🧬 biology

Coordination without communication: beyond optimisation and geometric Brownian motion

This paper introduces a physically grounded framework where population coordination emerges from information-constrained feedback in a partially observed stochastic system without direct communication or optimization, demonstrating that geometric Brownian motion and broader stochastic growth regimes arise as limiting cases of these conditional dynamics.

Original authors: G J Milburn, A K Ringsmuth

Published 2026-06-01

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: G J Milburn, A K Ringsmuth

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ⚕️ This is an AI-generated explanation of a preprint that has not been peer-reviewed. It is not medical advice. Do not make health decisions based on this content. Read full disclaimer

Imagine a crowded room full of people who cannot talk to each other. They can't send text messages, shout instructions, or hold a meeting. In fact, they don't even know exactly how many people are in the room.

Yet, somehow, the crowd starts to move in perfect unison. They grow together, shrink together, or jump in size all at once. How is this possible?

This paper proposes a new way to understand that kind of "silent coordination." It suggests that groups don't need to talk or plan to coordinate; they just need to react to a shared, slightly fuzzy signal.

Here is the breakdown of the paper's ideas using simple analogies:

1. The "Fuzzy Radio" Signal

Imagine every person in the crowd is wearing a radio. They can't hear the voices of the people next to them. Instead, they all tune into the same radio station.

This station broadcasts a signal that is related to the size of the crowd, but the signal is noisy. It's like trying to listen to a weather report while standing next to a loud construction site. Sometimes the signal says, "The crowd is huge!" and sometimes it says, "The crowd is tiny!" even if the crowd hasn't actually changed much.

The paper calls this the measurement signal. It's a shared piece of information that everyone gets, but it's imperfect.

2. The "Guessing Game" (Inference)

Since the people can't see the crowd, they have to guess the crowd's size based on that noisy radio signal.

If the signal gets louder, they guess, "Oh, more people must have arrived!"
If the signal gets quieter, they guess, "People must be leaving."

They don't know the true number of people. They only know their best guess based on the history of the radio signal. This is what the paper calls "inference."

3. The Feedback Loop

Here is the magic trick: The people use their guess to decide what to do next.

If they guess the crowd is growing, they might decide to invite more friends (or in the paper's biological example, reproduce faster).
If they guess the crowd is shrinking, they might leave or stop reproducing.

Because everyone is listening to the same radio and making guesses based on the same signal, they all react at the same time. This creates coordination without communication. They aren't following a leader; they are all following the same noisy weather report.

4. The Two Ways the Crowd Moves

The paper shows that depending on how "clear" the radio signal is, the crowd behaves in two very different ways:

The "Drifting" Crowd (Weak Signal): If the radio is very noisy (like static), the crowd's size changes slowly and randomly, like a leaf drifting in a gentle breeze. This is called diffusive behavior.
The "Jumping" Crowd (Strong Signal): If the radio signal is very clear, the crowd reacts instantly. One moment they are small, and the next, they suddenly "jump" to a huge size. The paper compares this to quantum jumps in physics, where a system suddenly snaps from one state to another.

5. The "Stock Market" Connection

The authors show that when the signal is strong and the feedback is right, the crowd's growth looks exactly like Geometric Brownian Motion (GBM).

You might know GBM from the stock market. It's the math used to predict how stock prices move. Usually, people think stock prices move that way because of complex human trading strategies.

The Paper's Twist: This paper says you don't need complex human strategies or "smart" agents. You just need a group of simple agents reacting to a shared, noisy signal. The "stock market" behavior emerges naturally from the physics of the signal and the feedback, not from people trying to optimize their profits.

6. The "Sacrifice" Example

The paper also looks at a scenario with two types of people: "Cooperators" and "Defectors."

Cooperators are like volunteers who do something risky or costly (like dying to release nutrients) that helps the whole group.
Defectors are the free-riders who just take the benefits without doing the work.

The paper shows that even without a moral code or a plan to "be nice," the group can naturally evolve a mix of these two types. If the "radio signal" (the feedback) is tuned correctly, the cooperators can thrive because their risky behavior actually helps the group grow faster in the long run, even if it hurts them individually in the short term.

The Big Takeaway

The main point of the paper is that coordination is a physical phenomenon, not just a social one.

You don't need to assume that people are "smart," "selfish," or "altruistic." You don't need them to have a plan. If you have a group of agents reacting to a shared, imperfect signal about the state of the world, complex patterns like cooperation, coordination, and even "market crashes" can emerge automatically.

It's like a flock of birds. They don't need a leader to tell them where to fly; they just need to react to the movement of the bird next to them. In this paper, the "bird next to them" is replaced by a shared, noisy signal that everyone is listening to.

Technical Summary: Coordination without Communication

Problem Statement
The paper addresses a fundamental gap in modeling coordination and cooperation within complex systems. Traditional approaches—ranging from game theory to evolutionary biology—typically rely on agents possessing explicit objectives, internalizing payoff functions, or having direct access to global system variables (e.g., prices, population density). These models often assume that agents optimize strategies or communicate directly to achieve collective outcomes. However, many biological and social systems operate under partial observability, where macroscopic states regulating behavior are not directly accessible to individuals. Instead, agents must infer global conditions from noisy, indirect signals. The authors ask whether coordination can emerge without explicit optimization, communication, or agency, solely through physical principles grounded in nonequilibrium thermodynamics and information-constrained feedback.

