# Idempotent Equilibrium Analysis of Hybrid Workflow Allocation: A Mathematical Schema for Future Work

arXiv: https://arxiv.org/abs/2508.01323 · HTML: https://mlsanity.com/paper/2508.01323/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 40 references that are in the map.
- First posted on arXiv 2 Aug 2025.

## Builds on (most cited first, 8 of 8)

- [GPTs are GPTs: An Early Look at the Labor Market Impact Potential of Large Language Models](https://mlsanity.com/paper/2303.10130/) (2023)
- [Which Economic Tasks are Performed with AI? Evidence from Millions of Claude Conversations](https://mlsanity.com/paper/2503.04761/) (2025)
- [The Turing Trap: The Promise & Peril of Human-Like Artificial Intelligence](https://mlsanity.com/paper/2201.04200/) (2022)
- [Future of Work with AI Agents: Auditing Automation and Augmentation Potential across the U.S. Workforce](https://mlsanity.com/paper/2506.06576/) (2025)
- [AI Safety Should Prioritize the Future of Work](https://mlsanity.com/paper/2504.13959/) (2025)
- [The centaur programmer -- How Kasparov's Advanced Chess spans over to the software development of the future](https://mlsanity.com/paper/2304.11172/) (2023)
- [Modeling the Centaur: Human-Machine Synergy in Sequential Decision Making](https://mlsanity.com/paper/2412.18593/) (2024)
- [Computational Sociology of Humans and Machines; Conflict and Collaboration](https://mlsanity.com/paper/2412.14606/) (2024)

## Abstract (arXiv)

The rapid advance of large-scale AI systems is reshaping how work is divided between people and machines. We formalise this reallocation as an iterated task-delegation map and show that--under broad, empirically grounded assumptions--the process converges to a stable idempotent equilibrium in which every task is performed by the agent (human or machine) with enduring comparative advantage. Leveraging lattice-theoretic fixed-point tools (Tarski and Banach), we (i) prove existence of at least one such equilibrium and (ii) derive mild monotonicity conditions that guarantee uniqueness. In a stylised continuous model the long-run automated share takes the closed form x^* = α / (α + β), where α captures the pace of automation and β the rate at which new, human-centric tasks appear; hence full automation is precluded whenever β > 0. We embed this analytic result in three complementary dynamical benchmarks--a discrete linear update, an evolutionary replicator dynamic, and a continuous Beta-distributed task spectrum--each of which converges to the same mixed equilibrium and is reproducible from the provided code-free formulas. A 2025-to-2045 simulation calibrated to current adoption rates projects automation rising from approximately 10% of work to approximately 65%, leaving a persistent one-third of tasks to humans. We interpret that residual as a new profession of workflow conductor: humans specialise in assigning, supervising and integrating AI modules rather than competing with them. Finally, we discuss implications for skill development, benchmark design and AI governance, arguing that policies which promote "centaur" human-AI teaming can steer the economy toward the welfare-maximising fixed point.