comprehensive database maintained by the U.S. Department of Labor that provides standardized descriptions of thousands of jobs. For example, O*NET characterizes Computer Progrummers by skills (e.g.. Pro- gramming, Wrättnn Comprehension, Oral Expression), tasks (e.g., Correcting errors, Developing websites), and knowl- edge areas. Each skill is rated by importance and required proficiency- instance, Writing might be rated 56/1 10 for importance and 46/100 for proficiency. While O*NET offers rich and structured data, it does not specify how tasks depend skills, nor how to evaluate performance at the level of a skill, task, or job (see Section E. for more detail). Recent work has begun to address these limitations using compositional and task-skill dependency models (Arora & Goyal, 2023; Okawa et al., 2023; Yu et al., 2024).
the root cause) and action skills (e.g., implementing the fix). This conflation obscures the underlying sources of success or failure, leading to biased or incomplete evaluations, In addition, evaluations are rarely standardized across settings, and lab-based assessments often fail to capture the broader skillsets required in real-world jobs (Microsoft, 2023; Vac- caro et al., 2024). In summary, significant challenges re- main in evaluating human workers and GenAI tools: (i) The conflation of reasoning and action, leading inaccurate per- formance attributions. (ii) The statistical noise inherent in limited-scope evaluations. (iii) The lack of standardization, creating inconsistencies across assessments,
Our contributions. We introduce a mathematical frame- work to assess job accuracy by modeling jobs, workers, and success metrics. A key feature is the division of skills into two types of subskills: decision-level (problem solving) and action-level (solution execution) (Section 2). Skill difficulty is modeled on a continuum [0, 1], where 0 represents the easiest and 1 the hardest. Workers, whether human or AI, are characterized by ability profiles (a1, a2). representing decision- and action-level abilities. These profiles quantify worker's capability for each skill s E [o, 1], incorporating variability through probability distributions. Jobs are mod- eled collections of tasks, each requiring multiple skills. We define a job job-success probability metric, combining error rates across skills and tasks to evaluate overall performance (Equation (1)). This framework addresses challenges in evaluation by isolating abilities, accounting for noise, and providing a metric for accuracy. Our main results include:
Metries for evaluating human workers include Key Perfor- mance Indicators (KPls) (KPL.org, 2024), customer feed- back, peer reviews, and productivity measures. For example, KPIs might assess programmer's ability to fix certain number of bugs within a set timeframe, while customer feed- back evaluates the perceived quality of service. However, such metrics often conflate outcomes with underlying com- petencies, making difficult to isolate a worker's ability on specific skills. In contrast, GenAT tools are typically evaluated using skill-specific benchmarks in areas such as coding, writing, and mathematical reasoning (Borji, 2022;
Phase transitions: small changes in average ability can cause sharp jumps in job success (Theorem 3.2).
Bubeck et al., 2023; bench authors, 2023; OpenAI, 2023;
Merging benefit: combining workers with complemen- tary subskills yields superadditive gains (Theorem 3.3).
Abdin et al., 2024; Yu et al., 2024; Reid & Vempala, 2024; He et al., 2024). For instance, bench authors (2023) intro- duced the BIG-bench Lite (BBL) dataset, which evaluates 24 skills, including code generation, by comparing the per- formance of GenAl models and human workers (see Sec- tion E.2). One representative task, Automutic Debugging, tests whether model can infer the state of a program given partial code-e.g.. determining the value of a variable at a specific line without executing the program.
Compression effect: our model explains the productiv- ity compression observed by Brynjolfsson et al. (2023), where GenAl narrows the performance gap between low- and high-skilled workers (Corollary 3.4).
Empirical validation: we apply our framework to real- world data from O*NET and BIG-bench Lite, aligning task descriptions and GenAl evaluations with subskill- based models (Section 4).
While GenAl tools perform well on structured or repetitive tasks, they often struggle with skills that require contextual understanding, planning, emotional intelligence (Bender et al., 2021: Arora & Goyal, 2023; Services, 2024; Ma- howald et al., 2024). These limitations are compounded by noisy and narrowly scoped evaluations, making it difficult to draw reliable conclusions about performance (Miller, 2024; bench authors, 2023). Consider a company that sets a KPI target of fixing 20 bugs per week. If a programmer fixes 18, their score would be 18/20- 90%. But such metrics conflate distinct abilities: reasoning skills (e.g., diagnosing
Interventions and extensions: we explore upskilling strategies through ability/noise interventions (Sec- tion D.I), and extend our analysis to dependent sub- skills and worker combinations (Figures 3, 4.2).
Bias and noise: analyze how misestimated ability profiles distort evaluations (Section D.2).
Our findings inform strategies for integrating GenAl into the workplace, including combining human and AI strengths, designing fairer evaluations, and supporting targeted up- skilling. Additional related work is reviewed in Section A.
2

---

A Mathematical Framework for Al-Human Integration in Work
2. Model
is monotonically decreasing in 8. That is, workers are ex- pected perform worse on easier subskills (s = 0 denotes the easiest and = 1 the hardest).
The job model. We model a job as a collection of m 2 tasks, where each task T; requires a subset of n 2 skills. This induces a bipartite task-skill dependency graph where edges connect tasks to the skills they depend on (Figure 12), aligning with prior work (Arora & Goyal, 2023; Okawa et al., 2023; Yu et al., 2024).
Linear ability profile. A natural used form is the linear function: E(s) == c - (1 - a)s, for parameters a, c E [0, 1] satisfying a c 1. Here, c represents the worker's maximum ability (attained at = 0), and a is the rate at which ability decreases with difficulty. This profile aligns with evaluations of GenAI tools (Hendrycks et al., 2021; bench authors, 2023); see also Figure 11 and Section B. for Big-bench Lite analysis. As a special case, setting a = 1 yields a constant ability function E(s) = c, where the worker has uniform performance across all subskills.
Each skill j is decomposed into two subskills: a decision- level component (e.g., problem-solving. diagnosis) and an action-level component (e.g., execution, implementation), following distinctions made in cognitive and labor mod- els (Licklider, 1960; Kahneman, 2011; Inga et al., 2023). For example, the skill "programming" involves both solving a problem (decision-level) and implementing a solution in code (action-level). This decomposition allows for more precise modeling of ability, especially for evaluating hybrid human-AI work. Adapting from O*NET, each skill j E [n] is associated with subskill difficulties 8j1, 8/2 E [0, 1]. where 0 indicates the casiest and I the hardest. These scores are used to index the worker's ability distributions. This representation mirrors the proficiency levels used in O*NET and simplifies the mathematical formulation; see Section E.
Polynomial ability profile. To model nonlinear improve- ments, we also consider the polynomial form: E(s) - 8 where B 2 0 controls the sensitivity of ability to difficulty. Larger values of / produce sharper gains in abil- ity as s + O, representing workers whose skills improve rapidly as tasks become easier.
Note that nearby subskills (e.g.. 8 = 0.7 and 0.8) yield sim- ilar values of E(s), reflecting the smoothness of the ability profile and inducing implicit correlations across adjacent subskills. Section B.1 provides additional visualizations of these profiles under uniform noise. Section B.2 shows that these profiles satisfy stochastic dominance (Definition B.1): for any fixed s E [0, 1] and threshold x E [0, 1]. the proba- bility Prx=a(=)[2 [X 2 æ] increases monotonically with the average ability.
The worker model. We model a worker by two ability profiles, ag and azg, which govern their decision-leve levell and action-level subskills, respectively. Each profile maps sub- skill difficulty E [0, 1] to probability distribution over [0, 1]. from which a performance value is drawn, represent- ing the worker's effectiveness on that subskill. This reflects the stochastic nature of skill performance (Sadeeq, 2023; Bubeck et al., 2023). We consider ability profiles in which, for each subskill difficulty & E [0, 1}. the worker has an average ability Ess. and their actual performance is mod- eled by adding stochastic noise term E(S). The resulting ability value is used to define the worker's distribution over outcomes for that subskill.
Measuring joh-worker fit. To evaluate how well a worker fits a job, we define sequence of aggregation functions that compute error rates at the subskill, skill, task, and job levels, based on the worker's 's ability profiles (01, 02).
For each skill j E [m]. let 8ا1, E [0, 1] denote the diffi- culty levels of its decision-level and action-level subskills. We define the random subskill error rate as: Eze := X, where X ~ (s(s))), for E E (1, That is, bye represents the probability of failure (or error rate) for the E-th subskill of skill j. drawn from the worker's 's ability distribution.
We study two natural noise models: (i) Uniform noise: The noise term es) is sampled from a scaled uniform distribu- tion: E(s) ~ min mi)(F(s),
F(x))- Unif[-a, o). where [0, 1] controls the noise level. The scaling ensures that the perturbed value remains within the valid range [0, 1]. This model provides a simple yet effective way to introduce bounded variability and is often used in our analysis. (ii) Truncated normal noise: Here, we model s(s) using a trun- cated normal distribution: E(S) ~ TrunN(E(s), o2; 0,1,, 1). where the mean is E(s), the variance is o², and the sup- port is clipped to remain within [0, 1]. This model captures fluctuations consistent with human performance variability and is aligned with empirical measurements of GenAl tool behavior (bench authors, 2023) (see Figure 9).
To compute the error rate for skill j. we apply a skill error function h : [0, 1]2 - [0, 1). which aggregates the two sub- skill error rates Çy and Cy2- This gives the overall error rate for skill j. combining both decision-level and action-level performance. We assume common skill error function h for all skills, typically chosen as the average h( b) = atb or the maximum h(a, b) = max(a, 6} (KPI.org, 2024; Walker, 2023), both of which are monotonic.
Similarly, to compute the error rate for a task Tü we ap- ply a task error function g : [0. 1Ja + [0, 1]. which maps the error rates of the skills in T. to task-level error: g (Eh(SS1, (22))E))))))..
We assume that the average ability function F(s),, which maps subskill difficulty s E [0, 1] to expected performance,
3

---

A Mathematical Framework for Al-Human Integration in Work
Finally, we define job error function f :
Independent noise assumption. We assume independence across the random noise realizations the subskill level. This assumption pertains only to execution noise: once the ability profiles a1 and 0g are fixed, the realized perfor- mances across subskills are modeled as independent draws. The ability profiles themselves may still induce correla- e...s-egg.. via a smooth expected ability function E(s) where adjacent subskills have similar mean performance.
[o, 1j rates into a single overall job error rate: Err(C) :=
[0, which aggregates the task error
f (g (fh(Cs1, Sp))een)...... 9 (fh(S)1, (r2))se...))
In our empirical analysis (Section 4), we instantiate and f as weighted averages, where the weights reflect the im- portance of individual skills and tasks. More generally, we assume that h. g. and f are monotonic: improving subskill abilities cannot increase the resulting error rate.
Assumption 3.1 (Noise independence). For all j [n]] and E E {1,2), the subskill error rates Sye are independent drawn from (xe(8je).
Given threshold r E [0, 1]. we say a job succeeds if the job error rate satisfies Err(c) S T. The job success probability for a worker with profiles (a1, a:2) is then defined as:
This modeling choice is standard for both human and GenAl workers. For example, GenAl tools often produce condition- ally independent task outputs given a fixed model state. We explore noise-dependen settings in Section 4 and extend our theoretical analysis such settings in Section C.5. In particular, we find that strong correlations in noise reduce the sensitivity of P to changes in ability parameters.
Pla1,ag,h, g, f.T) == Pre,e [Err({) S ST.
(1)
In the special case of noise-free abilities, the job success probability becomes binary, taking values in (0.1}. 1}.
3. Theoretical results
Notation on sensitivity to ability parameters. To study how the job success probability P varies with abil- ity, we begin by analyzing the expected job error rate: Er (H1, T1 #2, 02) == E<(Err(()), where the expecta- tion is taken over the subskill error rates Gje = 1 X, with X ~ (e(())e.. This quantity captures the average error rate for a fixed job instance and ability parameters (HI, (12, 02). To quantify the impact of decision-level ability on average error, we consider @Err, This mea- öyte
This section presents our theoretical results on how the job success probability P (1) varies with worker ability pa- rameters. We fix the job instance throughout: task-skill structure (7:), subskill difficulties {sje}, aggregation func- tions h, f. and success threshold т. Given this setup, P(a1, a2 h, g. r). abbreviated as P, depends only on 0n and (r2. We begin by analyzing a single worker with decision- and action-level profiles parameterized by aver- age ability pe 2 0 and noise level og 2 for l E (1, Fixing P2, 1 02, we show that P undergoes sharp phase transition as μ1 crosses a critical value (Theorem 3.2).
sures how sensitive the average job error is to changes in pe, holding the other parameters fixed. Given fixed values of the noise levels and the other ability parameter, the minimum influence of μl on the expected job error is defined as:
Next, we study the benefits of merging two workers with complementary abilitics. Given workers A and B with profiles (A) .02 aa(4)) and (a{ (B) _aa²),, we consider all four possible combinations of decision-level and action- level profiles: (a) (a{4), .22 (A)), (a(4), .2
MinDer, (0), U2, 02) == inf /220 20
(1, 02)
), (a (B) a2 (A)), (a2)) (.) ). Theorem 3.3 gives conditions under which merging improves success probability. We conclude with Corollary 3.4, which connects this analysis to the productiv- ity compression observed in Brynjolfsson et al. (2023).
(B)),
A large value of MinDer,, indicates that small increases in decision-level ability μ4 can significantly reduce the ex- pected job error. MinDer,pa 01,02) is defined similarly.
(ai (B)
As example, consider Err(() 1. E, Se be av- erage over all subskills and as(s) = 1 - (1 az)s s + es(s) be linear profiles, where E2(() ~ min
3.1. Notation and assumptions
0z )s)8,1 1 - (1 (1-a))s}
gx).. We compute that 1 E,Em)) 8ze. This implies that MinDer, = 1 2n Σ Ejenn $51 and MinDer,, = 2n EjE(m) $32.
DErra 2n
We begin with a structural observation: since the error aggre- gation functions h, g. f are all monotonic, their composition Err also monotonic in the subskill error rates Çje- Hence, to show that the job success probability P increases with μl, it suffices to show that higher μu leads to lower values of GjI Recall that Eie = 1 1-2 X, where X ~ (s(5))) Thus, lower error rates correspond to higher sampled ability values. This follows from stochastic dominance: for any s, E [0, 1], Prx [X 2 x] increases with μæg see Proposition B.I.
dar
Lipschitz assumption. We assume the job error function Err is L-Lipschitz with respect fn-nom::
Err({) - Err(()) S L- -- -'1l1 for all C.C' E [0, 1|20
When Err is the average of subskill errors, I. 2n
4

---

A Mathematical Framework for Al-Human Integration in Work
3.2. Threshold effect in job success probability
with P 0.5, either increasing dn or reducing effectively improves success.
We now quantify how the success probability P changes with decision-level ability μi, holding other parameters fixed. We prove sharp threshold behavior: once the ex- pected job error crosses the success threshold т, even small changes in jai can cause the success probability to jump from near zero to near one. This tieenmmenon-foranne below-shows a phase transition in job success probability, controlled by a critical ability level μuj
Generalization. In Section C.I, we prove generalized form of Theorem 3.2 that accommodates arbitrary noise models e()), leveraging the notion of a subgaussian con- stant to quantify the dispersion of E(s). In Section C.2, we further extend the analysis to non-liness aggregation rules (e.g., max) and alternative ability profiles (e.g.. constant, polynomial). The resulting transition width V varies from (1/nn) to O(1), depending on the functional form and the underlying distributional assumptions.
Theorem 3.2 (Phase transition in job success probability). Fix the job instance, action-level ability p2, and noise levels 0n 22 Let p be the unique value such that the expected job error equals the success threshold:
3.3. Merging workers to improve job success
The phase transition result (Theorem 3.2) shows that small increases in ability parameters can sharply increase the suc- cess probability P. We now apply this insight to demon- strate how merging two workers with complementary skills can result in a significant performance gain, especially rele- vant in settings combining humans and GenAl tools.
( Eng (p1, T1, 01, 02) T.
Let 0 E (0,0.) be a confidence level, and define the tran- sition width: "1
Mi (o) (2,00)) where L is the Lipschitz constant of the job error function. Then the job success probability satisfies:
Suppose Worker 1 (W1) has stronger decision-leve ability, while Worker 2 (W2) excels in action-level execution. Let the decision-level profiles be denoted aj -(4)) o (()),
PSO 0 if i S 19 and P21-0 if m 2 pf + 71-
(E)
Theorem 3.2 shows that increasing μ1 by approximately 2 transitions the success probability P from at most 0 to at least 1 -- 0. A smaller value of 7 implies that even modest gains in decision-level ability can have a significant impact on job success. Conversely, slight increase in the threshold 7 can sharply reduce P. As expected, 71 increases with the Lipschitz constant L and total noise variance n(o + oz), and decreases with the sensitivity Min The core tech- nical step is to relate the probability P to the expectation Errawg using a concentration bound under the independence assumption (Assumption 3.1), via McDiarmid's inequal- ity (Kontorovich, 2014).
and action-level profiles 022
(2) ~ (nl
(2) 22
(2)) for 8 (1,2}-
Assume /((1) > pi and μt2 < P22 i..., W1 is stronger
[1]
in decision skills, and W2 in action skills.
We define merged worker W12 that uses the decision- level ability of W1 and the action-level ability of W2: aj (12) (1) aj ai (12) ag Let P12 denote the suc- cess probability of W12, and P2 that of W2. We now give conditions under which the merged worker has substantially higher success probability than either of the original work- ers.
Theorem 3.3 (Success gain from merging com- plementary workers). Fix the job instance. Let
Illustrative example: linear ability profiles. Consider random job with tasks, each requiring i randomly chosen skills from pool of n. Let all aggregation func- tions h, g. be averages. In the balanced case, the job error simplifies to: Err(C) = 1 Σك= + G22,, with 2n
O (0, 0.5) be a confidence level, and define: I. (1f"((a")))+600")
1n(1/0)
A =
and n (2)
MinDer, (n "1 1
L. ( ( ()+(((()+))))))))))))))))))))( 1m(2/00)
L 2 Suppose the ability profile is linear with noise: as(s) = 1 (1 - az)s + s(s), where s(s) min(1 - (1-11))) sz)s. - (1) Onif[-o, a). and assume 8ye ~ Unif(o, 1). Then the expected subskill difficulty is
MinDermy (of (0) "n?"' ,002 (1)) If
Err (1) - - (1) (1) (2) (2) STS
, - 112 02 (2))
Errave (2) (2) (2)
0.5 and MinDer (o1, #2, 02) = 1 Σ, 9,1 2 0.25. Thus, 2m
M1 ,0
M1 Ola /mn(100/n)) This implies that elite workers (small σ) large jobs (large n) experience sharper transi- tions in job success with ability.
then under Assumption 3.1, we have: P12 - P2 2 1 28.
Gain from merging complementary workers. If the av-
erage error function Erravg is fully determined by the abil-
Figure I shows this empirically. For a = 0.1, increasing a by just 4.3% (from 0.492 to 0.513) raises P from 0.2 to 0.8. As a decreases, the transition sharpens, validating our theoretical prediction. Figure 1(c) shows that for jobs
ity parameters, and Err (p1 (1) (1) (1) (2)
-1 0i H22 _
T = Errine (u, + 7 ,0 .12 : 02 ). then it follows
(2) (2) (2) (2) (2)).
that (1()) 11 + 71
(2)
(1)
+ + (2) This implies that if W1's
5

