Path Dependence under Adaptive AI Delegation

Lingxiao Huang; Nisheeth K. Vishnoi

cs.CYMar 3, 2026

Path Dependence under Adaptive AI Delegation

Authors: Lingxiao Huang, Nisheeth K. Vishnoi

Organizations: Nanjing University · Yale University

Abstract

Repeated AI assistance can improve immediate task performance while reducing the skill available for future independent work. We develop a mathematical framework for this long-run tradeoff. The model tracks two state variables: a latent human skill level governing expected independent performance, and a delegation level representing the learner's evolving tendency to rely on AI. Skill changes through error-driven learning under practice and decay under delegation; delegation responds to observed performance, increasing when AI-assisted work appears to outperform independent work. We analyze the resulting dynamics and contrast them with fixed delegation. With fixed delegation, skill follows a one-dimensional learning-decay process with a single stable equilibrium. With adaptive delegation, the coupled system has two attracting equilibria separated by the stable manifold of an interior saddle. The existence and geometry of this separatrix require a global phase-plane analysis of the coupled dynamics. The system is path-dependent: small differences in initial skill or reliance can lead to different long-run outcomes. We use this characterization to show that AI assistance can improve short-run performance while producing worse long-run performance than a no-AI baseline. Increasing AI capability can enlarge the basin of attraction of the low-skill equilibrium, making delegation appear beneficial for longer while increasing the risk of eventual skill loss. The qualitative picture is observed to persist across alternative specifications. Together, these results show that the risk is not AI assistance itself, but the coupling between performance-driven reliance and use-dependent skill change.

Figures & tables

Figure 1: Overview of the repeated-task adaptation model. In each short time window $t$ , the learner faces many comparable tasks and delegates a fraction $p(t)$ of them. Evaluator feedback is aggregated across tasks and enters the skill-update and delegation-update mechanisms. The former combines error-driven learning under practice with non-use decay, while the latter updates reliance through relative performance feedback.

Panel of Figure 2 — Figure 2: Phase portraits for the adaptive-delegation dynamics. (a) Fixed delegation has a single attracting skill level, while adaptive delegation creates competing attracting regimes. (b) The stable manifold of the interior saddle separates the high-skill and low-skill basins. (c) Increasing AI skill shifts the basin boundary downward, expanding the low-skill basin. Default parameters are $(\theta_{a},\kappa,\Delta)=(0.5,3,2)$ unless varied.

Panel of Figure 3 — Figure 3: Plots illustrating the instantaneous performance loss over time with and without AI, and how the crossing point varies with respect to AI skill $\theta_{a}$ , with default settings $(\theta_{a},\kappa,\Delta,\theta_{0},p_{0})=(0.5,3,2,0.4,0.3)$ .

Symbol	Domain	Description
$\theta_{d}$	$[0,1]$	Default skill
$\theta_{a}$	$[0,1]$	AI skill
$\theta_{0}$	$[0,1]$	Initial skill of learner
$\theta(t)$	$[0,1]$	Learner skill at time $t$
$\theta(t;\theta_{0},p_{0})$	$[0,1]$	Learner skill at time $t$ with initial state $(\theta_{0},p_{0})$
$p_{0}$	$[0,1]$	Initial delegation level of learner

Table 1: Used notations

Panel of Figure 4 — Figure 4: Heatmap of the probability of converging to the high-skill equilibrium $(1,0)$ for SDEs ( 13 ) and ( 14 ) as a function of the initial state $(\theta_{0},p_{0})$ , with default settings of $(\theta_{a},\kappa,\Delta,\sigma)=(0.5,3,2,0.1)$ . Green indicates probability $0$ , and red indicates probability $1$ .

Panel of Figure 5 — Figure 5: Plots illustrating the closeness between the stable manifold $\psi(\cdot)$ and its approximation $\widetilde{\psi}$ , with default parameter settings $(\theta_{a},\kappa,\Delta)=(0.5,3,2)$ .

Panel of Figure 6 — Figure 6: Plots illustrating the relationship between the basin boundary $\psi(\cdot)$ and model parameters $\theta_{a},\kappa,\Delta$ , with default setting $(\theta_{a},\kappa,\Delta)=(0.5,3,2)$ .

Panel of Figure 7 — Figure 7: Plots illustrating how the basins of attraction change under learner- and teacher-side interventions, with default parameter settings $(\theta_{a},\kappa,\Delta)=(0.5,3,2)$ .

Appendix figures & tables4 assets

Supplementary material from the paper’s appendix.

Appendix

Variant	Status
Jagged AI performance	Exact reduction via effective loss
Biased perceived AI skill	Analytically characterized
Power- $z$ loss	Analytic nullcline deformation and simulations
Stochastic delegation updates	Simulation-supported
Multiple skills	Simulation-supported
Learning from AI during delegation	Analytic equilibrium shift and simulations

Appendix

Table 2: Status of robustness variants.

Panel of Figure 8 — Figure 8: Plots illustrating $z$ -th power loss with default setings $(\theta_{a},\kappa,\Delta)=(0.5,3,2)$ ; and the two-skill extension ( 26 ), with default settings $(\theta_{a},\kappa,\Delta_{1},\Delta_{2},w_{1},w_{2})=(0.5,3,1,3,0.5,0.5)$ . Skill trajectories can still diverge from nearby initial conditions, while $\theta_{1}(t)>\theta_{2}(t)$ holds for all $t\geq 0$ because the decay rate satisfies $\Delta_{1}<\Delta_{2}$ .

Panel of Figure 9 — Figure 9: Plots illustrating how the basins of attraction vary under learner- and AI-side extensions, with default settings $(\theta_{a},\kappa,\Delta)=(0.5,3,2)$ .

Window $t$	Delegation fraction $\widehat{p}_{t}$	Avg. independent loss $\widehat{L}^{H}_{t}$	Skill estimate $\widehat{\theta}_{t}$
1	0.20	0.09	0.70
2	0.24	0.1024	0.68
3	0.30	0.1296	0.64
4	0.38	0.16	0.60

Appendix

Table 3: Synthetic repeated-task data used only to illustrate the calibration procedure.

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Abstract

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