The Strategic Rate-Distortion-Perception Tradeoff: Bridging Blau-Michaeli Compression Theory and Crawford-Sobel Strategic Communication

Executive Summary

Consider an agent that observes a state and must produce a summary for a decision-maker under three simultaneous constraints: the summary compresses, the agent's incentives diverge from the decision-maker's, and the summary must conform to an institutional, social, or algorithmic norm. An executive summary must look like an executive summary; a language model's response must look like what the reward model scores highly. Shannon, Crawford and Sobel, and Blau and Michaeli each formalized one of these constraints. This paper studies them together.

It defines the strategic rate-distortion-perception function R(D, P, Q): the minimum mutual information required to achieve distortion D when encoder and decoder have misaligned objectives and the decoder's output must lie within distance P of a target distribution Q. Numerical computation of the Gaussian equilibrium shows that conformity pressure accounts for 22–72% of total receiver distortion across representative parameter regimes, with a sharp phase transition at the Crawford–Sobel babbling boundary.

The paper maps reinforcement learning from human feedback onto the framework: the language model is the sender, the human evaluator the receiver, the reward model's preferences the target distribution Q, the KL penalty the perception constraint P, and sycophancy the generative residual. The model predicts that sycophancy rises monotonically with training intensity, consistent with published findings that more RLHF training increases sycophantic behavior.

Key Contributions and Methodology

Theorem A generalizes the Blau–Michaeli rate-distortion-perception tradeoff to arbitrary target distributions Q, proving monotonicity, convexity, and rate elevation, with a feasibility boundary characterized via optimal transport.

Theorem B derives the closed-form Gaussian strategic RDP equilibrium and defines the generative residual Δ_gen = D_R(P) − D_R(∞): the excess distortion attributable to distributional conformity. Three worked examples instantiate it. In the first (σ_θ² = 1, b = 0.1), Δ_gen = 0.274, a conformity share of 22.2% of total distortion; with a far-off target and a tight constraint, conformity dominates.

Theorem C gives four sufficient conditions under which any organizational communication channel satisfies the strategic RDP tradeoff: positive rate, preference divergence, an enforceable distribution of acceptable outputs, and non-degeneracy.

Key Findings

• Generalized tradeoff: R(D, P, Q) is monotone and convex, and conformity to a target distribution raises the rate required for a given distortion
• Cost of conformity: 22–72% of total receiver distortion across representative Gaussian regimes
• Babbling boundary: a sharp phase transition in the generative residual at the Crawford–Sobel boundary
• Hierarchies: generative residual accumulates across layers, linearly when layers are independent and superlinearly when variance cascades
• RLHF mapping: sycophancy as the generative residual induced by the reward model acting as a perception constraint
• Illustrative fit: across five training regimes, predicted residuals track illustrative sycophancy rates from the literature (r = 0.993 in the paper's comparison); the paper presents this as illustration, not a controlled test

Key References