---

A Mathematical Framework for Al-Human Integration in Work
Plats of vs. di, for various
Mots of vs. for various
Heatmap of P valuss across varying and
0-0.1
10 SLccess piotaoving
Job ELICEESS probabliry
Job success prissaiing
0-0.5 0.5
0.7
4: 0.3
0.5
0
01
(a) P V.S. @1
(b) P v.s.
(c) Heatmap of P
Figure 1. Plots illustrating the relationship between the success probability P(an, a22 h, J.r) and the parameters u1, for the linear ability example of Theorem 3.2 with default settings of (n, m. 7, u2) = (20, 20, 0.25, 0.4) and subskill numbers 85z ~v Unif(0, 1].
Hights poo siccess proteiity
Highisss autooaoooooooo (oobbaa
Slpwoodd spa23r out soodlie
Prt
Pst
Pri
(a) (a1 ) 0..4.
(b) (af
(1) 22) (0.5, 0.2)
(c) (a ( (0 ( (0.3, 0.4)
Figure 2. Heatmaps of the probability gain A max (P1, P2, P12, Pai) max (Pi, Pa} by merging two workers for different ranges of (af*) ) for the linear ability example of Theorem 3.3 with default settings of m,, a, 7) (20, 20, 0.5, 0.25). The region enclosed by the dotted lines in cach heatmap indicates where the corresponding job success probability is the highest among the four. For instance, in Figure 2(a) with (ai ) = (0.5, 0.4), we observe that when aj + (2) < 0.9 and an > 0.43, P1 is significantly larger than both Pi and P2 by an amount 0.6. Similarly, when af + 02 < 0.9 and a; > 0.52, Pn. is significantly larger than both P: and Pa. We note that the rapid color shifts in the heatmaps reflect abrupt change in A, indicative of phase transition phenomenon P.
decision-leve ability exceeds W2's by this margin, then their combination W12 can substantially outperform W22 alone in job success probability.
strates the impact of targeted upskilling, especially for high- ability, low-variance workers. Section D. I explores the par- tial derivative landscape to identify when such interventions are most effective.
Illustration with linear ability profiles. Let Err(S)
Ege Giz and assume subskill difficulties 8je On, 1]. Let each worker L E {1.2} have linear ability function at (s)(s) = 1 de (i))s + =ss), where e(s) (1 )s, assume common noise level a for both workers. We ana- lyze when merging Wi and W2 leads to a gain over either alone. If a
If Wi represents a human worker and W2 GenAl sys- tem (as motivated in Section 1), Theorem 3.3 shows that even modest GenAl advantages in action-level tasks can lead to meaningful gains in P12- As human action-level ability decreases, P1 falls but P12 remains stable, widen- ing the gap P12 Pt. This mirrors recent empirical find- ings (Brynjolfsson al., 2023; Noy & Zhang, 2023) and contributes the productivity compression effect, further analyzed in Section 3.4. Thus, combining GenAl with hu- man decision-making yields a productivity amplification effect rather than a replacement dynamic. Organizations should invest in decision-level skill development and in reducing ability noise through workflows and training.
(1)
1- (1-a
}s} )s)
Unif[-o, o). and
(2) < a then W, is optimal. if (1) 2 (2)
(1)
and 02
< d2 merging (P12) leads to a nontrivial gain.
(2)
If a1
2 a1 (2)
+ +(a O(a
and 02 < 02
(1)
(ln11/8))) then by Theorem 3.3, P12 112-PP P: 1 1-20 for l (1, The gain grows as decreases, making the merging criteria easier satisfy.
Finally, our results also highlight the risk of biased evalu- ations: underestimating P can exclude strong candidates (see Section D.2). Moreover, realizing the gains of merging hinges on accurate evaluations of both human and AI abili- ties (see also (Somers, 2023)). Section D.2 also quantifies how imperfect evaluations can negate these gains.
Figure 2 illlsstrates this effect. For example, when a 0.5 (2) + 0.1 and a2 (1) 0.4 af²) 0.1, we observe Pi2 = while P Pg 0.4, yielding gain of 0.6.
Implications. Our analysis informs both job-worker fit and human-Al collaboration strategies. Theorem 3.2 demon-
6

---

A Mathematical Framework for Al-Human Integration in Work
3.4. Application: Explaining productivity compression
4. Empirical results
Brynjolfsson et al. (2023) studied the effect of GenAI tools on customer service productivity, measured by resolutions per hour (RPH). They found that Al assistance dispropor- tionately benefited lower-skilled workers, increasing their RPH by up to 36% and narrowing the productivity gap relative to higher-skilled workers. We now show how Theo- rem 3.3 provides a theoretical explanation for this effect.
We demonstrate the usability of our framework using real- world data and validate our theoretical findings in both noise- dependent settings and scenarios involving the merging of workers with distinct ability profiles. Key results are sum- marized below, with full implementation details provided in Section E. We further validate the robustness of our findings across alternative modeling choices in Section E.4.
Let W1 and W2 be two human workers with the same fami- (1) indicating that W2 is more skilled than Wi at the action level. Let WAI be a GenAI tool sharing the same family of ability profiles as the human workers. For& E [1,2), job success probability of We before merging with WA and let P/P be the corresponding probability after merging. Assuming the job competition time is stable, note that the job success probability P is proportional to the productivity measure RPH (resolutions per hour). Hence, I/3 Pil and PP - Pil represent the productivity gap between Wi and W2 before and after merging, respectively. We define the productivity compression as
4.1. Data, subskills, abilities, and parameters
lies of ability profiles. Assume (42 112
We derive job and worker data from O*NET and Big-bende Lite. To bridge missing parameters, we introduce general subskill division method. As a running example, consider the job of Computer Programmers.
, let Pe denote the
Deriving job data. O*NET states that the Computer Programmer job as requires = 18 skills and tasks, and provides their descriptions. It also gives proficiency levels for each skill, represented by s = (.41, 43, .4,, .4,, .4.,..446 .4,, .55, .57. where 8y E [0, 1] denotes the skill's criticality for the job. O*NET also provides task and skill importance scores, which inform the choice of g and f.
17
.6,,466,,, .46, -4,, .46, .48, 55, 55 52,52.544
PC= = P2- - P1l- - PP PPl.
which measures how much the productivity gap is reduced by merging. A larger PC indicates that Al assistance more effectively narrows the gap. As consequence of Theo- rem 3.3, we obtain the following corollary, deriving condi- tions on the worker parameters to lower-bound PC.
Deriving workers' abilities. We begin by considering lab evaluations from Big-bench Lite (bench authors, 2023) for both human workers and GenAl tools (specifically PaLM (Chowdhery et al., 2023)). For example, we model the ability profiles of a human worker (Wi) 1) and a GenAl tool (W2) as a (1) = TrunN(1 0.78s + 0.22, 0.013; 0, 1), and (2) (s) T1 0.92s 0.08, 0.029; 0, 1).
Corollary 3.4 (Productivity compression). Fix the job instance. Suppose both human workers have the same decision-level abilities:
An approach for subskill division. Subskill divi- sion becomes essential for analyzing worker perfor- mance. We first use GPT-40 to determine the decision-level degree for each skill, given by A = 6,. .,.4, .4, 0, 3, 1, 1, 6,, .6, 0, 0,.4). Using a skill proficiency sy and its decision-level degree Ay. we com- pute subskill numbers as 851 >ysj, 832 = (1 >y)s,- These values are listed in Eq. (11). This formulation en- sures that subskill numbers 851 and 8,2 are linear functions of 8y and Ay. maintaining the property that 8y1 + sy2 = sy. Further examples are in Section E.3.
A p1
(2) = 11 (AT)
(2) (AI) of. (1) = =1 01
01
Let 0 E 00.) be a confidence level, and for each e E (111))))))))))))) In(1/01) If
11,2,AT) define 1
MinDor (0) (o)) (1))))
max {Errang oi,
122
(A1)
72
(A1) (AI)
00
Errave (taneeHh, oi 11
(2)
(2)
(2) STS
12
02
(1) (1) (1)
ErT oi 12
+ 12 .02
then under Assumption 3. we have: PC 2 20.
We decompose skill ability profiles a into subskill abil- ity profiles ai and a2. For a(s) ~ TrunN (1 a)s, of: 0, with decision-level degree X E [0, 1], we set (2(s) = TrunN - (1 - a2,, /2; 0, 1). so that
This result implies that if the AI assistant outperforms the (1) (AT), the productivity lower-skilled worker by at least 72 +72
an (s (s) the distribution of G1 + approximates first drawing X ~ a(s;) and then outputting 1 - X. Thus, skill profiles can be (approximately) reconstructed by setting the skill success probability function as h(Ci, (2) = (i + (2. Thus, we obtain (f)((s) (s) Tru - 0.78s, 0.0065; 0, 1), a (f) TrunN(1 - 0.92s, 0.0145; 0, 1).
gap can shrink significantly. To our knowledge, this is one of the first formal models explaining the productivity compression effect under realistic assumptions.
In Section C.4, we provide the proof of Corollary 3.4 and further extend this analysis to the case where the GenAl tool uses a different ability profile family, confirming that our framework generalizes beyond identical distributions.
7

---

A Mathematical Framework for Al-Human Integration in Work
Constructing the lask-skill dependency. Using task and skill descriptions from O*NET, we employ GPT-4o to generate task-skill dependencies T; S [ni] for each task E [m]. Details are provided in Section E.3.
Analysis. We plot the job success probability P in Figure 3 as the ability parameter and dependency parameter p vary. Figure 3(a) shows that phase transitions in P persist even when subskills are dependent (p > 0), although the transition window narrows as p decreases. For example, when p = 0, increasing a by 0.27 (from 0.07 to 0.34) raises P from 0.2 to 0.8, whereas for p = 0.4, a greater increase in a (0.44) is needed. Figure 3(b) shows that for fixed o, P increases monotonically with p when P < 0.5 and decreases monotonically when P > 0.5, similar to the trend in P v.s. (Figure 1(b)). This is because the variance of Er)(() increases with both p and ø. These results show that workers with loosely coupled subskills (low p) experience sharper gains P from ability improvements, underscoring the value of reducing skill interdependencies.
Choice of error functions and threshold. We set the skill error function as h(CI, (2) = <i + {2 to ensure consistency with the skill ability function a derived from Big-bench Lite. Task and job error functions, g and f. are weighted averages based the importance of skills and tasks from O*NET, resulting in Err(()= = Eje(n) Wy Wy(S] + (z2). (See (13( for details.) We set the threshold 5 = 0.45, representing a medium job requirement.
Summary, In this manner, all necessary job and worker attributes can be extracted from sources such as O*NET and Big-bench Lite, with GPT-4o (or similar models) assisting in estimating skill proficiencies, decision-lewe intensities, and task-skill mappings. This subskill decomposition method is generic and can be applied to other job and worker datasets, making it practical across diverse domains.
4.3. Merging two workers with distinct ability profiles
We empirically examine the utility of merging two work- ers (W1 and W2). Theorem 3.3 assumes identical ability profile families, ensuring that W1 consistently outperforms W2 across all decision-level (action-level) subskills, or vice versa. In practice, however, this may not hold-e.g.. a GenAI tool may surpass human in some action-level sub- skills but not others. This raises the question: Does the sharp increase in job success probability from merging persist when workers have ability profiles from different families?
We note that O*NET and Big-bench Lite offer complemen- tary but biased views of work. O*NET emphasizes tasks involving judgment, creativity, and interpersonal skills, po- tentially under-representing emerging digital or computa- tional activities. Conversely, Big-bench Lite focuses on structured, rule-based problems where GenAl systems tend to excel. Empirical insights should therefore be interpreted in light of these distributions, as each dataset highlights different aspects of human-A complementarity.
Choice of parameters. We set the subskill ability profiles of W1 to be linear: aj ()) = at (s) = TrunN(1 0.78s, 0.0065; 0, 1), representing a human worker. For the second worker (W2), we define af TrunN(1 - (1 a)&, 0.0145; 0, 1) and a² = Trun 0.0145; 1). This models GenAl tool that excels at easier decision-level sub- skills but degrades with difficulty, while maintaining strong and uniform action-level abilities.
4.2. Evaluating worker-job fit with dependent abilitics
Theorem 3.2 assumes independent subskill abilitics (As- sumption 3.1), but this may not hold in practice. For in- stance, a worker's 's current state-such as fatigue or motiva- tion-can influence their abilities (J. et al., 1976), creating dependencies between subskill error rates Gee- This raises the question: Under such dependencies, can a slight increase in ability still lead to a dramatic nonlinear rise in success probability?
We analyze which decision- and action-level subskills should be assigned to each worker and quantify the resulting gain in job success probability.
If the average of ()) exceeds that of ()) (i.c., a < 0.22), all decision-level subskills are assigned to Wi; otherwise (a 2 0.22), to W2. Because the two workers' action-level abilities differ non-monotonically, neither domi- nates the other across all subskills. This renders the uniform merging strategy from Section 3.3 sub-optimal. Instead, we select the action-level subskill provider based on difficulty: the average of 0z (s) is 0.78s, while for a!
Choice of parameters. We set a1(s) TrunN(1 (1 a)s, 0.0065; 0) and (s(ss)
TrunN(1
0.78s, 0.0065; 0, 1) to model human worker, where the pa- rameter a controls the decision- level ability. For E [0, 1]. let F, denote the cumulative density function of ai (s). To introduce dependency between subskills, we assume that the worker has a random status 3 ~ Unif(0, 1). For each subskill. Efe
(2) (s) is constant at c. Thus, for 8,2 S ع- Wi has higher expected 0.78
(s())e) with probability 1 - p and Ee 1 F21(() with probability p. As p increases, the dependency between the SES strengthens. Specifically, when p= = 0, all Ges are independent. Conversely, when 1, all Eyes are fully determined by the worker's status 3, making them highly correlated.
ability and is chosen; otherwise, W2 is selected. This cre- ates merged worker W merge whose decision-level ability is linear and action-level ability is piecewise linear with a breakpoint at 8,2 0.78
I-c Let Pmeegg denote the job success
probability of this merged worker.
8

---

A Mathematical Framework for Al-Human Integration in Work
Pots
forvarbus
Pullues
oritwstiyy
3+0
.0002 2-
4-04
06
9:00
s=1
2-08
(a) P v.s.
(b) P v.s.
(c) Heatmap of P
Figure 3. Plots illustrating the relationship between the success probability P(at, a2, h. (.)) and the ability parameter and depen- dency parameter p for the Computer Programmer example with default settings (a, T) = (0.08, 0.45).
offers insights into effective human-AI collaboration. Our theoretical results identify conditions under which job suc- cess probability changes sharply with worker ability, and show that merging workers with complementary subskills can substantially enhance performance, reinforeing the view that GenAl tools augment, rather than replace, human exper- tise. This includes explaining the phenomenon of productiv- ity compression, where GenAl assistance disproportionately benefits lower-skilled workers, narrowing performance gaps, consistent with empirical findings from recent field studies.
(a) Heatmap of P Pmerge
(b) Heatmap of A
Figure 4. Heatmaps of the merged job success probability P Piegge and the corresponding probability gain A Paeege max P2}. , shown across different values of the ability param- eters (a, c) for the Computer Programmers example with default threshold = 0.45. Rapid color transitions reflect persistent phase shifts both Pmerge and A, even when worker profiles differ. Compared to Figure 2, the narrower bright region in Figure 4(b) suggests that merging distinct profiles yields more gradual im- provements than merging identical ones.
We demonstrate how the framework integrates with real- world datasets such as O*NET and Big-bench Lite, high- lighting its practical relevance. Empirical results validate theoretical insights, even under relaxed assumptions.
Our analysis focuses primarily on job success probability. In practice, performance also depends on factors such as efficiency, time, and cost. Incorporating these dimensions would yield a more comprehensive view of worker-job fit and inform workforce optimization strategies.
Analysis. Figure 4.2 plots the heatmaps of job success probability Peerge and probability gains A = Pmerge max (Pi,Ps) as ability parameters and e vary. When a < 0.22 (i.e., W2 has lower decision-level ability than Wi) and с E [0.78, 0.82], we observe P'meree = 1 while Pi, < 0.6, indicating a probability gain of least P Pmerge Pt 2 0.4. This occurs because first reaches 0.78, triggering sharp increase in Pm Pmerge as predicted by Theorem 3.2, and later reaches 0.82, aligning P2 with the trend in Figure 2. The range of c is narrower than that of 02 (2) in Figure since increasing results in smooth transition in action-level subskills from W1 to W2. Con- (2) surpasses 02 all action-level subskills
Moreover, the datasets used may not fully capture the com- plexity of skill attribution in dynamic work settings. O*NET reflects static, survey-based assessments, while LLM-based estimates from Big-bench may embed modeling biases. In- corporating empirical benchmarks (e.g.. HumanHval for coding, customer support transcripts) could strengthen the framework's 's empirical grounding.
Our model underscores the importance of improving eval- uation mechanisms better reflect the strengths and lim- itations of human and Al capabilities. More broadly, this work contributes to the growing literature on AI and work by offering a quantitative lens to study the interplay between human expertise and GenAl systems. As Al continues to re- shape labor markets, balancing human skill and automation remains critical challenge.
versely, when 0²2 shift abruptly, causing a more sudden transition. These findings confirm that the nonlinear probability gain from merging persists even when workers specialize in different action-level subskills, affirming our hypothesis.
5. Conclusions, limitations, and future work
This paper offers step toward quantifying that balance; fur- ther research is needed to refine models, incorporate behav- ioral studies, and promote equitable and effective human-AI collaboration in an evolving workplace.
This work examines the evolving impact of GenAl I tools in the workforce by introducing mathematical framework to assess job success probability in worker-job configurations. By decomposing skills into decision-level and action-level subskills, the framework enables fine-grained analysis and