Methodology
The authors propose a physically grounded framework based on partially observed continuous-time Markov processes. The core components of the model are:

Population Dynamics: The population size ( $n$ ) evolves as a birth-death Markov process. The transition rates ( $w^+_\alpha, w^-_\alpha$ ) are not fixed but are modulated by a shared, stochastic measurement signal.
Partial Observability: The true population state $n(t)$ $n (t)$ is "hidden" from individual agents. Instead, agents observe a stochastic signal $y(t)$ $y (t)$ (or measurement current $I(t)$ $I (t)$ ) which is correlated with the population state but subject to diffusive noise.
- The signal is modeled via an Itô stochastic differential equation: $dy(t) = \chi \bar{n}_c(t)dt + \sqrt{\kappa}dW(t)$ , where $\bar{n}_c(t)$ is the conditional mean population estimated by agents based on the observation history, $\chi$ is a rate constant, and $\kappa$ represents noise intensity.
Inference and Feedback: Agents use the observation history to update their estimate of the population state (conditional probability distribution $p_{n,c}(t)$ $p_{n, c} (t)$ ) via a Kushner-Stratonovich-type filtering equation.
- The transition rates are instantaneously adjusted based on the measurement current $I(t)$ , creating a macro-to-micro feedback loop.
- The joining/defection rate is defined as $\gamma(t) = \gamma(1 + \lambda I(t))$ , where $\lambda$ determines the feedback strength (positive for joining, negative for defecting).
Signal-to-Noise Ratio (SNR): The framework introduces the SNR as a critical parameter determining the quality of information available to agents. The measurement strength is defined as $\Gamma = \chi^2/\kappa$ .

Key Contributions and Results

Emergence of Geometric Brownian Motion (GBM): The authors demonstrate that GBM arises as a limiting case of the conditional dynamics. When the measurement strength is high and the conditional population statistics approximate a Poisson distribution (where variance equals the mean), the stochastic differential equation for the conditional mean population $\bar{n}_c(t)$ converges to the GBM form:
$d\bar{n}_c(t) = r\bar{n}_c(t)dt + \sqrt{\Gamma}\bar{n}_c(t)dW$
This provides a physical derivation of GBM from information-constrained feedback rather than assuming it as a primitive.
Regime Dependence on SNR: Varying the signal-to-noise properties produces a spectrum of stochastic growth behaviors:
- Weak Measurement ( $\Gamma$ small): The population dynamics are diffusive, and the conditional mean grows slowly.
- Strong Measurement ( $\Gamma$ large): The dynamics exhibit "jump-like" regimes where the conditional statistics deviate rapidly from Poissonian behavior, resembling quantum jumps.
- Despite these trajectory-level differences, the ensemble-average growth remains exponential in all regimes.
Coordination as Inference: The paper redefines coordination not as strategic alignment or optimization, but as the result of inference and feedback under partial observability. Agents align their behaviors simply by responding to a shared, noisy estimate of a global variable.
Extension to Multiple Phenotypes: The framework is extended to a two-phenotype model (cooperators and defectors). Cooperators self-destruct to produce a public good, while defectors survive. By introducing separate measurement currents for total population and cooperator mortality, the model shows how "sacrificial" behavior can be favored when rapid early growth confers an advantage, driven purely by the feedback dynamics of the measurement signals.
Connection to Peters and Adamou: The framework recovers the stochastic multiplicative growth model of Peters and Adamou (PA) in the appropriate limit. However, it offers a distinct physical interpretation: the difference between time-average and ensemble-average growth rates is a property of conditional dynamics under measurement, rather than a result of strategic pooling or sharing.

Significance and Claims
The paper claims to provide a physically grounded bridge between low-level dynamical systems and high-level descriptions of cooperation. Its primary significance lies in:

Removing Agency Assumptions: It demonstrates that collective order and "cooperation-like" outcomes can emerge without assuming agents have internal states, utility functions, or the ability to communicate. The behavior is driven entirely by the physics of measurement, inference, and feedback.
Reinterpreting Coordination Failures: The framework suggests that large-scale collective failures (e.g., in climate change or economic crises) may not stem from individual defection or antagonism, but from the limitations of the information channel (low SNR) linking micro-actions to macro-states.
Physical Basis for Preferences: It proposes that what is traditionally modeled as "preferences" or "payoff functions" can be understood as feedback laws modulating transition rates based on imperfect information.
Generalization of GBM: By showing GBM as a limiting case of a broader class of stochastic processes driven by measurement noise, the paper offers a more fundamental physical interpretation of multiplicative growth processes found in finance and biology.

The authors maintain a modest stance, noting that while the framework recovers known models (like PA's), it does so by deriving them from first principles of stochastic dynamics and information theory, thereby offering a new perspective on the mechanisms of coordination.