Learning increases growth and reduces inequality in shared noisy environments

Abstract

Stochastic multiplicative dynamics characterize many complex natural phenomena such as selection and mutation in evolving populations, and the generation and distribution of wealth within social systems. Population heterogeneity in stochastic growth rates has been shown to be the critical driver of wealth inequality over long time scales. However, we still lack a general statistical theory that systematically explains the origins of these heterogeneities resulting from the dynamical adaptation of agents to their environment. In this paper, we derive population growth parameters resulting from the general interaction between agents and their environment, conditional on subjective signals each agent perceives. We show that average wealth growth rates converge, under specific conditions, to their maximal value as the mutual information between the agent’s signal and the environment, and that sequential Bayesian inference is the optimal strategy for reaching this maximum. It follows that when all agents access the same statistical environment, the learning process attenuates growth rate disparities, reducing the long-term effects of heterogeneity on inequality. Our approach shows how the formal properties of information underlie general growth dynamics across social and biological phenomena, including cooperation and the effects of education and learning on life history choices.

Information synergy maximizes the growth rate of heterogeneous groups

“…In the limit of no cooperation, for all g , and no agents pool information. In this case, the dynamics of the model are similar to previous work ( 38 ).…”

“…This signal may represent several different processes such as sensory input or a lead retrieved from memory. With optimal parameterization of a model of the environment, , an agent’s optimal investment strategy leads to an average resource growth rate (over time) ( 38 ), where is the mutual information between environmental states E and the agent’s signals S . (We are working in units of units time , for simplicity.)…”

“…The average over time, η , and the resource temporal SD (volatility), σ , combine under Itô integration to give the actual growth rate . Maximizing this growth rate (as a positive quantity) entails maximizing η and minimizing σ , which at the individual agent level can be achieved by (Bayesian) learning over time ( 38 ).…”

“…Bayesian learning is the optimal strategy to incorporate new information from observed events into the estimate of conditional probabilities, such as those of environmental states given agents’ signals ( 38 ). Agents can also learn the synergy embedded in their environment in groups by collectively weighing their conditional observations across their individual signals.…”

“…By working jointly to predict characteristics of their environment ( 31 ) and gather resources, groups of agents can maximize their collective fitness even when each individual has very limited knowledge. In a setting where there are resource returns to successful prediction and behavior, information of the state of a statistical environment determines the fitness of the population ( 38 , 39 ), though there are questions about how such benefits emerge quantitatively ( 40 ). Here, we formalize the calculation of these social benefits in terms of the properties of information and show how maximizing knowledge complementarities (synergy) maximizes the long-term growth rate of collectives.…”