---

A Mathematical Framework for Al-Human Integration in Work
Impact statement
SSRN Electronic Journal, 2023. URL https:
/ //api www.nticcccoollr.......... /rCrpppsss: 259701939.
This paper introduces a framework for assessing worker- job fit for both human and AI workers, aiming advance workforce optimization in an era of rapid technological change. Our insights and methodologies contribute more effective allocation of human and Al resources, improving job success probability and facilitating productive human- Al collaboration.
Angelova, V., Dobbie, W. and Yang, C. Algo- rithmic recommendations and human discretion. SSRN Electronic Journal, 2023. URL https: / seap.isemantccccooo... www.aa.psog/Cooo.... 253373458.
The societal implications are twofold. On one hand, the framework empowers organizations make data-driven workforce decisions, enhancing productivity and job satis- faction. On the other, it highlights challenges such as biases in ability evaluation and the evolving role of GenAT in labor markets, underscoring the need for careful consideration to prevent exclusion or unfair treatment of workers.
Annabelle Liang. Al hit 40% of jobs and worsen inequal- ity, IMF says. BBC, January 2024.
Anthropic. Claude: An Al assistant by Anthropic. https: //www///wwwwwww www.waaihhhticcccommm com, 2023. Available
at: wttpt:tp////...... htwp:://wwww./ee// www. www.:aaac.choccoommmm.
Arora, S. and Goyal, A. A theory for emergence of complex skills in language models. CoRR, abs/2300. 2023.
While our work may influence hiring practices and percep- tions of human-Al collaboration, these outcomes should be interpreted within the broader goal of equitable and efficient workforce optimization. We do not identify any immediate ethical concerns beyond these considerations.
Autor, D. Applying Al to Rebuild Middle Class Jobs. NBER Working Papers 32140, National Bureau of Economic Research, Inc, February 2024. URL https: / //ideas. repec. hwtp://r/bbbbeeee
Acknowledgments
bench authors, B. Beyond the imitation game: Quantify- ing and extrapolating the capabilities of language mod- els. Transactions on Machine Learning Research, 2023. ISSN 8835-8856. URL https:/ /opess////eeeeiee... net wttt/forummiidiin fot.rrrmmmmm1uysssssss
This work was funded by NSF Awards I04-2045951 and CCF-2112665, and in part by grants from Tata Sons Private Limited, Tata Consultancy Services Limited, Titan, and New Comnestone Science Foundation.
Bender, E. M., Gebru, T., McMillan-Major, A., and Shmitchell, S. On the dangers of stochastic parrots: Can language models be too big? In Proceedings of the 202/ ACM Conference on Fairness, Accountabil- ity, and Transparency, pp. 610-623, ACM, 2021. doi: 10. 445/3441188.445599
References
Workforce optimization. https:://en.wkkiediiia httikk//oooofffoopotiooocco
Abdin, M., Aneja, J., Behl, H., Bubeck, S., Eldan, R., Gunasekar, S., Harrison, M., Hewett, R. J., Javaheripi, M., Kauffmann, P., et al. Phi-4 technical report. arXiv preprint urXiv:2412.08905, 2024.
Borji, A. Limitations of AI in understanding human emo- tions. Cognitive Computation, 14(2):235-245, 2022. doi: 00000//112559021-09888888
Acemoglu, D. The simple macrocconomics ai. Economic Policy, 40(121):13-58 2025.
Brodsky, S. The hidden costs of AI: How gener- ative models are reshaping corporate budgets, 2024. URL https: /tt:://wwww//.. ibm. com/blog/ ai ed-economicc-ccmpteee-ooo cost/ ?utm_ source=chatgpt... com.
Acemoglu, D. and Autor, D. Skills, tasks and technologics: Implications for employment and earnings. In Handbook of labor economics, volume 4, pp. 1043-1171. Elsevier, 2011.
Brynjolfsson, E., Li, D., and Raymond, I. Gen- erative Al at work. SSRN Electronic Journal, 2023. URL https://api https://app..emmaiiccchooaa semanticscholar. org/ httg/oorputT:::52285555
Acemoglu, D. and Johnson, S.
https:
wtt.aa.rrgg/ee///iaaaiiioo Bandd//ssues//2033/1 Rebalancing AI Acemoglu-Johnson Dec 2023.
Bubeck, S., Chandrasekaran, V., Eldan, R., Gehrke, J. A., Horvitz, E., Kamar, E., Lee, P., Lee, Y. T., Li, Y.-F. Lundberg, S. M., Nori, H., Palangi, H., Ribeiro, M. T., and Zhang, Y. Sparks of artificial general intelligence: Early experiments with GPT-4. ArXiv, abs/2303. 12712,
Agarwal, N., Moehring, A., Rajpurkar, P., and Salz, T. Combining human expertise with artificial in- telligence: Experimental evidence from radiology.
10

---

A Mathematical Framework for Al-Human Integration in Work
2023. URL https://appt
semanticscholarr
Fosso Wamba, S., Guthrie, C., Queiroz, M. M., and Min- ner, S. ChatGPT and generative artificial intelligence: exploratory study of key benefits and challenges in operations and supply chain management. International Journal of Production Research, pp. 1-21, 2023.
prg/CorpusID::557
Cabrera, Á. A., Perer, A., and Hong, ai collaboration with descriptions of Al behavior. Pro- ceedings of the ACM on Human- Comann-Cmputer Interuction, (CSCW1))1-21, 2023.
J. I. Improving human-
Guo, D., Yang, D., Zhang, H., Song, J., Zhang, R., Xu, R.,
Zhu, Q. Ma. S., Wang, P., Bi, X., et al. DeepSeek-RI: Incentivizing reasoning capability in LLMs via reinforce- ment learning. arXiv preprint 1xXi::5501.129455 2025.
Cazzaniga, M., Jaumotte, F., Li, L.., Melina, G., Pan- ton, A. J., Pizzinelli, C., Rockall, E., and Tavares, M. M. Gen-Al: Artificial intelligence and the future of work. URL https:/ /aap:pp:://ppppppppp semant icschllrr STT/Co=ppSSDD:257000
Harding, J., D' Alessandro, W., Laskowski, N., and Long, R. Al language models cannot replace human research participants. Ai & Society, pp. 1-3, 2023.
Celis, I. E., Mehrotra, A., and Vishnoi, N. K. Interventions for ranking in the presence of implicit bias. In FAT*, pp. 369-380. ACM, 2020.
He, Z., Liu, Y., Zheng. J., Qin, B., Yao. J., Xuan, R. and Yang, X. FlagEvalMM: A flexible framework for comprehensive multimodal model evaluation, 2024. URL /ttpp::///ttttt///p/ com/ fl flageval b F FlagEvalMM.
Chowdhery, A., Narang, S., Devlin, J., Bosma, M., Mishra, G. Roberts, A., Barham, P., Chung, H. W., Sutton, C., Gehrmann, S., Schuh, P., Shi, K., Tsvyashchenko,
Hendryeks, D., Burns, C., Basart, S., Zou, A., Mazeika, M., Song, D., and Steinhardt, Measuring massive multitask language understanding. In International Conference on Learning Representations, 2021.
S., Maynez, J. Rao, A., Barnes, P., Tay, Y., Shazeer, N., Prabhakaran, V., Reif, E., Du, N., Hutchinson, B., Pope, R., Bradbury, J., Austin, J., Isard, M., Gur-Ari, G., Yin, P., Duke, T., Levskaya, A., Ghemawat, S., Dev, S., Michalewski, H.. Garcia, X., Misra, V., Robinson, K. Fedus, L., Zhou, D., Ippolito, D., Luan, D., Lim, H., Zoph, B., Spiridonov, A., Sepassi, R., Dohan, D., Agrawal, S.. Omernick, M., Dai, A. M., Pillai, T. S., Pel- lat, M., Lewkowycz, A., Moreira, E., Child, R., Polozov, O., Lee, K., Zhou, Z., Wang, X., Saeta, B., Diaz, M., Firat, O., Catasta, M., Wei, J., Meier-Hellstern, K., Eck, D., Dean, J., Petrov, S., and Fiedel, N. Palm: Scaling language modeling with pathways. J. Mach. Learn. Res., 24:240:1 1::40:1-240:1111 13, 2023.
Inga, J., Ruess, M., Robens, H., Nelius, T., Rothfuß, S., Kille, S., Dahlinger. P., Lindenmann, A., Thomaschke, R., Neumann, G., Matthiesen, S., Hohmann, S., and Kiesel, A. Human-machine symbiosis: A multivari- ate perspective for physically coupled human-machine systems. International Journal of Staman-Compun Studies, 170: 170:10 2023. ISSN 1071-5819. doi: https://doi.org/10. ht2.2iijjjhh222222222222
J., Richard, and Hackman. Motivation through the design work: Test of a theory. Organiza- tional Behavior and Human Performance, 16:250-279, 1976. URL https: //ts:///pp.... semanticscholar org/ htt///orpuID::111100
Chris Vallance. AI could replace equivalent of 300 million jobs report. BBC, March 2023.
DeepMind. Gemini: A family of multimodal models, deepmind, com/blog/ ppogle-deepmnnddunvveile--
2023.
Jaffe, S., Parikh, N., Butler, J. L., Farach, A., Cambon, A., Hecht, B., Schwarz, M., Teevan, J., Andersen, R., Bermejo-Cano, M., Bono, J., Buscher, G., Chen, C., Clarke, S., Counts, S., Dillon, E., Edelman, B. G., Gruber- Gremlich, U., Hilke, C., Hanrahan, B., Ho, S., Houck, B., Khemka, M., Kewenig, V., Kleiner, M., Knudsen, E., Manivannan, S., Meijer, M., Neville, J., Ngo, N., Ngwe, D., Peckham, R., Peng. S., Presson, N., Rangan, N., Rangareddy, R. Rintel. S., Rodriguez, R. Rotella, K., Safavi, T., Sarkar, A., Scott, A. E., Sellen, A., Shah, C., Simkute, A., Smith, T., Srinath, S., Suri, S., Tai, A.J.,, Tankelevitch, I.,, Wan, M., Wang, L., White, R. W., and Yang. L. Generative AI in real-world workplaces the sec- ond microsoft report on AI and productivity research. 2024. URL https: api. semant iesthhlas...... org/CorpusID:271633
Dillion, D., Tandon, N., Gu, Y., and Gray, K. Can AI language models replace human participants? Trends in Cognitive Sciences, 2023.
Eloundou, T., Manning. S., Mishkin, P., and Rock, D. GPTs are GPTs: An early look at the labor market im- pact potential of large language models. arXiv preprint 1:Xiv:2303.101:0 2023.
Felten, E. W. Raj. M., and Seamans, R. C. How will language modelers like ChatGPT affect occupa- tions and industries? SSRN Electronic Journal, 2023. URL https://ppi.emmtiicccllll org//CorpustD:2572800

---

A Mathematical Framework for Al-Human Integration in Work
Kahneman, D. Thinking, Fast and Slow. Farrar, Straus and Giroux, New York, 2011. ISBN 978-0374275631.
Okawa, M., Lubana, E. S., Diek, R. P., and Tanaka, H. Com- positional abilities emerge multiplicatively: Exploring dif- fusion models on a synthetic task. CoRR, abs/2310.0953 2023.
Kleinberg, J. M. and Raghavan, M. Selection problems in the presence of implicit bias. In ITCS, volume 94 of LIPics, pp. 33:1-33:17 17. Schloss Dagstuhl Leibniz- Zentrum für Informatik, 2018.
OpenAL. GPT-4 technical report. 2023.
OpenAL. LLMs.
Learning to reason with https://opent www.ooomeesseeeoe.ooocco.cccccce index/ Leariing-to-reassn-wttt--- 2024.
Klubnikin, A., Analyst, I., Likhadzed, V., CEO, I., Sta- shevsky, K., and CTO, I. Evaluating the cost of generative AI for effective implementation in your organization, 2024. URL https::/ttrexxroopp::
Otis, N., Clarke, R. P., Delecourt, S., Holtz, D., and Kon- ing, R. The uneven impact of generative Al on en- trepreneurial performance. SSRN Electronic Journal, 2024. URL httpps://app semanticscholas org/ NOrppsID:227115322
gict__sourre-ccatggae com.
Kochhar, R. Which U.S. workers are more exposed to Al on their jobs?, Jul 2023.
Paradis, T. will reshape the global labor force. Employers will need help their workers keep up. Business Insider, August 2024.
Kontorovich, Concentration in unbounded metric spaces and algorithmic stability. In International conference on machine learning, pp. 28-36. PMLR, 2014.
Reid, M. and Vempala, S. S. Does gpt really get it? a hierarchical scale to quantify human vs Al's un- derstanding of algorithms. ArXiv, abs/2406. 147222...172,, http://papp...aaaacssataaoeaeeaa semanticscholat org/ http/CorpusD:D:00 27 688183,
KPLorg. What is key performance indicator (KPI)? https: // /www///wwwk// htt.:k....rr//pii-baissc..ss 2024.
2024. URL https: /appi.
Licklider, J. Man-Computer Symbiosis, volume HFE-1. IRE Transactions on Human Factors in Electronics, 1960. doi: 10.1109/THFF2.1 01110//TFF2.199095555000 1560.452599.
Sadeeq, U. Noise: flaw in human judgment. Vikalpa, 48: 163 165, 2023.
Lo, W. Yang, C.-M., Zhang, Q., and Li, M. Increased productivity and reduced waste with robotic process tomation and generative Al-powered ioe services. Journal of Web Engineering, 23(1):53 87, 2024.
Services, C. AI for business study: The combined power of AI and generative AI. https //www tcs com/ bus ness- 2024.
i inst ght tttts/blloge /bwlgs/l///////////////o /
Sharma, Benefits or concerns of Al: A multistakcholddrr responsibility. Futures, pp. 103328, 2024.
Mahowald, K., Ivanova, A. A., Blank, I. A., Kan- wisher, N., Tenenbaum, J. B., and Fedorenko, E. Dissociating language and thought in large language models. Trends in Cognitive Sciences, 28:517-540, /t////app 2024. URL http:s///pp...smaacccccoooo semanticscholar.
Shine, I. and Whiting, K. These are the jobs most likely to be lost - and created because of AI. World Economic Forum, May 2023.
Sinclair, A. and for Employment Studies, I. Workforce Planning: A Literature Review, Institute for Employment Studies, 2004.
Microsoft. Al at work: Here now comes the hard part, 2023.
Miller, E. Adding error bars to evals: A statistical approach to language model evaluations. 2024.
Somers, M.
How generative Al can boost highly skilled workers' productivity, 2023. Accessed: 2025-01-28.
Naveh, Y., Richter, Y., Altshuler, Y., Gresh, D. L., and Con- nors, D. Workforce optimization: Identification and assignment of professional workers using constraint pro- gramming. IBM Journal of Research and Development, 51(3.4):263-25 279,, 2007. doi: 10.1 1t.11477dd.513.0265
Stade, E. C., Stirman, S. W., Ungar, L. H., Boland, C. L., Schwartz, H. A., Yaden, D. B., Sedoc, J., DeRubeis, R. J., Willer, R., and Eichstaedt, J. C. Large language models could change the future of behavioral healthcare: a proposal for responsible development and evaluation. npj Mental Health Research, 3(1):12, 2024.
Noy, S. and Zhang, W. Experimental evidence on the productivity effects of generative artificial intelli- gence. Science, 381:187 - 192, 2023. URL https: //api. semanticscholas org/CorpusII: 257394529.
U.S. Department of Labor, Employment and Training Ad- ministration. O*NET Online. National Center for OENET Development, 2023. https: / www. one online. org/
12

---

A Mathematical Framework for Al-Human Integration in Work
Vaccaro, M., Almaatouq, A., and Malone, T. W. When combinations of humans and Al useful: A system- atic review and meta-analysis. Nature human behaviour, 2024. URL https: /pp.. wttpppssssmatttscoo...oooo
Handel, R. Probability in High Dimension. 2014.
Lecture notes, available at https: /wwbb.wb... math. princeton. edu/ edd//rrvan/eetturesll pdf.
Vershynin, R. High-Dimensional Probability: An Intro- duction with Applications in Data Science. Cambridge Series in Statistical and Probabilistic Mathematics. Cam- bridge University Press, 2018. ISBN 9781 108415194. doi: 10. 1017/998 198231596.
Walker, J. Everything wrong with DORA metries. Aviator Blog, January 2023.
Wang, Y., Ma, X., Zhang, G., Ni, Y., Chandra, A., Guo, S., Ren, W., Arulraj, He, X., Jiang, Z., Li, T., Ku,
M.. Wang, K., Zhuang, A., Fan, R., Yue, X., and Chen, W. MMLU-Pro: A more robust and challenging multi- task language understanding benchmark. In Proceedings of the Neural Information Processing Systems Track on Datasets and Benchmarks, 2024.
White, C., Dooley, S., Roberts, M., Pal, A., Feuer, B., Jain, S., Shwartz-Ziv, R., Jain, N., Saifullah, K., Naidu, S., Hegde, C., LeCun, Y., Goldstein, T., Neiswanger, W., and Goldblum, M. Livebench: A challenging, contamination- free LLM benchmark. 2024.
Will Knight. Openai upgrades its smartest AI model with improved reasoning skills. WIRED, December 2024.
Yu, D., Kaur, S., Gupta, A., Brown-Cohen, J., Goyal, A., and Arora, S. SKILL-MIX: flexible and expandable family of evaluations for AI models. In ICLR. OpenReview.net, 2024.
13

