Coordination Requires Simplification: Thermodynamic Bounds on Multi-Objective Compromise in Natural and Artificial Intelligence

TL;DR

The paper addresses how information-processing systems coordinating multiple agents and objectives face intrinsic thermodynamic constraints that force information loss and focal-point simplification. It develops Thermodynamic Coordination Theory (TCT) by deriving lower bounds on the coordination protocol length $L(P)$ that scale as $L(P) \ge N \\bar{K} \\log \\bar{K} h(\\rho) + \\binom{N}{2} \\frac{d(d+3)}{2} \\log(1/\\varepsilon)$, leading to an $O(N^2 d^2)$ growth, and demonstrating that findability pressure dominates accuracy via $U(s) = \\Omega[A(s)] \\cdot P(\\text{coordinate on } s)$ with a divergent ratio $\\frac{\\partial U/\\partial F_i}{\\partial U/\\partial A}$. It further introduces the coordination temperature $T_{co}$, a renormalization-group picture that drives model complexity toward a focal point $K_0$, and estimates energy costs for metastable coordination. The framework suggests that, across AI, organizations, and biological and social systems, coordination pushes toward robust, findable solutions at the expense of perfect accuracy, with broad implications for AI alignment, system design, and understanding of cultural and economic dynamics.

Abstract

Information-processing systems that coordinate multiple agents and objectives face fundamental thermodynamic constraints. We show that solutions with maximum utility to act as coordination focal points have a much higher selection pressure for being findable across agents rather than accuracy. We derive that the information-theoretic minimum description length of coordination protocols to precision $\varepsilon$ scales as $L(P)\geq NK\log_2 K+N^2d^2\log (1/\varepsilon)$ for $N$ agents with $d$ potentially conflicting objectives and internal model complexity $K$. This scaling forces progressive simplification, with coordination dynamics changing the environment itself and shifting optimization across hierarchical levels. Moving from established focal points requires re-coordination, creating persistent metastable states and hysteresis until significant environmental shifts trigger phase transitions through spontaneous symmetry breaking. We operationally define coordination temperature to predict critical phenomena and estimate coordination work costs, identifying measurable signatures across systems from neural networks to restaurant bills to bureaucracies. Extending the topological version of Arrow's theorem on the impossibility of consistent preference aggregation, we find it recursively binds whenever preferences are combined. This potentially explains the indefinite cycling in multi-objective gradient descent and alignment faking in Large Language Models trained with reinforcement learning with human feedback. We term this framework Thermodynamic Coordination Theory (TCT), which demonstrates that coordination requires radical information loss.

Figures (1)

• Figure 1: Thermodynamic Coordination Theory: Key Relationships. (a) Information scaling showing $\log_{10}$ of coordination protocol length $L(P)$ as a function of number of agents $N$ and objectives $d$. Cyan dashed line marks human working memory limit (100 bits), demonstrating how even modest multi-agent coordination exceeds human capacity. Contours show information growth in orders of magnitude. Parameters: mean agent model complexity $\bar{K}=10$ bits, conflict resolution precision $\varepsilon=0.1$. (b) Distribution of solutions across agent space. Each cell represents one agent and the color represents which solution(s) they can find and agree to implement. Findable solutions (green+yellow) form extended connected regions reaching into corners; accurate but "invisible" solutions (red+yellow) exist as isolated islands. Yellow regions indicate where both solution types are available. The green+yellow network forms a spanning cluster enabling system-wide coordination, while red+yellow regions remain disconnected. This collective marginal utility increase drives selection for findable solutions over more accurate one, connecting to topological constraints from Arrow's impossibility theorem (see Supplementary Section 1). (c) Coordination temperature states visualized through variance in agent models. Each circle represents one agent's internal model state. Circular clusters from left to right show: ordered state with tightly aligned models (low $T_{\mathrm{co}}$), critical coordination state with moderate variance, and disordered state with maximal model divergence (high $T_{\mathrm{co}}$). (d) Minimum thermodynamic work required to cool system from initial temperature $T_1$ to final temperature $T_2 (<T_1)$. Bottom axis shows temperature ratio $T_2/T_1$; top axis shows absolute final temperature $T_2$ assuming $T_1=1$. Curve height represents work in bits. Critical temperature $T_{c,\mathrm{co}} \approx 0.128$ determines stability: systems with $T_2 < T_{c,\mathrm{co}}$ (green region) can maintain coordination without additional energy input; systems with $T_2 > T_{c,\mathrm{co}}$ (red region) require continuous resource expenditure for metastable coordination. Parameters: $N=50$, $\bar{K}=20$, $K_0=10$ bits.