The entropic basis of collective behaviour

In this paper, we identify a radically new viewpoint on the collective behaviour of groups of intelligent agents. We first develop a highly general abstract model for the possible future lives that these agents may encounter as a result of their decisions. In the context of these possible futures, we show that the causal entropic principle, whereby agents follow behavioural rules that maximise their entropy over all paths through the future, predicts many of the observed features of social interactions between individuals in both human and animal groups. Our results indicate that agents are often able to maximise their future path entropy by remaining cohesive as a group, and that this cohesion leads to collectively intelligent outcomes that depend strongly on the distribution of the number of future paths that are possible. We derive social interaction rules that are consistent with maximum-entropy group behaviour for both discrete and continuous decision spaces. Our analysis further predicts that social interactions are likely to be fundamentally based on Weber’s law of response to proportional stimuli, supporting many studies that find a neurological basis for this stimulus-response mechanism, and providing a novel basis for the common assumption of linearly additive ‘social forces’ in simulation studies of collective behaviour.

💡 Research Summary

The paper introduces a novel perspective on collective behaviour by applying the causal entropic principle (CEP) to groups of intelligent agents. The authors construct an abstract model of all possible future states that agents may encounter, representing these futures as a branching tree. Each branch generates a random number of new options according to a fixed probability distribution, modeled as a Galton–Watson (GW) or Yule process. The key parameter is the extinction probability α, which captures the chance that a branch terminates without further options.

Using this framework, the authors derive the distribution of the ratio R = n_A/(n_A+n_B) of future paths behind two competing choices A and B. When the numbers of future paths are independent exponential variables, R is uniformly distributed on (0,1); the possibility of extinction adds Dirac masses at 0 and 1 weighted by α. The probability that a group of N agents selects A is then a weighted sum of binomial distributions integrated over P(R). This yields a markedly higher tendency toward consensus than would be expected if agents chose independently at random.

The analysis extends to K > 2 options. The proportions of i.i.d. exponential variables follow a beta distribution β(1, K‑1), leading to a general formula for the consensus bias that grows with both group size N and the number of alternatives K, while still decreasing as K becomes very large.

A particularly striking result is the emergence of a Weber‑law–like interaction rule. Conditioning on A agents already committed to option A and B agents to option B, the probability that a new agent will choose A is

 P(A | A,B) = (A + 1)/(A + B + 2).

This matches Weber’s law with a single pseudo‑observation, providing a mechanistic link between CEP and the proportional response observed in many animal and human groups. Consequently, the familiar “social forces” used in crowd‑simulation models can be derived from an entropy‑maximisation principle rather than being an ad‑hoc construct.

The authors discuss how the model predicts strong cohesion and consensus formation, especially when α is non‑zero, and they illustrate these predictions with toy examples and analytical calculations. Limitations include the current inability to model conflicting preferences or heterogeneous sub‑groups, which the authors acknowledge as avenues for future work.

Overall, the paper demonstrates that maximizing future‑path entropy offers a parsimonious, mathematically grounded explanation for a wide range of collective phenomena, bridging statistical‑mechanics concepts with behavioural ecology and social physics.