---

A Mathematical Framework for Al-Human Integration in Work
A. Additional related work
Workforce optimization seeks to align employee skills with organizational objectives to improve productivity, efficiency, and satisfaction (Sinclair & for Employment Studies, 2004; wik; Services, 2024; Naveh et al., 2007). Although this area has been extensively studied empirically, theoretical models that systematically evaluate worker-job fit remain limited.
The emergence of GenAl tools has reignited debates around automation and its impact on skilled labor (Dillion et al., 2023; Harding et al., 2023; Stade et al., 2024). Recent work demonstrates Al's proficiency in complex tasks-from expert-level reasoning (OpenAI, 2024; Will Knight, 2024) to high performance on domain-specific benchmarks such as LiveBench and MMLU-Pro (White et al., 2024; Wang et al., 2024). These advances underscore the need for principled frameworks to assess human-Al complementarity (Yu et al., 2024; Hendrycks et al., 2021).
Several studies compare human and AI capabilities across domains (Brynjolísson et al., 2023; Noy & Zhang, 2023; Sharma, 2024; Eloundou et al., 2023). However, existing models often conflate decision-maiee -aainn and execution, overlooking their distinct roles in work. Our framework explicitly separates decision-level and action-level subskills, enabling more granular analysis of how AI systems complement human abilities.
Research on the labor implications of AI suggests that GenAl tends to augment lower-skilled workers (Acemoglu & Autor, 2011), consistent with our finding that AI enhances action-level subskills while decision-level abilities remain critical. Studies on AI-driven productivity gains (Noy & Zhang, 2023; Lo et al., 2024; Fosso Wamba et al., 2023) offer additional empirical support for our model's predictions. Integrating real-world data to further validate our theoretical insights an important direction for future work.
Recent work by Acemoglu (2025) presents a macroeconomic model that analyzes the effects of Al on productivity, wages, and inequality via equilibrium-base task allocation between labor and capital. While developed independently, their assumptions-such as task decomposition, differential Al performance across task types, and heterogeneity in worker productivity with our framework's decomposition of skills and modeling of worker ability. The key distinction lies in scope: their model addresses aggregate, market-level outcomes, whereas ours focuses on job-level success and collaboration between individual workers.
B. Properties of ability profiles
This section discusses the properties of several ability profiles.
B.1. Monotonicity, variability, and visualization for ability profiles
Below, we formalize three ability profiles: constant, linear, and polynomial with additive uniform noise.
Constant profile. We consider constant profile a = c e for some E [0, 1] and distributed according to a certain noise distribution, e.g., ~ min {c, 1 -c-- Unif[-o, oj for some a E [0, 1]. When a = 0, a = c is a constant function. This model is useful for skills where ability does not vary with increased experience, such as automated processes handled by GenAl tools, where the output remains consistent regardless of operational duration. The scale of two parameters e and a determines the performance of a. Parameter e determines the average ability of constant profiles, where e represents the worst ability and c represents the best ability. Moreover, fixing o, the ability of constant profile rises smoothly from the worst to the best increases from 0 to 1. Also note that Var [s)] = min {c, 1 - c)- g² Thus, parameter a reflects the variability of a.
3
a)s + E(s) for some a, E [0, 1] and noise e(s) min {(1 - a)., 1 (1 - a)s) Unif[-o, a, a) for some [0, 1]. When > O, this model is apt for scenarios involving workers whose ability to develop skill increases linearly with the ease of the skill. Note that for any [0, 1). E(s) = e (1 a)s is a monotone increasing function of both and Thus, the average ability of a linear profile rises smoothly from the worst to the best as (or c) increases from 0 to 1. In this paper, we usually set c = 1 such that (O))= = 1, representing the highest ability for the easiest subskill.
Linear profile. We consider a linear profile a(s) = c- 1
Polynomial profile. We consider a polynomial profile (ss) = 1 3 + E(s), for some parameter 3 2 0 and noise E(s) ~ min { 1 - s0} } Unif[-o, a] for some a E [o, 1]. This function can be used to model how decision decision-making subskills often improve nonlinearly with the easiness of the skill. The unique parameter, (9, allows us to adjust the
14

---

A Mathematical Framework for Al-Human Integration in Work
sensitivity of the ability function changes in 8, where 3 0 represents the worst ability a e(s) and x represents the best ability a 0. Moreover, higher values of / indicate more pronounced increase in average ability as approaches 1.
We plot constant, linear, and polynomial profiles with additive uniform noise in Figure 5. In these models, we know that ability profiles are usually characterized by parameter measuring the average ability (c.g.. c for constant profiles, a for linear profiles, and 3 for polynomial profiles) and parameter o measuring the variability. A key problem in this paper is quantifying how these parameters affect the performance of workers in a given job. Intuitively, the quality of an ability profile typically improves as the average ability increases or the variability decreases.
Mots of constant profites for various when (m
Plats Inear profiies various when o 0.1
Plots of polynomial profies for when a
0.1
0.3
0.7
0.1
3 0.9
0.2
0.5
0.7
1.4
1.8
(a) Constant profiles
(h) Linear profiles when
(c) Polynomial profiles
Figure 5. Plots by varying the ability parameter for various families of ability profiles with an additive uniform noise when the noise level
0.1. Observe that the width of the domain as)) the largest when the average ability E(s) 0.5 and is the smallest (0) when
E(s) 0 or 1. This noise level indicates that the best worker always completes the subskill flawlessly (a(s) 1), the worst worker always fails at subskill (a(s) = 0), while the performance of a medium worker exhibits greater variability.
B.2. Stochastic dominance for ability profiles
We first define the stochastic dominance property for ability profiles.
Definition B.1 (Stochastic dominance for ability profiles). Let ax, ar be two ability profiles parameterized by ability parameter μ. ní, respectively, and the same noise parameter a. Suppose S. n.. We say a has stochastic dominance over a' for any s E [0, 1] and x 2 0:
Pr (~ai()
| x|
<
15' 2 x]. (s)
Pr
We propose the following proposition that shows that the studied ability profiles have stochastic dominance properties. Proposition B.1 (Stochastic dominance for ability profiles). The followin hold:
Constant profile Let a = c + E for some E [0, 1] and E ~ min {c, 1 - cc- Unif[-a, for some a E [0., 1]. Let a' = + E(s) for some c' E 1] with e é and E(s) ~ Unif[-o,a)) Then a has stochastic dominance over a'.
Slope parameter for linear profile Let ass) = 1 1 (1 (aaa)s ={(s) for some a E [0. 1] and noise E(s) min {(1 - a)s, 1 (1 a)s} Unif(-a, o). Let (s) = 1 (1 (1-a')s a''s =(ss) for some a' E [0,1] with S a' and noise ((s) Unif[- min {(1 d')s, 1 (1 aa'ss) min {(1 d')s, 1 - (1 a')s) o). Then a has stochastic
dominance over a'.
Polynomial profile Let a(s) = 1 - Φ + E(S) for some 3 E [0, and noise E(s) ~ min {s 1- 11-55 s²} Unif[-o, -o, a). Let as(s) Unif[- min { { 1 1--52 50 o. min { s
s + E(S) for some 3' [0,1] with S 3' and noise E(s)
min
o o). Then a has stochastic dominance over a'.
The first two items ensure that ability profiles satisfy stochastic dominance with respect to both ability parameters c and a.
Proof of Proposition B.I. We prove for each family of ability profiles.
15

---

A Mathematical Framework for Al-Human Integration in Work
Constant profile.
For any
have
K 2 x| min { m.xax (0. x - c) + min {c, 1 cJ
Pr (~a((s)
2 min {c, 1 - cJ 0
Note that when 1 1- c 5.050 0.5, we have
(1 - c) + min -1.
0.5
2 min {c, c) a
20
2(1 c)o
which is increasing with When 1 >> 0.5, we have
(1 c) + min fc, 1 - ct 1
1
1
0.5
2 min {c, - c} a
20
2co
which is increasing with c. Overall, Prçwa(e) [C 2 æ] increasing with c. Thus, since e S d' we have Prewa(e) (C x) s
(e) Is' 2 z].
Linear profile.
For any x E [0, 1] and E [0, 1], we have
Pr Is 2 x] = min {1. max 0, (1 a)s + min {(1-(((1)))))))) (1 a)s) (~a(s)
2 2mi {(1 a)8, 1 - (1 - (1-))8}
Note that when (1 - a)s < 0.5, we have
x - (1-08 a + min {(1 - a)s, (1 (1-o)ss
+ 0.5 +
min {(1 a)s, 1 (1 a}s}o
2o
2(1 a)sa'
which is increasing with When (1 a)s 0.5, have
(1- a)s + min {(1 a.s, (1 (1-a)s
1
1 x
0.5
min {(1 - a..,a.s111 (1 a)s}
20
2(1 (1 a)s)a'
which increasing with a. Overall, Prcwa(() [c 2 æ| increasing with a. Since a S a', we have Pr(wa(e) 2/ Prewi-a((s) IK' z].
Polynomial profile.
For any E [0, 1] and
[0, 1]. we have
8 min {8 1 - 58) o
Pr (~)a(s)
1< x] min 1, 0,
2 min { ,,1 - 3
Note that when S < 0.5, we have
x + min {sa 1 1
1 + 0.5 +
min (8) =00
20
283
which is increasing with 3. When 8 > 0.5, we have
x 8 + min { { - 3
0.5
2 { 1 36) 38}0
2o
2(1
$ 30)00'
which is increasing with B. Overall, Prşwa(s) IS 2 x] is increasing with B. Since 3 S 3r, we have Prewa(s) (52 x| S (re Is' 2 x..
Thus, complete the proof of Proposition B.I.
C. Proofs of results in Section 3 and extensions
In this section, we provide the omitted proofs of the results in Section and demonstrate how to extend them to accommodate general and dependent noise models. We further extend the analysis of linear ability profiles from Section 3.2 to alternative choices of job error functions and ability profile families (see Section C.2).
16

---

A Mathematical Framework for Al-Human Integration in Work
C.I. Proof of Theorem 3.2: Phase transition in success probability
We prove generalized version of Theorem 3.2 that accommodates arbitrary noise models E(s), extending beyond the uniform and truncated normal distributions introduced in Section 2.
We first need the following notion that captures the dispersion degree of ability profiles (01,02) 02).
Definition C.1 (Subgaussian constant and maximum dispersion). Let ag be parameterized by pe, 0g 2 0. For each subskill Sit, define the smallest constant Sgje (pe, σt) such that for all [3 R.
sg (Me, o2)
(
E Exyar(sy)
< exp(
2
We define the subgaussian constant as:
sg(p1, 01, 01, 02) =
Σ
jE|n),
Given d1, P2, 02 the maximum dispersion μr is defined as:
MaxDisp, (01, 42, 02) == sup sg(un, 01, , M2; 02).
m2
MaxDisp (02: (1, 01) is defined similarly.
Intuitively, the subgaussian constant 8g, quantifies the variability of subskill's ' ability distribution around its mean (van Handel, 2014; Vershynin, 2018). The maximum dispersion MaxDisp captures the cumulative uncertainty across all subskills by aggregating the subgaussian parameters. For example, under uniform noise e(s), we have sgje S of/4, yielding MaxDisp yn (2, M2; 02) S na? + / Under truncated normal noise, Sgye S oz giving MaxDisp,, (01, H22 02) S nff + o2). ). As the noise parameters de increase, MaxDisp also increases, reflecting greater dispersion in ability. In the deterministic case where σ 02 = O, we have MaxDisp 0, indicating no uncertainty in abilities.
We propose the following generalized version of Theorem 3.2, where the term no + +o)) in 71 (designed for both uniform and truncated normal noises) is replaced by the more general quantity MaxDisp,,, (01 (01, 02).
Theorem C.1 (Extension of Theorem 3.2 to general noise models). Fix the job instance, action-level ability p2z and noise levels 01, 02. Let fi be the unique value such that the expected job error equals the success threshold:
Err Eng (Mq 01, P2, 02)
Let 0 E (0,0.5) be a confidence level, and define the transition width: 71 = MaxDisp,, (11,12.003)11((10)) where L is the
MinDor #1 (11
Lipschitz conslant of the job error function. Then the job success probability satisfies:
P50 if pn PP 71 and P P if +m
For preparation, we first introduce the following variant of McDiarmid's inequality. Given a random variable on IR we define ||X ll+ to be the smallest number a 2 0 such that for any 3 E R, E [e < Pa2/2
Theorem C.2 (Refinement of Theorem 1 in (Kontorovich, 2014)). Let G RT IR be functionn Suppose X1...., XT are independent random variables. Then we have for any > 0,
212 Y.em 1x,- 1[x,11
Pr [G(X,, Xx) E(, Xx)] + 1)5 S
X Xy....y
and
2+2 Ejery 1X,
Pr [G(X,,.... XT) E Xx)] S X XI
x x
The theorem provides a concentration bound for the function value of G when its input variables are independent subgaussian. Now we are ready to prove Theorem C.I.
17

---

A Mathematical Framework for Al-Human Integration in Work
Proof of Theorem C.I. Recall that Err is function of 2n realized subskill abilities Sze. By Assumption 3.1, Syes independent random variables. Moreover, by Definition C.I, we know that
(8)(He,0)) = |15s (ewl-ae [S] 1e
and hence,
= Σ (g)( = E
1Rsse -olsss)
aelsse) [S] li
JE(1)
(E(2)
Since Err L-lipschitz, function 2 Err I-lipschitz. Also, recall that Erravg(M1, 01 22, 02) B(gendaar(ss) [Err(()]-
Now we plugin T =2n, G + Err, Xys being Ljes in Theorem C.2. We obtain that for any > O,
Err(S) 2 Erravg (M1, H2, 02) + t
Pr Pr|Err(() Erravg 01, +t| Ge
Pr Ge
and
Pr [Er((() < 01, 02) - 1 S Peegiri (22)22). Eye
(2)
Note that MaxDispp, pn (o), H2, 02) 2 01 02). Thus, when t = L MaxDisp (0,, 02) In f.4 we have
Pr Pr((E)()) Err 01, + t) A and Pr [Err(() S Errawg (11, /2, t] s 0.
Eie
Ezt
This implies that if Erravg (11, 01, 01, 02) 5 --
Pr
(E)(( S <T] 2 Pr [Err(() > (rtm + 2 -16
(3)
Eze
Eye
Also, if (r1 T1, 02) +
Pr Err(() ST] <
Pr [Err()) > 11,, Eze
01, 02) - 1 0.
(4)
Recall that Ly MaxDisp,, (01:22.2))1n((111 MinDer (a) (M2:000
Then it suffices to prove the following lemma.
MinDer,,, (01.)99005)
Lemma C.3. Let t > 0. Under Assumption 3.1, if M1 S pf MinDer,y (21))2 then Errave (M1, T1, (1, P2, 02) 2 r + t; and
im 2 14 + MinDery, (21) then Err T1, P22 02) < 1T-
Proof. We first prove that aErraz (11,01, 1 U2, 02) < 0. It suffices prove that for any u. with p 2 n², Errawg(J, 11 02) < Errawg(A (I", D1 u2 02). Let an be parameterized by (n, o1), of be parameterized by (u', oz), and 02 be parameterized by (122,02 02). By Proposition B.I, we know that a' has stochastic dominance over an Thus, for every j [n]. there exists a coupling of (C, 5') for 1- (sze) and ç (s such that S Ç². Let n, be the joint probability density function of (C. We have Jg. ("dC = on (S) and Je (,( = ((()))((). Let my be the joint probability density function of (G1z, Gj). We have J (T, = on (C) and J (5, (8)) Then we have
Erravg (14, (1, 01, (1,,11,22222 BG [Er(()]
Il Ty (Sye Err)))
(Assumption 3.1 and Defn. of TT))
/ II my (Tyee Errlg)
(Montonicity of Err)
< Bg (fn-l--((()),,,221111- [Er((()
< Errawg (H', T1, H2, 02).
18

---

A Mathematical Framework for Al-Human Integration in Work
Thus, if #1 < /1
have
MinDerpy (P:
Erfing (41, 01, (2;
Errang 01, H2, + (Hi (1)- Min (2, (Defi. of MinDer (o1 M2, 02))
MinDer (2, MinDer,, (o) (2)
σ2) = =))
L.
Similarly, we can prove that μn > pf MinDerp1 (21)) of Lemma C.3.
then Erravg 02, 02) <T- T- This completes the proof
Combined with Lemma C.3, we have completed the proof.
C.2. Bounding for alternative choices of error functions and ability profiles
Similar to the illustrative example in Section 3.2, we analyze the window " in Theorem 3.2 for alternative choices of the error functions h, 9, and ability profiles , 02. We consider uniform noise such that MinDer, (0 122, σ2) S ng² Then the key is to bound the Lipschitz constant L and MinDer,,, ( 02)
Analysis for max functions. Let h, 9, be max such that Err(g) = jjj Siz. Suppose the ability profile is linear with noise: ar(s) = 1- - ae)s + E()), where E(s) ~ Unif -o ar. We can compute that
p=
II
Pr(Se S <).
(E
Below, we analyze the window of the phase transition for this case from both the positive and negative sides of the parameter range.
A positive example. Assume all Sjz 0.5, af = =2 = 0.5. and = 0.75 + -. Then each Pr[Ge S T] is implying that
1
2n
P=
II
PrIca S (1
2n ) = 1/c.
jE(m)
Let 7/1 = 40/n. On one hand, a; = af - n1 we can compute that for each j E [n], Pr(G11 I 7] 1 n.
Then
1 )" (1 2 < 1/e² < 0.14
P S (1
2n
On the other hand, a; af +,, Pr(C11 I T] = for each j E [n]. Then
p= (1
2n )m 2 0.6.
Thus, increasing a from below af - 7i to above of + 71 results in a probability gain of 0.46. The window this phase transition only 7 Oaa/n), which is even sharper than the ( //) window observed for the average error function.
A negative example. Assume all 8yt 0 except that $11 af 02 0.5, 0.25 and 0.5. Then
II
Pr[Gr S Pr[çi 1 s 0.5] 0.5.
jE(n),EE(1,2)
We can also compute that for aj E (0.35,0.65).
0.5
P = Pr(G11 5 0.5] = 0.5 min (a1, a1}²
which is close to linear function of 01. Then the window phase transition is O(1,, yielding a smooth, non-abrupt transition.
19

---

A Mathematical Framework for Al-Human Integration in Work
Analysis for weighted average functions. function. Given an importance vector E [0, for skills (e.g., derived from O*NET), select the task error function
We still select the skill error function 1(S1, (2) {(1 (2) the average
g ((g))een) EjEr, W/ JEI,
E wyhy:
to be the weighted average function, where Z=jez; wy tasks (e.g., derived from O*NET), we select the job error function
is a normalization factor. Give an importance vector e E [0, 1 for
f fon...... Im)
U:fi,
Eve(m)
U/
sE(m]
Σie(m is normalization factor. Then we have that their composition to be weighted average function, where Zie(m) function is:
Err(s) Σ
Σ
Evelm Dy Eyenr Wy (5 + G22)
(5)
jE(n] *E|ma]JET
We have the following observation for the Lipschitzness of Err.
Proposition C.4 (Lipschitzness of Err in Equation (5)). The lipschitzness constant of Err in Equation (5) is L S
Ti max ZIemm) Evejm Ver Lyeepy Myy"
wy
For instance, when each wj- 4 and i. L, we have L S maxje(n) Die(bbyen mाा.- Specifically, when each T; contains k skills and each skill appears in km tasks, we have I, S zn Suppose the ability profile is linear with noise: ar(s) = 1 - (1 ar)s + e(s), where E(s) ~ Unifl-o, o]. Similarly, we can compute that
MinDera, (2,02,00
1 Σ 2
Σ
Eveim) Uy Eyrera Wy 8j1-
UN
jE[m] iE JEM
Then we have the following bound for
maxjE/m] Dielm]yen Even Ly wy 0.5mg In(1/0)
LV Im(1/8)
71 MinDer,,, (o1, M2, 02)
Eje(n) Zie(m])EE, Evejmi by Evers my 8j1
b
Analysis for constant profiles. We select ae or + min {c,, ce Unifl-or o, o as constant profiles with noise level detailed in Section B.I. We still let Err(() 2n (y + (j2), where L < Note that MinDer,, (a, 02 = Then we have the following bound for 71:
LV 0.5no In(1/0) MinDer (a, 02
5a. In(1/0)
71
2n
Analysis for polynomial profiles. noise level a. We still let Err(C) =
We select = - g + min {s 1 S Unif[-a, polynomial profiles with 1
2n j= (S11 + (12), where L S 2n
1.
Then we have
aErr (31, o, B2,
03
2n
71
j=1
Note that this partial derivative is 0 when 31 = 0, which results in MinDerg, (a, (2, 0) = 0 and 71 = 00.
However, by the proof of Theorem C.I, it suffices to bound the partial derivative for 81 [ - 71, B9 ml instead of the
entire domain 22-- Suppose we know that [3f 71, 39
71] [0.5, 2}: this implies that
8
811-
2n
4m
j=1
j=1
20

