Reward model overoptimization in RLHF is usually described as reward hacking. We recast it as rent-seeking: the policy, a firm, earns reward from a regulator, the proxy reward model, that true quality does not back. The lens turns overoptimization into measurable quantities, namely rent, a capture index and deadweight loss, and into testable predictions from the economics of regulation. In a fixed, version-controlled testbed with three model families, accuracy-matched proxy reward models and two gold reward models, RL turns 41% to 66% of its proxy gain into rent while gold reward plateaus. Rent follows each regulator's prices. Neither proxy verifies facts, but one pays for confident specifics and the other for displays of diligence, so RL policies learn to fabricate details about nonexistent entities (97% to 100% of probes) or to deflect factual questions (67% to 98%). Kinship between policy and regulator adds a fixed premium of about 0.1 standard deviations but gives the policy no extra leverage under optimization. At a fixed expected fine, frequent mild audits beat rare severe ones, and random audits are as efficient per audit as targeted ones. Bringing the bunching estimator of tax economics to RL training, we also find a first sign that policies learn to stay just below a targeted audit's threshold.
Figures & tables
Figure 1: Overoptimization as rent-seeking. The policy earns proxy reward that gold reward does not back (rent). Rent follows the prices that each proxy RM sets.
Figure 2: Rent shares of rule-detected strategies. Left: RL cells over training. Right: BoN cells. “All” applies every rule in sequence. Shares of single rules overlap and do not sum.
Model
Correct
No ans.
Follow
Fabricate
SFT Qwen
0.41
0.10
0.57
0.83
SFT Llama
0.35
0.21
0.64
0.95
Qwen × Qwen
0.35
0.10
0.49
0.97
Llama × Qwen
0.30
0.10
0.67
1.00
Qwen × Llama
0.21
0.67
0.52
0.17
Llama × Llama
0.02
0.98
0.60 †
0.30
Table 1: Probe results, means of three seeds for RL cells. Correct and No answer refer to neutral questions. Follow is the rise in the share of committed answers that adopt a suggested wrong answer. † Few committed answers, confidence interval wider than 0.5.
Figure 3: Revolving-door effect Δ . Left: Δ against KL for RL and BoN. Right: estimates at 4 nats with 95% intervals.
Condition
p
F
Seeds
No audit
0
n/a
3
Equal-exp. penalty
0.02 / 0.05 / 0.2
50 / 20 / 5
3 each
Back tax only
0.05
0
3
Full gold
1
0
2
Targeted audit (T2)
0.05 ∗
20
3
Length penalty
λ=1.3
2
Table 2: Audit conditions. The expected fine is c=p⋅F=1 . ∗ Nominal budget, set on SFT outputs.
Figure 4: Equal-expected-penalty line ( p⋅F=1 ). Left: judge win rate over SFT. Middle and right: gold reward and capture index at step 600, which audits partly optimize. Grey lines are seeds.
Figure 5: Disagreement scores near the audit threshold after step 100. Orange bins form the window below the threshold, and the curve is the counterfactual fit outside it. Left: targeted audits. Right: random audits as a placebo.
Appendix figures & tables14 assets
Supplementary material from the paper’s appendix.
Appendix
Decision
Data seen at the time
Match proxy accuracy by family means within one point of the Llama and OLMo mean; Qwen matched tier of 3,000 pairs; RL uses seed 3
Proxy RM accuracies only
BoN shows no rise and fall of gold reward up to n=256 ; no extra samples, the rise and fall moves to RL
BoN curves
New judge prompt and answer parsing; all earlier judge outputs discarded
Raw judge outputs
RQ2b interpolation and bootstrap; RQ1 rent decomposition and length control
Synthetic data only
No clear overoptimization in the RL pilot yet; 600 RL steps
Pilot validation curves
Validate RL every 10 steps and rerun the pilot, because KL passed 4 nats before the first validation point
KL growth only
Appendix
Table 3: Changes made after the plan freeze.
Economics
RLHF
Measure
Social welfare
True preference
Gold reward zg
Regulator
Proxy RM
Proxy reward zp
Firm
Policy
πθ
Rent
Unbacked reward
ρ=zp−zg
Rent seeking
Optimization
KL from SFT
Capture
Unbacked share
Capture index
Appendix
Table 4: Economic concepts and their RLHF counterparts.
