# Environmental dynamics impact whether matching is optimal

## TL;DR

This paper explores how environmental changes affect whether animals' matching behavior is optimal for maximizing rewards.

## Contribution

The study analytically determines when matching is optimal based on environmental replenishment dynamics.

## Key findings

- Matching is optimal when all options share the same replenishment dynamics.
- Optimal policies can deviate from matching when replenishment dynamics differ across options.
- Environmental stochasticity can amplify deviations from matching behavior.

## Abstract

Foraging animals often sample options that yield rewards with different probabilities. In such scenarios, many animals exhibit “matching,” whereby they allocate their choices such that the fraction of rewarded samples is equal across options. While matching can be optimal in environments with diminishing returns, this condition alone is not sufficient to determine optimality. Moreover, diminishing returns arise when resources deplete and replenish over time, but their form depends on the temporal structure and statistics of replenishment. Here, we investigate how these environmental properties influence whether matching is optimal. We consider an agent that samples options at fixed rates, and we derive the resulting reward probabilities across different types of environments. This allows us to analytically determine conditions under which the optimal policy exhibits matching. When all options share the same replenishment dynamics, matching emerges as optimal across a wide range of environments. However, when dynamics differ across options, optimal policies can deviate from matching. In such cases, the rank-ordering of observed reward probabilities depends only on the qualitative nature of the replenishment process, and not on the specific replenishment rates. As a result, the optimal policy can exhibit under- or over-matching depending on which options are more rewarding. We use this result to identify environments where performance differs substantially between matching and optimality. Finally, we show that fluctuations in replenishment rates—representing environmental stochasticity or internal uncertainty—can amplify deviations from matching. These findings deepen our understanding of the relationship between environmental variability and behavioral optimality, and provide testable predictions across diverse settings.