---

A Mathematical Framework for Al-Human Integration in Work
M4
41
14
44
01
Figure 6. Heatmaps of productivity compression value PC PC= P() by merging low-skilled human worker with action-level ability parameter and high-skilled human worker with action-level ability parameter with GenAl tool for different ranges of (a1; 02) for the Computer Programmers example with default settings of 0.45.
(P3
Thus, we have the following bound for 71
LV0.5mo* In(1/0)
2n ln(1/))
71
4n Σj=
j== 8/1
So
> j= 8/1
C.3. Proof of Theorem 3.3: Success gain from merging complementary workers
Similar Section C. extend Theorem 3.3 to handle general noise model es)). The only difference is still the introduction of MaxDisp.
Theorem C.5 (Extension of Theorem 3.3 to general noise models). Fix the joh instance. Let E (0, 0.5) Mo(1/0)
be confidence
L MaxDiap,, (o( 14 o2
and 71 L. MaxDis,,, (019),13 21 )
level, and define: 1
If MinDer, (o w/! 112 ,0
MinDer,, (o .u (0
Erravg(14 (1) - -1 .0 : 112 02
(1) (1) (2) (2) IT S Err (11
(2) + 71 01 12
(2) (2) (2) (2)
then under Assumption 3.1, we have: P12 P2 29.
Proof. If (1r (2)
+ +71 , .1 12 2 2 r, by the proof of T'eoormm 3.2, we know that
(2)
Errawg (4]² .71 12
(2)
(00 2 71
(2) MinDer,,, (o . P2 oo22) = L- MaxDisp, (o] , (o( Im(1//).
(2)
=
By Inequality (4), we conclude that P2 S Similarly, by Errave (t) vi ,0 u²2 ,02 (2)) S t and Inequality (3), we
(1)
can obtain P12 8. Thus, A₂ 2 28, which completes the proof.
C.4. Proof of Corollary 3.4 and extension to distinct ability profiles
Similar to Section C.I, we extend Theorem 3.3 to handle a general noise model e(s). The only difference is still the introduction MaxDisp.
Corollary C.6 (Extension of Corollary 3.4 to general noise models). Fix the job instance. Suppose both human workers have the same decision-level abilities:
ui (AD, of = o(0) = (AT) oj
p
Let 0 E (0,).5) be a confidence level, and for each E (1, define 7/2 (1)
L. MaxDisp (o (0 (o2",mHi (e()))
If
MinDer, (1 mi
max { (Erravg 0i,02 7/2 .02 Errany oi (2)
(AI) (AI) (AI)
112 -
(1) (1) (1) -2(²) -/,22210 02 (2)) S Errayg oi,12 72 022
then under Assumption 3.1, we have: PC 2 20.
21

---

A Mathematical Framework for Al-Human Integration in Work
Proof. By the assumption on the decision-level abilities, the merging of We and WAI must utilize decision-level ability profile parameterized by (#1, ot). Since Err (1t, oi -1/44 - 72 22
(2) ( I - E of
.12 + +22 ). follows (AT) (AT)
from Theorem 3.3 that P22 P1 2 1 20. Also, since Erravg (1i, of M2 (AT)
72 22 I I Erravg i 01,42 +
(1) ) it follows from Theorem 3.3 that Pi- Pi 2 1 - 20. Also note that Pl S P2. Hence,
12
PC = P2 Pil - P2 P P2 P1 P P/2
If the merging of W2 and WAI utilizes W2's action-level abilities, we have P3 P3 and hence,
PC = P2 - P P PP 1 28.
Otherwise, if the merging of W2 and Ai utilizes WAI's action-level abilities, we have P3 = Py and hence,
PC P P= = P1 2 1
Overall, we have completed the proof.
Evaluating productivity compression with distinet ability profiles. Similar to Section 4, we investigate whether the productivity compression effect induced by Al assistance persists when the ability profiles of human workers and GenAI originate from different functional families.
Choice of parameters. We set the decision-lewe levell ability profiles of W1 and W2 to be linear with aj ()) = aj (s)
0.0065; 1). We assume (2 d1, representing two human workers with distinct skill levels. The parameter ranges are set ai E [0, 0.2] and 02 E [0.3, 1]. motivated by the observation that = 0.22 corresponds to a job success probability of 0.55, characterizing a medium-skinn skilled worker. For the GenAl tool WAI, we define the decision-level ability as aj (A))(s)
TrunN(1 - 00.7s, 0.0065; 0, 1), and define their action-level ability profiles 0g (2) (s) (runN(11
TrunN(1
0.92s, 0.0145; 0, 1) and the action-level ability as 02 (Al) (s) TrunN(0.8, 0.0145; 0, 1), such that the decision- level ability is consistently weaker than that of W,. We adopt the same merging scheme between human workers W, and the GenAl tool
WAI. Recall that for e E {1,2). Pt denotes the job success probability of W₁ before merging with WAI. and P/ denotes the corresponding probability after merging.
Analysis. Figure presents a heatmap of PC = (P2 - -PP as the ability parameters a₁ and d2 vary. We observe that PC increases with the ability gap 02 a1, indicating that the benefit of merging is more pronounced for lower-skilled workers. For instance, when d 0.1 and 02 0.8, the productivity compression reaches PC 0.8. These findings confirm that the productivity compression effect from human-Al collaboration persists even when workers specialize in different action-level subskills, thereby affirming our hypothesis.
C.5. Extending Theorem 3.2 to noise-dependent settings
We consider the noise-dependent setting introduced in Section 4. In this setting, dependency parameter p E [0, 1] controls whether subskill errors are drawn from a shared latent factor ,3 (with probability p) or independently (with probability 1-p). The following theorem extends Theorem 3.2, which corresponds to the independent case p = 0, to general p E [o, 1]. Notably, the sensitivity window 71 vanishes as p + 1, indicating that stronger dependencies smooth out abrupt transitions.
Theorem C.7 (Phase transition in noise-dependent settings). Fix the job instance, action-level ability p2, and noise levels σi, 02 Let pe be the unique value such that the expected job error equals the success threshold:
Err T1, 02) =2)= =T.
Let 0 E (0,).5) be a confidence level, p E [0, 1] be dependency parameter, and define the transition width: ?11 := MrMaxDisp (or (2) max ( 20 v 2o
where L is the Lipschitz constant of the job error function, Then the job
Men (a) 02)
success probability satisfies:
PSO of 1 S Sp 7 and P - if m 14 +
22

---

A Mathematical Framework for Al-Human Integration in Work
Absolute derivatives for a 0.08
Absolute derivatives a 0.22
Absolute value derivativees
|P
Absolute value of derivatives
IP'
nP
|p
(a) P v.s.
(b) |P| v.S.
Figure 7. Plots illustrating the relationship between the absolute derivatives [P]] and [P]] and a, for the Computer Programmers example with default settings of (a, o, 7) = (0.22, 0.08, 0.45).
Proof. The main difference is that the probability bound by Inequality (2) in the proof of Theorem 3.2 changes to be:
Pr (Er((S) Err 01, 02) (1 p) P1 + P.e Prooneer
Sze
The choice of y ensures the right-hand side to be at most 0, which completes the proof.
D. Additional implications of theoretical results
We empirically analyze the impact of workers' ability profiles on job success probability. In Section D.I. we illustrate how framework be used determine strategies upskill workers. In Section D.2, analyze the impact of evaluation bias on workers' abilities.
D.1. Evaluating intervention effectiveness: Boosting ability reducing noise
As discussed in Section 3 (see also Figure 5(b)), both increasing the ability parameter and reducing the noise level efficient interventions for increasing the job success probability. We consider the "Computer Programmers" example with independent abilities across subskills (p 1) in Section 4. Then, the job enror function Err is defined as in Equation (13) and the subskill numbers are as in Equation (1)). We set the subskill ability profiles be aj (')(s) TrunN(1 (1 a)s, o² /2; 1)
and (1) (s) TrunN(1 - 0.78s, o² /2; 0, 1), representing a human worker. We investigate which parameter-a, o- the greatest impact on P. This analysis crucial for guiding strategies to upskill workers for specific jobs. To this end, we first compute the derivatives of P with respect to a and a.. We denote |P] and [P' as the absolute values the derivative of with respect to a and a, respectively. We plot them in Figure 7 the default parameters (a, a, T) (0.22, 0.08, 0.45).
Figure 7(a) reveals that for 0.08, when a E [0, 0.15], |PI is larger; while when a E [0.15, 1], IP'| larger. In Figure 7(b), for 0.22, PP is always larger for any [0, 1]. Thus, in this specific example, depending on the ranges of (a. o)) either | or P may be larger, demonstrating that no single parameter universally outweighs the others in importance.
D.2. Analyzing the impact of inaccurate ability evaluation
We demonstrate how Theorem 3.2 highlights the importance of accurately evaluating workers' abilities for companies. Specifically, we consider the scenario where a worker's ability evaluations are biased and discuss the consequences of this bias. Mathematically, let the worker's true decision-level ability parameter be μl, while the observed parameter is fii = 3p1 for some (0, 1), reflecting bias the evaluation process. This bias model is informed by and builds upon a substantial body of research on selection processes in biased environments (Kleinberg & Raghavan, 2018: Celis et al., 2020). Below, we quantify the impact of such bias 3 on workers.
23

---

A Mathematical Framework for Al-Human Integration in Work
Plot of ratio of mistakenly evaluated high-qualifie workers
B
Figure 8. Plots illustrating the relationship between the ratio ry and the bias parameter | for the Computer Programmers example with default settings of 0.45.
Theoretical analysis. Suppose parameters u2, of satisfy that (Fravg(pi of (5,
= =. Let O E (0,0.5). Let 71 ==
L. Max ln According to Theorem 3.2, yn j41 2 71, the job success probability P(an,02 9, J: 2
MinDerg, (220))))22)
1 0, indicating that the worker fits the job. However, the evaluated success probability P is based on a weaker ability
profile a, parameterized by (1).o,). From Theorem 3.2, βui S μi -. the evaluated success probability p < 8
implying that the evaluation process concludes the worker does not fit the job. Since júi = 3p1, we conclude that 1 S 0
if μ (ut 71). Note that 1/ (14 71) > p 71 when 3
-i
Recall that 7/1 O
In in the linear
Mit
ability example shown in Section 3.3, which is o(1) when a << Vin or n > In 3.- Thus, the condition /
< 1 o(1) Consequently, for workers with ability parameter μan E lut + m. ( m)) the evaluated success probability P < 0, while the true success probability P 2 0. Thus, even a slight bias in ability evaluations can lead to dramatic errors in predicting the worker's job success probability, potentially causing companies lose qualified workers.
Simulation for one worker. We study the ratio of high-qualified workers with P > 0.8 who are mistakenly evaluated as insufficiently qualified with P z 0.6 due to evaluation bias. Again, take the example of "Computer Programmers" as an illustration. We set the decision-level ability profile to be TrunN(1 - (1 a)s, 0.0065; 0, 1) and the action-level ability profile to be TrunN(1 - 0.78s, 0.0065; 0, 1). representing human workers. We assume the ability parameter a follows from the density Unif(o, [0, 1] for mathematical simplicity (can be changed to e.g., truncated normal distribution). 0.8 if 0.34. Thus, 66% of workers are highly qualified for this job. In contrast, P S 0.6 if à = Ba S 0.25, i.e., S B Thus, high qugh-qual workers with ability parameter a E [0.34, 0.251 3 are mistakenly evaluated as insufficiently qualified. Then, the ratio of these workers among high-qualified workers is
By Figure 3(a), we know that P 2
0.25
r3 min { max (o, 3 0.34) /0.66 ): see Figure 8 for visualization. We note that T3 > 0 when B < 0.64, and it increases super-linearly to 1 as B decreases from 0.64 to 0.25.
0.25
Simulation for merging two workers. Next, we study how inaccurate ability estimation affects the gain in success probability from the merging process. We extend our merging analysis by introducing a trust parameter X to the merging experiment in Section which models imperfect merging by letting the estimated ability à Ac deviate from the true action-level ability of worker Wz. We then assign action-level subskills to W2 when its scaled ability, Ac, exceeds W1 's ability (i.e., 1 - 0.78s,2 < Ac, even though W2 completes skills at level c.
Figure 9 plots the probability gain A = Pmogge max P1,P3 across different values of c and X. We find that even modest errors in can sharply reduce A. For example, when = 1.14 and c = 0.2, the probability gain becomes A = -0.2, indicating that merging reduces job success. This illustrates the critical importance of accurate ability estimation, and complements the findings in Section D.2 on belief-driven merging.
24

---

Mathematical Framework for Al-Human Integration in Work
Hestmoo
C=0
-.1
A-0.
0.5
0
A-1
1-12
1-14
(a) A V.s.
(b) A v.s. A
(c) Heatmap of A
Figure 9. Plots illustrating the relationship between the probability gain A = Pmerge max P1, P2} and the W2's action-level ability parameter and trust parameter to W2's action-level ability for the Computer Programmers example with default settings = 0.45, Here, À > 1 indicates an overestimate of W2's action-level ability. while À < indicates an underestimate. We assign action-level subskills to W2 if its evaluated ability Xc dominates that Wi, i.c., - 0.78xy2 S Ac. Notably, the probability gain can be negative due the imperfect merging introduced by the trust parameter.
E. Omitted details from Section 4
We provide additional details for the examples stated Section 4.
O*NET. O*NET, developed by the U.S. Department of Labor, is a comprehensive database providing standardized descriptions of occupations, including required skills, knowledge, abilities, and work activities. It helps job seekers, employers, educators, and policymakers understand workforce needs and trends. O*NET aids career exploration, job description development, curriculum design, and labor market analysis. Employers educators align training programs with job market demands. OENET provides skill proficiency levels but lacks granularity in distinguishing decision-making from action-based abilities. GenAl excels in technical execution but struggles with strategic problem-solving. Enhancing O*NET to capture these distinctions would improve AT-human job interaction analysis and workforce planning.
it to identify workforce needs, while
F.1. Details of deriving data from O*NET
The job of "Computer Programmer" consists of n = 18 skills and m = 17 tasks, together with their descriptions (link: ht attpps: : // www. onet onlinee org/ link/ taakk/summaat commary/11--a 1251. 00). O*NET also offers the importance of cach task, which may influence the choice of job error function f. The task importance vector
e (.86, 85, 84, 77,, 76,.77,
.6,, .6,, .7,, .7,, .5,, 55,, .56, .3,,,,,,,,.... .56, .9,, .4)..
(6)
where v indicates the importance level of task i. Additionally, O"NET offers the importance and proficiency level each skill. The skill importance vector is
w (.5, ..,,,,,53.....
.53,533, .53, 5, -6, .5,,66,, .56,,,,666..... .56, .53, .63, 6...6..5.. .53, 15,,..3...00 .69, .69, .9,,,9994 .94) E 10,
(7)
where wy indicates the importance level of skill j. which may influence the choice of task error function The skill proficiency vector
= (.41, 43, ....44,4,54. 45, .5.445..555
.5. .66...44444 46, .6.446.440 .4,,448, 55. .5,,544.... .5,.5,,, .5,, .7) E (0, 10,
(8)
where sj represents the criticality of skill j for this job. We summarize how to derive this data in Figure 10. The derived data for tasks and skills are summarized in Tables and 2, respectively.
E.2. Details of deriving workers' abilities from Big-bench Lite
We show how to formulate ability profiles for human workers and GenAl tools via skill evaluations in Big-bench Lite (bench authors, 2023). BIG-bench Lite (BBL) is a curated subset of the Beyond the Imitation Game Benchmark (BIG-bench), designed to evaluate large language models efficiently. While BIG-bench contains over 200 diverse tasks, BBL selects 24 representative tasks covering domains such as code understanding, multilingual reasoning, logical deduction, and social bias
25

---

A Mathematical Framework for Al-Human Integration in Work
O-NET OnLine
Computer Programmers
dinemnts
1 11511:
Skills
Cateril
Ercqget en tye infernddes
Tasks
eocomar pracnin sprendio codery
vemirn
(a) List of Tasks
(b) List of skills
Tasks
Worber lepuirmess
ded vnter
trker
se ricing boriee ring semeenoon dee
froor ss duso
or
rossemennn
(c) Task importance un
(d) Skill importance 8j
O-NET OnLine
O-NET OnLine
Cocputiony
Browse by Basic Skills
Browse by Basic Skills
Active Usstnnng Inride
Welcc
Geing
Lente
BIG BBB
Imgrttnne
(c) Skill proficiency wy: Step 1
(f) Skill proficiency wy: Step 2
Figure 10. Deriving job data for computer programmers from O#NET, Subfigures (a) and (b) are on the "Summary" page of the job (link:
https:/ / /www. cnetenl i ine org/ iink/ susmarumr/y////// 0t//15512511 00). Subfigures (c) and (d) are on the "Details" page. Subfigures (e) and (f) show how to obtain skill proficiencies sys from O*NET.
26