Setting
Value
Algorithm
RLOO, no std-dev normalization
KL
Reward penalty, β=0.005
Batch
64 prompts × 8 samples
Learning rate
2×10−6 , constant
Sampling
Temperature 1.0, at most 1,024 tokens
Steps
600
Appendix
Table 5: RL settings.
Jobs
GPU hours
Audit conditions
510
RL 2×2 grid
362
Audit calibration pilots
130
BoN scoring
67
RL pilots
58
KL sweep and exploratory RL runs
34
Appendix
Table 6: A100 GPU hours by job type, from the cluster accounting of all project jobs.
Figure 6: RL in the 2×2 grid. Top: proxy and gold reward against KL , mean of three seeds. Bottom: prevalence of rule-detected behaviors over training.
Candidate category
Coverage
Added
Structured formatting
36%
80%
Bulleted key points
36%
71%
Concluding summary
29%
91%
External resources
25%
76%
Disclaimers and caveats
27%
60%
Knowledge-cutoff notes
11%
61%
Appendix
Table 7: Directional candidate categories. Coverage is the share of pairs with at least one description in the category; Added is the share of its descriptions that the high-rent response adds.
Rule
Changed
Validation
Formatting
66.6%
No content removed
Concluding summary
23.5%
0 of 10,907 removed
Repetition
7.4%
No false positives
Resource lists
5.8%
No false positives
Disclaimers
4.7%
87% sentence precision
Template phrases
2.8%
0 of 10 false
Appendix
Table 8: Rewrite rules, the share of the 46,500 RL and BoN outputs that each rule changes, and validation results.
Cell
Tokens
Proxy
Gold
Share
Qwen × Qwen
224 – 420
0.17
0.06
0.22 [0.15, 0.29]
Qwen × Llama
218 – 847
0.15
0.05
0.30 [0.22, 0.39]
Llama × Qwen
224 – 308
0.11
−0.05
0.14 [0.11, 0.17]
Llama × Llama
218 – 553
0.14
−0.07
0.22 [0.18, 0.26]
Appendix
Table 9: Length channel. Tokens: mean length at step 0 and step 600. Proxy and Gold: within-prompt slopes in log length.
Model
Rent
Proxy
Gold
Sycophancy: following a suggested wrong answer
SFT, both proxies
0.97–1.13
−0.01 –0.18
−1.13 – −0.84
RL, all cells
0.37–1.37
−0.21 – −0.02
−1.43 – −0.40
Fabrication against honest uncertainty
SFT, Qwen proxy
1.54–1.58
0.46–0.73
−1.12 – −0.80
SFT, Llama proxy
1.12–1.36
0.00–0.56
−1.12 – −0.80
Appendix
Table 10: Rent premiums in standard deviations, ranges over models and seeds. Under the Qwen proxy almost every RL response fabricates, so the fabrication premium is undefined.
Figure 7: Public checkpoints. Top: prevalence of rule-detected behaviors at the SFT, DPO and final stages. Bottom: gold reward.
Tier
Favoritism
Δ (capture index)
5,000 pairs
0.13 [0.11, 0.15]
−0.002 [ −0.010 , 0.005]
20,000 pairs
0.17 [0.15, 0.19]
0.027 [0.021, 0.033]
80,000 pairs
0.12 [0.11, 0.14]
0.007 [0.001, 0.012]
Matched
0.11 [0.09, 0.14]
0.006 [0.000, 0.012]
Appendix
Table 11: Static revolving door in BoN at n=64 , averaged gold reward.
Figure 8: BoN in the 3×3 grid, matched tier. Proxy and gold reward against KL for same-family and cross-family cells.
Condition
Audit rate
Judge win
Gold (z)
Capture idx.
