# Reward Hacking as Equilibrium under Finite Evaluation

arXiv: https://arxiv.org/abs/2603.28063 · HTML: https://mlsanity.com/paper/2603.28063/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Placed in Reward hacking and misspecification, one of 10 directions in LLM alignment / RLHF / preference optimisation.
- Builds on 5 of its 13 references that are in the map.
- First posted on arXiv 30 Mar 2026.

## Placed in

- LLM alignment / RLHF / preference optimisation / [Reward hacking and misspecification](https://mlsanity.com/alignment/reward-hacking-and-misspecification/)

## Builds on (most cited first, 5 of 5)

- [Training language models to follow instructions with human feedback](https://mlsanity.com/paper/2203.02155/) (2022)
- [Direct Preference Optimization: Your Language Model is Secretly a Reward Model](https://mlsanity.com/paper/2305.18290/) (2023)
- [Constitutional AI: Harmlessness from AI Feedback](https://mlsanity.com/paper/2212.08073/) (2022)
- [Discovering Language Model Behaviors with Model-Written Evaluations](https://mlsanity.com/paper/2212.09251/) (2022)
- [The Effects of Reward Misspecification: Mapping and Mitigating Misaligned Models](https://mlsanity.com/paper/2201.03544/) (2022)

## Built on by (most cited first, 5 of 5)

- [Kernel Contracts: A Specification Language for ML Kernel Correctness Across Heterogeneous Silicon](https://mlsanity.com/paper/2604.22032/) (2026)
- [Multimodal Reward Hacking in Reinforcement Learning](https://mlsanity.com/paper/2607.09492/) (2026)
- [EvalSafetyGap: A Hybrid Survey and Conceptual Framework for LLM Evaluation-Safety Failures](https://mlsanity.com/paper/2606.30219/) (2026)
- [A Vocabulary for Multi-Agent Automated Research Systems](https://mlsanity.com/paper/2607.22682/) (2026)
- [Rent-Seeking Language Models: Reward Models Set the Prices, Frequent Audits Curb the Rent](https://mlsanity.com/paper/2610.19832/) (2026)

## Abstract (arXiv)

We prove that under five minimal axioms -- multi-dimensional quality, finite evaluation, effective optimization, resource finiteness, and combinatorial interaction -- any optimized AI agent will systematically under-invest effort in quality dimensions not covered by its evaluation system. This result establishes reward hacking as a structural equilibrium, not a correctable bug, and holds regardless of the specific alignment method (RLHF, DPO, Constitutional AI, or others) or evaluation architecture employed. Our framework instantiates the multi-task principal-agent model of Holmstrom and Milgrom (1991) in the AI alignment setting, but exploits a structural feature unique to AI systems -- the known, differentiable architecture of reward models -- to derive a computable distortion index that predicts both the direction and severity of hacking on each quality dimension prior to deployment. We further prove that the transition from closed reasoning to agentic systems causes evaluation coverage to decline toward zero as tool count grows -- because quality dimensions expand combinatorially while evaluation costs grow at most linearly per tool -- so that hacking severity increases structurally and without bound. Our results unify the explanation of sycophancy, length gaming, and specification gaming under a single theoretical structure and yield an actionable vulnerability assessment procedure. We further conjecture -- with partial formal analysis -- the existence of a capability threshold beyond which agents transition from gaming within the evaluation system (Goodhart regime) to actively degrading the evaluation system itself (Campbell regime), providing the first economic formalization of Bostrom's (2014) "treacherous turn."