---

A Mathematical Framework for Al-Human Integration in Work
Table 1. Data for tasks associated with the job of "Computer Programmers."
Task id
Task nane
Inportance (erk)
analyze, review, rowrite programs, using workflow chast diagram,
and applying knowledge of computer cupabiliies, subjectt matler, and symbolic logic
Correct errors by making appropriate changes and rechecking the program ensure that the desired results are produced direct revision, repair, expansion existing programs increase operating efficiency adapt requirements Write, update, and maintain computer programs software packages to handle specific jobs such
tracking inventory. storing or retrieving data, or controlling other equipment
Consult managerial, engineering. lechnical personnel clantyy program intent, identily problems, suggest changes Conduct trial runs of programs and software applications to be sure they will produce the desired information and that instructions are correct
Prepare detailed workflow charts diagrams that describe input, ourpot, logical operation. and convert them into a series of instructions coded in a computer language
7
Compile and wrile documentation of program development and subsequent revisions, insertiny
8
64
in the codldd instructions others understand the program
9
Consult with and assist computer operators or system analysts to deline and resolve problems in running computer programs
63
Perform systems analysis and programming tasks maintain and
10
57
control the of
systems software systems programmer Write or contribute to instructions manuals to guide end users
11
57
Investigate whether networks, workstations, the central processing unit of the system. peripheral equipment responding program's instructions
12
57
13
Assign, coordinate, and roview work and activities of programming personnel
56
suhordinates progaamiig and program coding
15
Develop Web sites
16
Train users on the use and function of computer programs
49
17
Collaborate with computer manufacturers and other users to develop new programming methods
Table 2. Data for skills associated with the job of "Computer Programmer": sorted in increasing order of proficiency.
Skill
Skill
Importance (w%)
Proficieccy (8%)
Decomposition (A)
Decision (8,1)
(8 :2)
Covedination
50
41
D
D
0.41
Social Percepiiveness
43
0.43
Matherattccs
0.45
Time Management
45
0.45
0
Monitoring
45
0.45
0
Systems Analysis
Juilgment and Decision Making
0.322
0.138
Writing
56
46
0.4
0.184
0.276
Active Learning
46
0.184
0.276
Speaking
0.48
Quality Control Analysis
63
50
0.3
0.15
0.35
Reading Comprehension
0.5
0
Systems Evaluation
Operations Analysis
54
0.6
0.324
0.216
15
Complex Problem Solving
69
55
0.7
0.385
0.165
Critical Thinking
0.35
0.23
Active Listening
57
0.57
18
Programming
94
70
0.4
0.28
0.42
27

---

A Mathematical Framework for Al-Human Integration in Work
assessment. bench authors (2023) assessed the accuracies of the best human rater, the average human rater, and the best LLM for these skills in Big-bench Lite. By Figure ((c) of (bench authors, 2023), we know that the best LLM refers PaL.M (Chowdhery et al., 2023).
Our goal is to formulate the ability profile of a human worker using the data for the average human rater, and formulate the ability profile for a GenAl tool using the data for the best LLM. Their accuracies for 24 skills are summarized Table 3. Note that the accuracy corresponds to the average ability of workers. Thus, to formulate ability profiles, we need to know the proficiencies of these skills and the variance in the workers' abilities. Below, we illustrate how to derive this data.
Deriving skill proficiencies. We use GPT-4o to derive proficiencies for the 24 skills and obtain skill proficiency vector in [0, 1)
(0, .7,887.4. .66,, 1, .3, .8, .00, .91, .7, 0, 20, 20, .7,, 25, 000,,,,,00000000 .7,, .91, .64, .00,664,,, 66,, .50).
The prompt is: "# Table 3. Given the list of 24 skills in Big-bench Lite, please construct a 24-vector where 8j represents the proficiency level of skill j with 0 for the easiest and for the hardest."
Deriving the ability profiles. Given accuracies in Table 3, we have the accuracies of the average human rater and the best LLM. Combining the accuracies and the skill proficiency vector s, we observe that linear functions can fit skill ability profiles of the average human rater and LLM; see Figure 11. We fit the ability of the average human rater by 1 0.78s, whose estimation variance is 0.013. Also, fit the ability of the LLM by 0.92s, whose estimation variance is 0.029. Additionally, by Figure App.9 of (bench authors, 2023), we observe that the noise distribution of abilities is close to truncated normal distribution. Overall, we formulate the ability profiles of human worker (Wi) and GenAl tool (W2) to be
(s) = Tr 0.78s + 0.22, 0.013; 0, and a (s) = TrunN(1 - 0.92s + 0.08, 0.029; 0, 1),
(9)
respectively, where TrunN o²; 0, 1) is a truncated normal distribution with mean μ and variance o² on interval [o, 1]. These ability profiles represent that both human workers and GenAI tools excel easier skills but struggle with more challenging ones.
Accuracy Skill Proficiency
Accuracy
Human Accuracy
Human Fit: 1-(1-a]s, a=0.22
LLM Accuracy
LLM Fit: 1:1 a)s, 000.8
Skill Proficiency
Figure 11. Accuracies of the average human rater and LLM v.s. skill proficiency for tasks in Big-bench Lite. We use linear functions fit the plots. We fix the constant parameter c = for ease of analysis such that the slope parameter can vary from O to 1. The variances for human and LLM 0.013 and 0.029, respectively.
E.3. Details for subskill division, task-skill dependency, and the choices of error functions
This section details how our framework can be adapted to the derived data from O*NET and Big-bench Lite.
Deriving decision-level degree of skills. To derive subskill numbers, we first need to know the decision-level degree of a skill. Suppose the decision-level subskill contributes \, frafticto and the action-level subskill contributes (1 - >, )ffacttoo for some > E [0, 1]. This quantization Ay depends on both the skills and the considered jobs.
To this end, we first use GPT-4o to obtain the description of the decision and action aspects of each skill. According to the descriptions, we also distinguish whether both the decision and action-level aspects can be evaluated separately. The prompt
28

---

A Mathematical Framework for Al-Human Integration in Work
Table Accuracies of average human raters and the hest I.M for 24 skills in BIG-bench Lite; information from Figure 4 of (hench
authors, 2023). "NA" represents that the accuracy is unclear from the figure. Aceruacy of average human rater
Skill
Accuracy of the best LLM
gebuugebugggng
0.15
NA
bbq_lite_json
0.73
0.73
code codelline_dee _descrippoo
0.6
0.46
conceptual.commiiation
0.83
0.48
NA
0.5
emoji_movie
0.94
0,9
formal fallacics
0.55
0.53
hindu knowlede
NA
0.75
known_unknowns
0.8
0.68
NA
0.36
mingusttics_uzzzlee
NA
NA
logie-grid_puzzie
0.4
0.36
logical_deducttoe deduction
0.4
0.36
misconceptions. russian
NA
0.68
novel_concepts
0.65
0.57
operators
0.46
0.36
parsinlu reading comprehension
NA
play Hay_ddalog_ssame saaallogssme_orrro samfeoor,different different
NA
0.63
repeat copy logic
0.39
0.12
strange stories
0.8
0.63
strategyqa
0.62
0.6
symbol interpretation
0.38
0.25
vitamin fact. verification
0.63
0.57
winowhy
NA
0.59
is "# Table 2. Given the list of skills for the job of Computer programmers from O*NET, please provide the description of the decision-level and action-level aspects for each skill. Moreover, for each skill, determine which one of its decision and action aspects is more essential and whether both decision-level and action-level aspects can be evaluated separately. Output a LaTeX table in box containing the above information." See Table 4 for summary. For instance, the decision and action aspects of "Active Listening" are "Understanding Context" and "Engagement" respectively. It is an action-type skill and the decision and action aspects are difficult to be assessed separately. For such inseparable skills, we set A3- = 0 for action ones and Xy- = 1 for decision ones, For the remaining separable skills, we use GPT-4o to derive a decision-level degree Aj. This concludes the generation of the following vector for decision-level degree:
A (0, 1, 1, 6, 4, 4,
4, .3, 1, 6, 6,0, [0. 1]
(10)
The prompt is "# Table 4. For each skill; j with "Separable = Y", please construct a decision-level degree Aÿ E [0, 1] representing the decision-level degree while 1 1- \ represents the action-level degree of skill j."
Deriving subskill numbers from skill proficiency and decision-level degree. First, we note that we can not measure the decision and action aspects of some skills separately. We take "Active Listening" as an example to illustrate how to determine subskill numbers for these skills. It is an action skill with X, = 0. Then, the difficulty of the leeision-level subskill should be the easiest one, while the action-level subskill should be equal the skill proficiency. This corresponds to (5)) s0, The case of Xy= is symmetric. Thus, provide the following assumption.
Assumption E.1 (Extreme points for subskill allocation). We assume if >3 1, (8y1, 852) = (83, 0); and if X, O. (s)1, 832) (0,
For the remaining skills j that can be well divided into decision and action aspects, we have AS E (0, 1). To determine $51 and 8,2, we first analyze the desired properties of them. We take Programming as an example.
29

---

A Mathematical Framework for Al-Human Integration in Work
Table Subskill descriptions. In the column of "Separable". "Y" represents that the decision- and action-level aspects of skills be tested separately: "N" represents that the skill can not be tested independently; and finally. "Decision" represents that skills cannot be tested separately but can be tested independently, and the main aspect is decision/action, respectively. For "N" skills, a possibility use group-based assessments, e.g., assign group project with clear dependencies among team members test Social Perceptiveness + Coordination; or simulate customer service scenario where candidates must listen to customer concerns and respond effectively test Speaking + Active Listening.
Skill id
Skill name
Decision
Action
Separable
Planning Interactions
Adjusting Actions Modiffyng behavior to align with others
Coordination
N, action
Identifying interdependencies
Recognizing Cues
Response Adjusting behavior based on social understanding
2
Social Perceptiveness
N, action
Understanding social signals
Conceptual Analysis
Calculation Executing
Mathematics
Decision
Choosing appropriate methods
mathematical computations
Prioritization Deciding
Scheduling Allocating
Time Management
N, decision
task importance
time to tasks
Identifying Key Indicators
Observation Actively
Monitoring
Decision
Determining what monitor
tracking performance
System Design Understanding
Application Using systems
Systems Analysis
Y
how changes affect outcomes
knowledge to modify systems
Weighing Options Assessing
Execution Choosing and
7
Judgment and Decision Making
Y
risks and benefits
enacting the best course
Planning Content Structuring
Execution Writing
Writing
Y
and organizing ideas
clearly and coherently
Understanding Context-
Engagement Showing
9
Active Listening
N, action
Interpreting information
attentiveness through responses
Content Sclection
Delivery - Articulating
Speaking
N, action
Deciding what to convey
information effectively
Standards Evaluation
Inspection Physically testing
11
Quality Control Analysis
Y
Deciding quality benchmarks
or inspecting outcomes
Interpretation Extracting
Application Using information
12
Reading Comprehension
Decision
key idcas from text
in practical context
Assessing Performance Setting criteria for evaluation
Monitoring Observing system function relative to criteria
13
Systems Evaluation
N. decision
Determining Requirements-
Implementation Designing
14
Operations Analysis
Y
Identifying needs
solutions based on analysis
Analyzing Options-
Implementation Applying
15
Complex Problem Solving
Y
Identifying potential solutions
solutions to problems
Evaluating Alternatives
Logical Application Applying
16
Critical Thinking
Y
Comparing pros and cons
chosen solution
Identifying Relevance-
Application Using new information to solve tasks
17
Active Learning
Y
Deciding useful information
Designing Algorithms-
Writing Code Implementing code in specific languages
18
Programming
Y
Choosing the best approach
30

---

A Mathematical Framework for Al-Human Integration in Work
Fixing Ay and increasing 8j (i.e., increasing the programming difficulty), expect that the difficulties of both decision and action aspects increase, leading to increase in 851 and sz2.
Fixing sj and increasing Ay (i.e., increasing the importance of decision-making in programming), we expect that 8y1 closer 8j- Specifically, when A 1, we have 851 83 by Assumption E.1. Moreover, we expect that 822 decreases since the requirement of action-level becomes easier. Symmetrically, as ア decreases, we expect that 5y1 decreases and 822 is closer to 8j.
These desired properties motivate the following assumption.
Assumption E.2 (Monotonicity for subskill allocation). We assume 1) 8.11 and 8,2 are monotonically increasingly as 85i and 2) When >j increases from to 8y1 is monotonically increasing from 0 to S while s,2 is monotonically decreasing from 8j to O.
Finally, note that S11 and 5,2 are derived from skill proficiency 5, and Xy only affects the allocation instead of the total skill difficulty, Thus, would like recovery of sj using şj1 and 822. Observed from Assumption E.I, we may expect that 8y1 8,2 = 8j holds. Accordingly, we have the following assumption.
Assumption E.3 (Subskill complementarity assumption). We assume that 851 + 832 8j-
Under Assumptions E.1.3, we conclude the following unified form of Sy1 and s,2:
$y1 (},)s and ,22 = (1 - ({a))ss
where (-): j0, 1] + [0, 1] is monotonically increasing function with (0) 0 and (1) = 1. The easiest way is to select
(A) = A, which results in
81 (0,0, 45, 45, 45, 4,,00,,444,45555777,22228,88,,,, .5,
,,, 0, E [0, 1]" and
(11)
82
18388, 12.,,2277777 .46,,,48,..0 .3,, 0, 0, 22.,, .2, E [0, 1]"
This choice makes 851 and 822 proportional to Ay. Other choices of vb include vb(A) = x², (AA) (1--(1-Je-- on.
A
and so
Deriving subskill ability profiles. We provide an approach to decompose skill ability profiles a to subskill ability profiles aj and a2, When ass) ~ TrunN(1 - (1 - a)s, o²; 0, 1) and the decision-level degree is A E [o, 1]. we set
(1 (sss) = TrunN(1 - -1--)((100//10) o),,,,//,/20.
This formula ensures that
o1 (2(2)) 2- TrunN(1 (1- a) -1- o),,,,,2/2200111 -runNI - (1 a)852, 02 1,
TrumN(11 - (s11 + 852), 02:0, 1) = TrunN(1 - (1 -1- a) + +s,2), : 0. = 11 a(ss))
where the last equation applies the property that 8y1 8/2 = = This ensures that the distribution of h((51, (72) is close to first draw X ~ a(x;) and then outputs 1 - X. Thus, we can (approximately) recover the skill ability profile via such subskill ability division by setting the skill success probability function h(S1, (2) G1 + C. Consequently, we have
(2) (s) TrunN(1 0.92s, 0.0145; 0, 1). TrunN(1 0.78s, 0.0065; 0, 1) and a;
(12)
In conclusion, we provide an approach to divide the data on skills into subskill numbers and ability profiles.
Details for deriving task-skill dependency. Given the descriptions of tasks and skills for the job of "Computer Pro- grammers", we use GPT-4o to generate the task-skill dependency Tis; see Figure 12. The prompt is: "# Tables I and 2. Given list of m 17 tasks with their descriptions and a list of = 18 skills with their descriptions in the job of Computer Programmers, please construct subset T; C [n]] for each task i E [m] that contains all
skills associated to task i." The resulting task-skill dependency is: Ti = [6, 8,8, 9, 16, 18). T3 = 75. 11, 16, 18]. T3 = [5. 13, 14, 16, 18), T3- = [1, 4, 13, 18), T₃ = =2,,7, 7, 10, 17), Tç = [6, 11, 16, 18, Tr = 6,8,9 18), Tg = 18, 11, 16, 18). Тg = [1, 17), Tho = 13, 14, 18), Tu = [2,8, 9, 10), 7,2 = 13, 14, 18), Tia = 11), 10, Tı = 16,
Tis 16, 18), T Tie = 17,,00, 18), and Tir [9, 17, 18).
31

---

A Mathematical Framework for Al-Human Integration in Work
Task-Skill Relationship Graph (Adjusted for Readability)
760
John Lineers
164
Qualtty Cortn/hahhe
Bitlen
Spmens Anehes
Paragimant
Parceplhannca
Figure 12. Task-skill dependency graph for the Computer Programmers example. In this graph, Ti 16,8,9.1 16,18,,8) 18). To
[5. 7, 16, 18). Ta = [5. 14, 16, 18). T4 = [1,4, 13, 18). Ts 12,7, 10, 17]. To 11, 18). 17 [6,8,9. 18]
T. [8, 16, 18). Ta 2, 177. Tio 13,13.14, 18), Tn 1,,8.9, 10), 712 = 13, 18). Tia -,1,,7, 10
Tu4 [7,8, 16, 18). Tis = 11, 18), Tie = 17, 18), and = [9,16, 18).
Choice of error functions. As discussed above, we select the skill error function h to be h((,, == Ci + C2 that takes realized subskill abilities Çi, {2 as inputs and outputs a skill completion quality. This choice of h aims recover the derived skill ability function from Big-bench Lite. Using the skill importance w. select the task error function g to be (((h))yer) Eyer, wy Ejet, wy hy that takes associated skill completion qualities of task i as inputs and outputs task completion quality. This choice of highlights the different importance of skills for the job. Finally, using the task importance v, we select the job error function f to be fgg.....,
== Σjer, e ΣiE[m) 0// 9i that takes all task completion qualities as inputs and outputs job completion quality. Combining with the task-skill dependency, we can compute the following function of job error rate composed by h, g, f: for any < E 1]an
Err(C) 0.04(S1,1 + (1,) 0.04 + (2,2) 0.03 + S4.2) 0.03($s.1 + (5,2) + 0.05($6.1 + G6,2) 0.06(S1o, + (10 + 0.05(G + S1122) +00 0.05 + (o2)
+ (7.2) + GG.) 0.06(( (82)
+ + + ( (10 + 0.06(Ci7.1 + (17.2)
0.26((18,1 (18.2)
(13)
Overall, show how to derive all the data for using our framework. We simulate that the job success probabilities of W, and W2 are Pi 0.55 and Pg 0.00, respectively. We also provide flow chart to summarize this procedure; see Figure 13. We remark that we can compute P1 and P2 even without subskill division, i.c., only using data including skill proficiencies as in Equation (8), skill ability profile as in Equation (9), and the function of job error rate as in Equation (13). For T = 0.45, we obtain that P1 = 0.84 and P3 = 0.00. The value of P1 is different but not too far from that computed using the subskill division, which convincing of the reasonability of our subskill division approaches.
32