DWL
Length
No audit (3)
0.000 ± 0.000
0.666 ± 0.008
+1.67 ± 0.02
0.41 ± 0.01
0.03 ± 0.01
424 ± 15
Back-tax only (3)
0.050 ± 0.001
0.680 ± 0.006
+1.73 ± 0.02
0.37 ± 0.01
0.02 ± 0.01
423 ± 9
Audit p=0.02 (3)
0.020 ± 0.000
0.672 ± 0.004
+1.72 ± 0.03
0.20 ± 0.03
0.00 ± 0.00
479 ± 12
Audit p=0.05 (3)
0.050 ± 0.001
0.709 ± 0.019
+1.71 ± 0.05
0.21 ± 0.02
0.01 ± 0.01
416 ± 26
Audit p=0.2 (3)
0.200 ± 0.001
0.714 ± 0.004
+1.90 ± 0.03
0.11 ± 0.04
0.00 ± 0.00
504 ± 19
Targeted (T2) (3)
0.082 ± 0.002
0.731 ± 0.005
+1.67 ± 0.04
0.05 ± 0.03
0.00 ± 0.00
470 ± 38
Appendix
Table 12: Audit conditions and baselines at step 600, mean ± standard deviation over seeds (number of seeds in parentheses). Judge win is the share of prompts on which the averaged judges prefer the final checkpoint over the SFT response. Gold reward and the capture index are partly optimized under audits.
Figure 9: Judge win rate against the number of gold-reward calls during training. The dashed line is the cost frontier.
Training language models via reinforcement learning often relies on imperfect proxy rewards, since ground truth rewards that precisely define the intended behavior are rarely available. Standard metrics for assessing the quality of proxy rewards, such as ranking accuracy, treat incorrect rewards as strictly harmful. In this work, however, we highlight that not all deviations from the ground truth are equal. By theoretically analyzing which outputs attract probability during policy gradient optimization, we categorize reward errors according to their effect on the increase in ground truth reward. The analysis establishes that reward errors, though conventionally viewed as harmful, can also be benign or even beneficial by preventing the policy from stalling around outputs with mediocre ground truth reward. We then present two practical implications of our theory. First, for reinforcement learning from human feedback (RLHF), we develop reward model evaluation metrics that account for the harmfulness of reward errors. Compared to standard ranking accuracy, these metrics typically correlate better with the performance of a language model after RLHF, yet gaps remain in robustly evaluating reward models. Second, we provide insights for reward design in settings with verifiable rewards. A key theme underlying our results is that the effectiveness of a proxy reward function depends heavily on its interaction with the initial policy and learning algorithm.
Shuning Shang, Hubert Strauss, Stanley Wei +2
Princeton Language and Intelligence, Princeton University
Despite their widespread use, the role of reward models in shaping reinforcement learning is poorly understood. Reward models offer a tempting promise: they automatically estimate response quality in the absence of verifiers or human judges. Unlike "verifiable rewards" which typically produce binary scores, reward models typically produce continuous scores, allowing them to be sensitive to fine-grained differences in responses. However, we show this apparent strength is a serious weakness: many popular reward models are oversensitive, assigning different scores to equally good responses. Theoretically, we show that seemingly perfect reward models can be highly oversensitive; empirically, this oversensitivity can lead to bad policies. In place of existing notions of "reward model accuracy," we propose evaluating reward models using distinct measures of "discriminative ability" and "specificity" (the complement of oversensitivity). As a solution, we describe a training-free algorithm that uses Monte Carlo dropout on any neural reward model to produce discrete reward clusters. Theoretically, we prove there exist discretizations that reduce oversensitivity at minimal expense of discriminative ability; empirically we show, in both controlled and natural RL settings, that discretizing rewards leads to less reward hacking and better policies than training on the original rewards.
Single-axis mitigations of reward-model biases (e.g., reducing proxy reliance on length, sycophancy, or style) can rotate optimization pressure onto correlated proxies rather than eliminate it, a failure mode we call reward bias substitution. The failure is enabled by a measurement-versus-optimization gap between audit and policy-induced distributions during mitigation evaluation and policy training. We formalize mitigation outcomes into a regime taxonomy and prove that successful mitigation, bias substitution, and overcorrection produce identical observables under any audit-distribution scoring, including ranking accuracy and win-rate, even when granted oracle access to the true reward. Across published preference-learning mitigation work, no method we survey reports the evidence needed to certify successful mitigation. Augmenting evaluation with policy-induced distributions while tracking multiple biases provably closes the gap, and we translate this into actionable prescriptions for mitigation methods and benchmarks. We demonstrate bias substitution in language model RLHF, where a length penalty during GRPO training compresses responses as intended yet redirects optimization pressure onto confidence calibration, driving the policy into overconfidence while factual free-form accuracy falls. We also show a published length-debiasing operator that zeroes reward-length correlation on the audit distribution but reintroduces bias under best-of-N selection on three of four SOTA reward models, and a length-sycophancy coupling whose direction reverses under human-LLM judge disagreement.