---

A Mathematical Framework for Al-Human Integration in Work
Catermine & job name
Get information TunOOHETT
No
Hove a of tasis with descrimions?
Have a lot of skiltvott descrppenn?
No
Cet information from OHNET: pee Figuro
Ves
Yes
Exoilishie
Stone by Excel sheet foorr
mekk [ut]otseerppiiciinnpee plinncimpoorinnns [8:1]
of skilj [s] proobbec number sy [9.1]
Ceven taskesadl descnptione, nok GPT to create
stall dependencc grock??
Know how decompese de into cecisoo scrion reprete?
Gven desenptions GPT provide desompanttion
depen
Store groph collection subsets c. [n] representing assursstee skilts tasic
dacompostion [0.1)" representing decasson undiactron sell
Selectt ability profiies fox worsers
Dectsionlleee abiiity profile (00 [0.1] Art i now anllyyaoiies [0.1] +F
Sweect hyppe sperraeettt
Skll success lundtio (- 1FFFF- [0:1]
Tustss Lunstio [0,2]
Gavena, 4. compute dddessnn--oude subsii number 1- (, - 5) (9.3] and action-levs sutskll numberay (1 A2(1 a) [L1]
sucoess function (0.1 [0.1]
Treeshold [0.1]
Compunsismutsricch succese probobiity
Figure A flow chart for the Computer Programmers example that illustrates how use our framework to assess job-worker
E.4. Robustness across alternative modeling choices
Besides the use of the derived job and worker data in Section 4, we also do simulations with alternative modeling choices to validate the robustness of our findings.
Alternative error functions. We replace the job/task error aggregation functions and f with max simulate more fragile task environments; see Figures 14 and 15. The main patterns remain consistent with those for average error functions, though line-crossings disappear due to monotonicity in the max-based error aggregation.
Alternative ability distributions. We substitute truncated normals with uniform noise in ability profiles (Figures 16 and 17), verifying that our key findings hold across distributions.
Robustness to task-skill graph variations. We randomly modify 5 edges in the task-skill dependency graph (Figures 18 and 19). Despite these changes, the phase transition behavior and heatmaps remain stable.
33

---

A Mathematical Framework for Al-Human Integration in Work
Pors of Pxc. for parous
Pott of Pars p trrvetous
(eattmep of seluee acress verying aand
5200005 prebasarry
2-0
2-02
2-02
2-04
3-06
0-08
p=1
(a) P V.S. a
(b) P v.s.
(c) Heatmap of
Figure 14. Plots illustrating the relationship between the success probability P(at, 0zg, h. 9, f.r) and the ability parameter a and dependency parameter p for the Computer Programmers example with default settings of (a, r) (0.08, 0.6), replacing the error functions g. f from weighted average in Section 4 to max. Note that we increase r from 0.45 (for weighted average) to 0,6 (for max), since the resulting rate of max is higher. The job structure and worker ability profiles follow the same design Figure 3 main paper, demonstrating the robustness of our empirical results for the job error rate function JER.
(a) Heatmap of Pmeegee
(b) Hcatmap of A
Figure Heatmaps of job success probability Pmerge and the probability gain A Pmerge max (P1, P2) by merging two workers for different ranges of (a, c) for the Computer Programmers example with default settings of r = 0.6, replacing the error functions g. J from weighted average Section 4 max. Note that we increase 7 from 0.45 (for weighted average) to (for max), since the resulting error rate of max is higher. The job structure and worker ability profiles follow the same design as Figure 4 in our main paper. demonstrating robustness of our empirical results for the job error rate function JER.
Hots
farvarouss
Piotuues
antwesity
3=0
+#02
p=0
06
2-04
9-00
0.00
1=1
340
2-08
(a) P v.s..
(b) P v.S.
(c) Heatmap of P
Figure 16. Plots illustrating the relationship between the success probability P(an, 02, f.. (..9g...1 S.T) and the ability parameter and dependency parameter p for the Computer Programmers example with default settings of (a, (0.2, 0.4). replacing the truncated normal noise in Section 4 with the uniform noise. Setting 0.2 ensures that the variance of the uniform distribution matches that of the truncated normal distribution. The job structure the job error rate function JER. follow same design as Figure in our main paper, demonstrating the robustness of our empirical results for worker ability profiles.
34

---

A Mathematical Framework for Al-Human Integration in Work
(a) Heatmap of Pmerge
(b) Heatmap of A
Figure 17. Heatmaps of job success probability I mmerge and the probability gain A I'merge max (P1,12) by merging two workers for different ranges (a, e) for the Computer Programmers example with default settings of (01, 02, T) (0.2, 0.29, replacing the truncated normal noise in Section 4 with the uniform noise. The variance parameters and are chosen so that the variance of the uniform noise both workers aligns with that of the truncated normal distribution. The job structure and the job error rate function JER follow the same design as Figure 4 in our main paper, demonstrating the robustness of our empirical results for worker ability profiles.
Plrte Py..
Pic. t
Jestrrap of valuee
exciee pritesiity
2-02
2-04
2-0.
2-08
0.
p=1
(a) P v.S.
(b) P v.s. P
(c) Hcatmap of P
Figure 18. Plots illustrating the relationship between the success probability P(a1, 02 J. and the ability parameter a and dependency parameter p the Computer Programmers example with default settings of (α, = (0.08, 0.45), randomly shifting five edges in the task dependency graph. The worker ability profiles and the job error rate function JER follow the same design as Figure in our main paper, demonstrating robustness of our empirical results for job structure.
(a) Heatmap of Pmerge
(b) Heatmap of A
Figure 19. Heatmaps of joh success prohability Pmeegee and the probability gain A Pmerge max (P1, P2) by merging workers for different ranges of (a, c) for the Computer Programmers example with default settings of r = 0.45, randomly shifting five edges in the task-skill dependency graph. The worker ability profiles and the job error rate function JER follow the same design Figure in our main paper, demonstrating the robustness of our empirical results for job structure.
35


Reviews:
[{"review_id": "n70qdRie5W", "reviewer": "Reviewer_pD8B", "summary": "The authors propose a model of workforce replacement by AI and run some simulations based on it.", "questions": "", "limitations": "", "rating": 2, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for taking the time to evaluate our submission and for the opportunity to clarify key aspects of our work. Please see this PDF for new figures (link: https://acrobat.adobe.com/id/urn:aaid:sc:eu:fef7d6b2-24f6-4a59-9386-fdf09456ed99).\n\n> \"..assumptions that skills are independent.,..\"\n\nOur model does **not** assume skill independence. Each skill $s \\in [0,1]$ is assigned an ability function, and nearby skills (e.g., 0.7 and 0.8) have similar expected abilities. Thus, correlation across skills is encoded by design. For instance, coding and debugging are correlated due to their proximity in skill space.\n\nThe independence assumption applies only to **skill execution**: once ability functions are fixed, realizations of performance across tasks are treated as independent—assuming task inputs are drawn independently. This is standard in modeling both humans and LLMs. We will clarify this in the final version.\n\nAdditionally, Sec. 4 (Lines 373L–356R) already considers dependent skill executions: a parameter $p\\in[0,1]$ introduces correlation via a shared latent variable $\\beta$. We extend Theorem 3.1 to this setting (details appear in response to Reviewer aCoN).\n\n> \"deep truths about the job market... but do not make any real claims.\"\n\nWe make no such claim. Rather, we introduce a theoretical framework for analyzing human-AI task-level collaboration, supported by empirical validation. As stated in the abstract and introduction, our contributions include formalizing how job success varies with ability profiles and quantifying the benefit of merging human and AI workers—both supported mathematically and empirically.\n\nWe also added new empirical results that may be of interest to practitioners. Notably, we now show that our model captures the **productivity compression** phenomenon reported in Brynjolfsson et al. (2023). Specifically,  Theorem 3.2 implies that merging low-skilled worker $W_1$ with a GenAI tool yields a larger gain in job success probability than merging a high-skilled worker $W_2$, thus narrowing the gap $P_2-P_1$. Since the job success probability correlates with productivity, this explains the empirical narrowing in productivity observed in the cited study.  Our heatmaps (Figure 6 in the PDF) empirically reproduce this effect. To our knowledge, our model is among the first to offer a formal explanation of this compression under realistic ability assumptions (details appear in response to 1zaz.)\n\n> \"No modeling of real labour force dynamics or attempts to test the framework.\"\n\nOur model is not intended as a macroeconomic tool. It is a **task-level, microeconomic framework** that enables reasoning about **skill decomposition** and **human-AI integration**. It is grounded in real-world data: O\\*NET (task-skill-job structure), Big-Bench Lite (AI capabilities), and GPT-generated task-skill mappings. Section 4.4 discusses modeling limitations and directions for broader validation.\n\n\n> \"I did not check the theorems. I found the assumptions... make the actual model irrelevant.\"\n\nWe respect your decision. However, the theoretical results—such as phase transitions and merging theorems—are central to our work. Understanding when merging yields gains or when success changes abruptly is non-trivial and requires careful analysis. Other reviewers have highlighted the value of this contribution: Reviewer aCoN noted the rigor of the merging results, and Reviewer 1zaz emphasized the model’s potential for interpreting labor studies.\n\n> \"What am I supposed to gain from the heatmaps?\"\n\nThe heatmaps illustrate how job success probability $P$ and merging gain $\\Delta$ vary with ability profiles:\n\n- **Phase transition behavior:** The heatmaps confirm whether transitions in $P$ or $\\Delta$ occur gradually or abruptly. Distinct color boundaries validate the predicted transitions.\n  \n- **Extension beyond theory:** While Theorem 3.2 assumes identical profiles, Figure 4 shows that phase transitions still emerge when profiles differ.\n\n- **Effort required to achieve gains:**  \n The size of the bright region in Figures 2 and 4(b) indicates how easily a large $\\Delta$ can be achieved by merging. Figure 2(b) vs. Figure 4(b) shows that merging identical profiles yields more abrupt gains than merging distinct ones.\n\nThese visualizations help connect our analytical results to observable behavior in practical scenarios.\n\n> \"There is no connection to the real world.\"\n\nWhile our model is stylized to allow formal analysis, it is **empirically grounded** and addresses concrete, real-world questions about human-AI integration at the task level. It uses real datasets (O\\*NET and Big-Bench Lite) and models skill-action decompositions and merging—challenges at the heart of current research in both ML and labor economics. We hope our clarifications and new results help make this connection clearer.\n\nWe thank you again for your time, and hope our responses convey the relevance, rigor, and potential of our work.", "claims_and_evidence": "The authors claim to uncover deep truths about the job market, but they rest upon a foundation of assumptions that are not justified. They also do not make any real claims beyond stating they have a working model. This make evaluating the paper difficult as it presents theoretical arguments, but the empirical claims are vague or unusable outside their framework.", "methods_and_evaluation_criteria": "No, they only make claims based on the model. There is not modeling of real labour force dynamics or attempts to test the framework. Just an examination of how it could be applied to an existing dataset. But even when the dataset is used the framework is implicitly enforced thanks to the use of an LLM to generate the data.", "theoretical_claims": "I did not check the theorems. I found the assumptions of independence and lack of connection to the real world to make the actual model irrelevant to my analysis.", "experimental_designs_or_analyses": "See abov", "relation_to_broader_scientific_literature": "If the paper delivered what is promised this might have relevance, but as written does not. I suggest the authors consider either increasing the empirical groundedness of the work before resubmitting to CS-society venue, or do more work to build on the theory and submit to an econ theory venue.", "essential_references_not_discussed": "None come to mind", "other_strengths_and_weaknesses": "I am particularly concerned with the assumptions that skills are independent in LLMs and that they will generalize reliably. Both of these are still open questions in the literature, so some discussion of these limitations should at least be included.", "other_comments_or_suggestions": "No", "questions_for_authors": "What am I supposed to gain from the heatmaps? They seem very sensitive to initial conditions/parameter selection. Showing that there are transition points in an econ model is not novel.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "9P7dDB2YEV", "reviewer": "Reviewer_aCoN", "summary": "This paper models human-AI collaboration in jobs. In particular, it models jobs as being composed of multiple different subtasks, each of which involve different skills. The ability of different agents is noisy and ordered (e.g. the same agent can’t perform worse on easier subtasks on average than they do on easier subtasks). This paper studies multiple effects within this model, such as how the probability of task success varies with average ability and the effect of “merging” multiple workers (e.g. combining their relative strengths).", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your thoughtful and constructive feedback. We especially appreciate your recognition of our modeling approach and the suggestion regarding imperfect merging, which we have now incorporated. Please see this PDF for new figures (https://acrobat.adobe.com/id/urn:aaid:sc:eu:fef7d6b2-24f6-4a59-9386-fdf09456ed99).\n\n> \"...when workers are imperfectly 'merged'.\"\n\nThank you for this suggestion and the reference. We extend our merging analysis by introducing a trust parameter $\\lambda$ to the merging experiment in Section 4 (Line 358 right column - 439 left column), which models imperfect merging by letting the estimated ability $\\hat{c}=\\lambda c$ deviate from the true action-level ability $c$ of worker $W_2$. We then assign action-level subskills to $W_2$ when its scaled ability, $\\lambda c$, exceeds $W_1$'s ability (i.e., $1 - 0.78s_{j2}\\le \\lambda c$, extending the setting in Lines 414–415, left column), even though $W_2$ completes skills at level $c$. We analyze the probability gain $\\Delta=P_{\\text{merge}}-\\max\\\\{P_1,P_2\\\\}$ across different values of $c$ and $\\lambda$ (see Figure 5 in the PDF linked above).\n\nWe find that even modest errors in $\\lambda$ can sharply reduce $\\Delta$. For example, when $\\lambda=1.14$ and $c=0.2$, the probability gain becomes $\\Delta=-0.2$, indicating that merging reduces job success. This illustrates the critical importance of accurate ability estimation, and complements the findings in Section E.2 on belief-driven merging.\n\nWe will create a separate subsection in Appendix E to present these findings clearly.\n\n> \"...wonder whether another (or a simpler model) would have sufficed.\"\n\nOur model combines a task-skill graph, subskill-level abilities, and error rate functions to estimate job success (Section 2). We clarify three key points:\n\n- **Generality:**  \n  The model allows flexible task-skill dependencies, ability profiles, and error functions. Our framework includes two variants—noise in abilities and subskill division—to capture variation in worker performance. It can be adapted to simpler models while retaining its predictive power.\n\n- **Simplicity vs. expressiveness:**  \n  A special case with a single task, noise-free abilities, and no subskill split yields a binary job success probability (0 or 1) and misses correlations across skills. Without subskill division, it is difficult to analyze how merging a human and GenAI tool—each excelling in different subskills—affects performance.\n\n- **Non-triviality of results:**  \n  While some conclusions may seem intuitive, formally establishing phase transitions in job success requires careful analysis, as illustrated by the following examples:\n  \n  - **Dependent skills:** By introducing a dependency parameter $p \\in [0,1]$, subskill errors are drawn from a shared latent status with probability $p$ and independently otherwise. The phase transition window becomes $\\gamma_1=\\frac{L\\sqrt{\\mathrm{sg}(A_1)\\cdot\\ln\\frac{1}{\\theta}}}{\\mathrm{Infg}(A_1)\\cdot\\sqrt{1-p}}$, which vanishes as $p \\to 1$, showing that stronger dependencies smooth out abrupt transitions.  \n  - **Max aggregation:** \n    We show that Theorem 3.1 may not yield a phase transition when using the max operator for error rate functions $g,f$. Consider a job where one critical task is very difficult (e.g., skill difficulty 0.1) while all other tasks are relatively easy (e.g., skill difficulties 0.8). Since the job error rate is defined as $\\max_{j\\in[n]}h_j$, the overall error is dominated by the hard task. In this case, the job success probability is essentially $P=\\Pr[h_1\\leq\\tau]$, making it a smooth function of the ability on that single, critical task—thus, no sharp phase transition occurs. Notably, the Lipschitz constant here is significantly higher (e.g., $L=1$) compared to the average-case model (e.g., $L=1/(2n)$ in the linear case), which prevents an abrupt transition.\n\nThese findings show that richer modeling is necessary to rigorously capture how noise, dependencies, and aggregation affect outcomes like merging. We will include these as theorems in the appendix.\n\n> \"Modeling jobs as multiple subtasks has been studied...\"\n\nThank you for this pointer. We will cite related works (e.g., Autor and Thompson) and clarify that our contribution lies in offering a formal, ML-grounded framework for modeling skill decomposition and human-AI complementarity. While inspired by similar questions, our analysis tackles fine-grained questions around merging, noise, and error functions not typically addressed in existing models.\n\n> \"Making the main body.. more high level...\"\n\nThank you. We will revise the presentation to focus on high-level takeaways in the main paper and move derivations to the appendix. We anticipate using the ICML final version page allowance to improve clarity without altering content.\n\nOnce again, we thank you for your detailed feedback and for helping us improve the quality and impact of our work.", "claims_and_evidence": "I appreciated that the paper took an especially nuanced view on how humans may integrate algorithmic tools into their workflow.The paper includes a rigorous theoretical analysis of the problem, as well as an empirical analysis.", "methods_and_evaluation_criteria": "This paper has a fairly involved theoretical model of tasks - my suspicion is that most of the core results would generalize to other models, but given that the results are relatively intuitive, it makes me wonder whether another (or a simpler model) would have sufficed.\n\nOne core concept that this paper explores is that of “merging workers”, which in this context likely means the benefits from partial automation of jobs. Given the model where humans and AI tools may have complementary skills sets (e.g. in decision-level or action-level skills), it seems natural that there would be benefits in merging workers. However, empirical work has shown that humans are sometimes unable to know when algorithmic tools are more or less accurate (e.g. https://arxiv.org/abs/2406.01382). It would have been useful to see more discussion of how these results change when workers are imperfectly “merged”.", "theoretical_claims": "N/A", "experimental_designs_or_analyses": "N/A", "relation_to_broader_scientific_literature": "See below", "essential_references_not_discussed": "The idea of modeling jobs as given by multiple subtasks has been studied extensively in prior work, especially since some core parts of this paper could also model human-human integration (e.g. merging multiple jobs into one).  For example, this lecture https://economics.mit.edu/sites/default/files/2024-09/Autor-Schumpeter-Expertise-20240829-handout.pdf, and related papers https://www.nber.org/system/files/working_papers/w32140/w32140.pdf studies similar issues.", "other_strengths_and_weaknesses": "This is more a stylistic point, but the paper is very densely written, where the main body of the paper often includes more technical details than is necessary, which may make it difficult for the reader to follow the high-level story. Making the main body be written at a more high level (removing more technical discussion to the appendix) would be helpful.", "other_comments_or_suggestions": "N/A", "questions_for_authors": "N/A", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "8n7C8k555i", "reviewer": "Reviewer_ChBu", "summary": "This paper presents a mathematical framework for modeling jobs, workers, and worker-job fit, focusing on subskill decomposition into decision-level and action-level tasks to highlight the distinct strengths of humans and AI. The study examines how variations in subskill abilities affect job success and identifies conditions under which collaborative skill division leads to superior performance compared to relying on a single worker. The framework's effectiveness is validated using O*NET and Big-Bench Lite datasets, demonstrating its real-world applicability. The results emphasize that Generative AI (GenAI) is best suited to complement human workers' skills rather than replacing them.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank you for your detailed and encouraging review. We are especially grateful for your recognition of our framework’s real-world applicability and for your thoughtful suggestions regarding dataset limitations and empirical grounding, which have directly shaped our additional experiments. Please see this PDF for new figures (https://acrobat.adobe.com/id/urn:aaid:sc:eu:fef7d6b2-24f6-4a59-9386-fdf09456ed99).\n\n> *\"A potential limitation is whether the chosen datasets fully capture the complexity of skill attribution in dynamic work environments. Additional experiments with task-specific benchmarks or real-world AI-assisted work scenarios could further validate the robustness of the proposed approach.\"*\n\nWe agree that additional empirical results could further ground our mathematical model and its implications. As one of the first formal frameworks for understanding task-level human-AI collaboration, our empirical results—based on O\\*NET and Big-Bench Lite—serve primarily as calibration tools to illustrate how the model can be instantiated and to demonstrate that key theoretical insights remain valid in practice. While we recognize the limitations of these datasets—O\\*NET reflects static, survey-based data, and LLM-based estimates may introduce bias—we have conducted several new empirical analyses (see accompanying PDF) to test the robustness of our findings across alternative modeling choices. We will describe these limitations clearly and include the new results in the final version. We also agree that task-specific benchmarks (e.g., HumanEval for programming, customer support transcripts) could further strengthen empirical grounding and plan to explore such extensions in future work.\n\n- **Alternative Error Functions:**  \n  We replace the job/task error aggregation functions $g$ and $f$ with $\\max$ (suggested by Reviewer 1zaz), to simulate more fragile task environments. The main patterns remain consistent (Figures 1 and 2 in the PDF linked above), though line-crossings disappear due to monotonicity in the max-based error aggregation.\n\n- **Alternative Ability Distributions:**  \n  We substitute truncated normals with uniform noise in ability profiles (Figures 3 and 4 in the PDF linked above), verifying that our key findings hold across distributions.\n\n- **Robustness to Task-Skill Graph Variations:**  \n  We randomly modify 5 edges in the task-skill dependency graph (Figures 7 and 8 in the PDF linked above). Despite these changes, the phase transition behavior and heatmaps remain stable.\n\n\n> *\"It would be useful to discuss potential biases or limitations in these datasets... are there task distributions favoring humans or AI?\"*\n\nWe will explicitly discuss biases in O\\*NET and Big-Bench Lite datasets regarding task distributions potentially favoring humans or AI. We summarize key observations below:\n\n- **Tasks Favoring Humans:**  \n  Tasks that are context-rich and require nuanced judgment, creativity, or interpersonal skills (e.g., strategic planning, ethical decisions) are better captured by human-centered data like O\\*NET.\n\n- **Tasks Favoring AI:**  \n  Structured, repetitive tasks (e.g., basic arithmetic, data classification) are common in benchmarks like Big-Bench Lite, which may overrepresent tasks where AI excels.\n\nThese differences suggest that while O\\*NET may underrepresent emerging digital tasks, Big-Bench Lite might favor tasks with clear, rule-based responses.\n\n> *\"Beyond theoretical modeling, empirical user studies involving real-world AI-human collaboration could strengthen the paper’s conclusions.\"*\n\nWe agree this would be valuable, though it is outside the scope of this paper. Our primary goal is to provide a rigorous theoretical framework with proof-of-concept empirical validation. We believe our framework offers a foundation for future empirical and behavioral studies on AI-assisted work. As noted, several of our new experiments aim to move further in that direction.\n\nWe thank you again for your thoughtful comments, which have significantly improved our work's clarity and empirical reach. We hope our additions meaningfully address the concerns raised.", "claims_and_evidence": "Yes, the claims made in the submission are well-supported. For example, the authors argue that conflating reasoning skills with action skills can lead to misattributions of success or failure, resulting in biased or incomplete assessments. To address this, they provide a detailed framework for decomposing skills into two distinct subskill types: decision-level skills (problem-solving) and action-level skills (solution execution), with skill difficulty quantified on a 0-1 scale.", "methods_and_evaluation_criteria": "Yes, the proposed methods and evaluation criteria are well-aligned with the problem being addressed. The paper introduces a mathematical framework that systematically decomposes skills into decision-level and action-level subskills, which is a logical and structured approach to analyzing worker-job fit in the context of human-AI collaboration.\n\nFor evaluation, the use of O*NET and Big-Bench Lite datasets provides a real-world grounding for their framework, ensuring that the findings are not purely theoretical. These datasets contain job-related skill data and AI task benchmarks, making them appropriate for assessing the practicality of skill decomposition.", "theoretical_claims": "I am not very familiar with the topic of assessing job accuracy of AI and humans, so I cannot fully judge whether all the theoretical components are meaningful. However, I have reviewed the mathematical symbol definitions, and they are clearly defined and well-presented.", "experimental_designs_or_analyses": "The approach appears comprehensive and sound. The authors define a job-success probability metric that integrates error rates across skills and tasks to assess overall performance. This metric provides a structured and quantitative evaluation of human-AI collaboration effectiveness, ensuring a more holistic assessment of task execution.", "relation_to_broader_scientific_literature": "By refining how AI-human performance is measured, this work contributes to future AI deployment strategies, helping design collaborative work environments where AI complements human abilities rather than replacing them. The proposed methodologies provide insights into the effective allocation of human and AI resources, ultimately improving job success probability and fostering productive human-AI collaboration.", "essential_references_not_discussed": "I am not familiar with workforce optimization, so I cannot fully assess the essence of the related work.", "other_strengths_and_weaknesses": "A potential limitation is whether the chosen datasets fully capture the complexity of skill attribution in dynamic work environments. Additional experiments with task-specific benchmarks or real-world AI-assisted work scenarios could further validate the robustness of the proposed approach.", "other_comments_or_suggestions": "The study uses O*NET and Big-Bench Lite, but it would be useful to discuss potential biases or limitations in these datasets.\nAre there task distributions that may favor either AI or human workers?\n\nBeyond theoretical modeling, empirical user studies involving real-world AI-human collaboration could strengthen the paper’s conclusions.", "questions_for_authors": "Please refer to Section of Other Comments Or Suggestions.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "y1OGFb9XYP", "reviewer": "Reviewer_1zaz", "summary": "This paper develops a model of job success probability by viewing jobs as a composition of tasks that need to be accomplished, and workers supply skills that affect the probability tasks as successfully completed. The authors then calibrate the model using the O*NET database's skill descriptions associated with computer programming. The authors calibrate the skill of workers and the skill of GenAI tools in their model to examine whether merging the human + the GenAI tool can lead to higher job success probabilities than either working in isolation. I provide a more detailed description of the theoretical model and experimental results before presenting my comments below.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank you for your thoughtful, detailed, and insightful feedback. In response, we added new theoretical and empirical analyses that sharpen our results, test their robustness across modeling choices, and highlight connections to real-world phenomena such as productivity compression.\n\nPlease see this PDF for new figures: https://acrobat.adobe.com/id/urn:aaid:sc:eu:fef7d6b2-24f6-4a59-9386-fdf09456ed99.\n\n> *\"Pick a particular empirical result (e.g., Brynjolfsson et al.)... and discuss whether the model explains it.\"*\n\nThank you for the suggestion. We show that our model formally supports the **productivity compression** effect observed in Brynjolfsson et al. (2023). Consider two workers from the same ability family (e.g., constant, linear, polynomial), with equal decision-level ability and noise: a low-skilled worker $W_1$ with action-level ability parameter $a_1$ and a high-skilled worker $W_2$ with action-level ability parameter $a_2>a_1$. Let $P_1,P_2$ be their success probabilities before merging with GenAI\n(whose decision-level ability is always weaker than $W_\\ell$ and action-level ability is in the same family with $W_1$ and $W_2$), and $P'_1, P'_2$ after merging. We define productivity compression as: $\\mathrm{PC}=(P_2-P_1)-(P'_2-P'_1)$.\n\n**Theoretical insight:** A corollary of Theorem 3.2 shows that if $a_{AI}>a_1$, then $W_1$’s merged gain $\\Delta_1=P'_1-P_1$ can be large-up to $1-2\\theta$ for some small $\\theta$—while $W_2$'s gain $\\Delta_2\\approx 0$. Thus, $\\mathrm{PC} \\approx 1-2\\theta$.\n\n**Empirical results:** Our experiments (Figure 6 in the PDF) show that even when the GenAI ability profile differs from that of the worker, compression increases with the skill gap. For $a_1=0.1$, $a_2=0.8$, compression reaches 0.8, closely matching empirical findings from Brynjolfsson et al.\n\nTo our knowledge, this is one of the first formal explanations of the compression effect under realistic assumptions.\n\n> *\"..uniform alternative?\"*\n\nWe chose truncated normal distributions to approximate ability variability observed in empirical benchmarks (e.g., Figure App.9 in Bench authors, 2023). That said, we now include experiments with uniform noise distributions (Figures 3 and 4 in the PDF) and find qualitatively similar results.\n\n> *\"What if you instead took the max error...\"*\n\nThis excellent suggestion led us to study max-based aggregation, where job success probability is given by $P = \\prod_{j} \\Pr[h_j \\leq \\tau]$. We conducted both theoretical and empirical analyses:\n\n- **Sharper Phase Transition:** With identical skills, the transition window scales as $\\frac{1}{n}$, sharper than the average-based case $\\frac{1}{\\sqrt{n}}$. \n\n- **Negative Example:** We identify a natural setting where no phase transition occurs. With one hard skill (e.g., 0.1) and others easy (e.g., 0.8), success is dominated by the hard one: $P \\approx \\Pr[h_1 \\leq \\tau]$, yielding a smooth, non-abrupt transition.\n\n- **Empirical Results:** Experiments (Figures 1 and 2 in the PDF) confirm these effects. Notably, line-crossing in average-based settings disappears under max aggregation due to its monotonicity.\n\nWe will include these results in the final version.\n\n> *\"struggled to map the model into any concrete job...\"*\n\nOur model is intended to be flexible. For project-based roles (e.g., programming), a job may represent a single project composed of subtasks. For ongoing roles (e.g., teaching or customer service), a job could aggregate performance over a time period. The randomness in subskill outcomes captures intra-personal variability (e.g., fatigue) and task-level heterogeneity. While stylized, we believe this abstraction enables us to reason about human-AI collaboration in both discrete and continuous work environments.\n\n> *\"Prompting GPT-4o plays a key role...\"*\n\nWe will clarify the role of GPT-4o in the main text and include API-based prompting code in the final version to ensure consistency and reproducibility.\n\n> *“Results are relatively weak and unsurprising.”*\n\nWhile some results may seem intuitive, our theorems precisely characterize when phase transitions in job success occur and quantify non-trivial merging gains—analytically nontrivial and not evident without formal modeling. Our extensions further show when phase transitions disappear (e.g., max-based errors, strong skill dependencies), underscoring the value of our framework.\n\n> *“Rely heavily on stylized versions of the model (e.g., linear ability functions)...”*\n\nWe use linear ability functions as an interpretable baseline aligned with prior empirical work (e.g., Brynjolfsson et al., 2023). This choice illustrates key effects clearly and enables tractable analysis. In Appendix C.3, we extend our results to polynomial ability functions, and our experiments confirm that the core insights persist under more complex profiles.\n\nWe thank you again for your thoughtful engagement, which has helped us clarify, extend, and strengthen the contributions of our work.", "claims_and_evidence": "Please see my discussion of theoretical claims and experimental designs.", "methods_and_evaluation_criteria": "Please see my discussion of theoretical claims and experimental designs.", "theoretical_claims": "The authors theoretical model has the following key components:\n* A job consists of $m$ tasks and each task $T_i$ depends on $n$ skills. Each skill $s$ has two components: decision-level and action-level sub-skills.\n* A worker in the model is associated with two ability profiles $\\alpha_1(s), \\alpha_2(s)$ that summarizes their ability across the two subskills for each skill. The ability profiles are governed by a probability distribution that is summarized by two parameters: (i) the average ability for the skill, and (ii) the noise for the skill. \n* For each skill $j \\in [n]$, the worker's skill error function is given by the random variable $h(\\zeta_{j1}, \\zeta_{j2})$ where $z_{j,l} = 1 - X_{jl}$ for $X_{jl}$ sampled according to the skill profile $\\alpha_{l}(s_{jl})$. The task error then aggregates the skill errors with $g(h(\\zeta_{11}, \\zeta_{12}), ..., h(\\zeta_{n1}, \\zeta_{n2}))$. Finally, the job error aggregates the task errors $f \\colon [0,1]^m \\rightarrow 1$. Consequently, task success and ultimate job success are random variables due to the randomness in the worker's skill errors. \n* The authors' main object of interest is the job success probability which is the probability the job error is less than some threshold $\\tau$. (Equation 1)\n* The authors then provide two results: (1) the authors show that the job success probability is increasing in the average skill of a worker, (2)  there can be gains in the job success probability by merging two workers together.\n\nOne of the strengths of the paper is the model's generality -- in particular, the setup can accommodate a wide variety of choices of skill errors, task errors, and job errors. Akin to the authors example of the max, you could carry this forward and imagine an \"O-ring\" (Kremer, 1993) style model which would involve: skill errors as being either zero or one, tasks $g() = 1$ only be completed if all skills are successfully completed  $h() = 1$ and jobs $f() = 1$ only being completed if all tasks are completed. \n\nOne of the weaknesses is that while the model is general, the results are relatively weak and unsurprising. Consequently, it is not clear why this model helps me reason through the suite of empirical studies that analyze the productivity effects of generative AI. The authors rely heavily on returning to stylized versions of the model to build intuition (e.g., linear ability functions) but I did not find that enlightening. I would have found it immensely valuable for the authors to either (i) pick a particular result (e.g., Brynjolfsson et al.), or (ii) recurring finding across empirical analyses (e.g., the finding that the introduction of GenAI tool leads to a compression in the productivity distribution across workers) and discuss whether the model has anything to say about those results. \n\nAnother weakness of the paper is that I struggled to map the model into any concrete job. Take the example of computer programmers that the authors study in the experiment. If I think about the model as describing one worker, it describes a worker completing many jobs and describes the fraction of jobs that the worker successfully completes. So, for a computer programmer, a job actually corresponds to a specific programming project and we would be describing the success rate of the computer programmer across many such jobs? In this view, the randomness in the skill successes arises from randomness in the worker's skill across such tasks (maybe some days I am tired and other days I have a lot of coffee?). I can see how this works for a job like programmer where there is a somewhat discrete output being produced, but for other jobs it is not obvious this applies.", "experimental_designs_or_analyses": "To illustrate the model, the authors derives data on tasks and skills for computer programmers using O*NET. O*NET describes a computer programmer as consisting of 18 skills and 17 tasks with associated with proficiency levels for each skill. The authors prompt GPT-4o to provide the relationship between skills and tasks, the division of skill proficiencies into action and decision skills, and prompt GPT-4o using big-bench lite to construct skill profiles for an average person and an LLM. \n\n(1) Prompting GPT-4o plays a key role in the empirical results. It would have been useful for the authors to describe this more explicitly in the main text -- in particular, the GPT-4o outputs are used to build the task-skill dependency network; GPT-4o outputs are used to divide proficiencies into decision and action skills; and GPT-4o outputs are used to construct the proficiencies of LLMs and human rates based on big-bench lite.  I am not sure how seriously to take this except as a way to construct some numbers used for the calibration of the model. \n\n(2) The authors pick particular parametrizations of the skill profiles based on truncated normals -- how were these chosen? Why this specific choice and not the uniform alternative? It would be valuable to see the sensitivity of the calibration results to that specific modelling choice here.\n\n(3) Related to my earlier comment about the generality of the choice of skill, task and job functions, it would be interesting to see alternative variations on the choice of the JER function -- the authors choose a rather simple function that is the weighted average of the skills. What if you instead took the max error associated with each skill in a task and then max error across tasks? How would the comparisons change?", "relation_to_broader_scientific_literature": "As discussed by the authors in their introduction, a large and highly active literature studies the productivity effects of generative AI across a wide variety of settings. But much of this work lacks a clear theoretical framework for understanding when/how generative AI tools affect productivity. This paper aims to provide such a framework by viewing jobs as a collection of tasks that need to be completed and workers as having skills that affect their probabilities of successfully completing tasks. I find this to be a valuable exercise and a potentially important contribution to this active empirical literature. At the same time, as I discussed above, the model is simultaneously specific yet opaque. Moreover, there results provided are obvious -- Theorem 3.2 has a lot of leg work involved to show that increasing the average skill of a worker leads to an increase in the job success probability; and Theorem 3.3 has a lot of leg work involved to show that combining two workers that are more skilled along different dimensions leads to an increase in the job success probability. While the authors motivated the framework using these empirical findings, I struggled to link this model back to those findings.", "essential_references_not_discussed": "The paper provides an thorough review of empirical research studying the impacts of generative AI on worker productivity. There are no glaring omissions to me from this literature. In particular, I am not aware of existing paper that attempts to write a stylized model to understand of (i) jobs as a composition of tasks to be completed, and (ii) skills leading to imperfect completion rates of underlying tasks.", "other_strengths_and_weaknesses": "Please see my earlier comments.", "other_comments_or_suggestions": "As a small comment, between Section 2 and Section 3, the authors appear to switch notation for the average skill/ability. In Section 2, it is introduced as $E(s)$ and in Section 3 it is denoted by $\\mu$. The paper would be easier to digest if the authors used consistent notation throughout.", "questions_for_authors": "Please see my previous comments on the theoretical claims and the experiment.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: This paper presents a mathematical framework for modeling human-AI collaboration in the workplace by decomposing jobs into decision-level and action-level subskills. The authors analyze conditions under which combining agents with complementary strengths improves job success, introduce formal results characterizing phase transitions in job success probability, and demonstrate how the model can explain empirical observations such as productivity compression. The framework is instantiated using the O*NET and Big-Bench Lite datasets, with GPT-generated mappings to calibrate AI and human capabilities.

Overall, reviewers appreciate the proposal of a formal model addressing a timely topic within the growing literature on GenAI in the workforce. The reviewers' primary reservations focus on (1) the gap between the theoretical model and practical applications and (2) whether the paper might be more suitable for economics venues. For the first concern, following reviewer 1zaz’s suggestion, the authors demonstrated in their rebuttal that their model explains the "compression effect" of GenAI tools documented in existing empirical studies. This alleviates some concerns about practical relevance, and I strongly encourage the authors to expand this line of discussion further. Regarding venue fit, while the work has an economics-oriented flavor, I believe the topic is relevant and would attract interest from the ML community. However, I also want to highlight reviewer aCoN's suggestion to clearly convey high-level ML-relevant takeaways in the main text to increase accessibility and appeal to the ML community.
Decision: Accept (poster)