Compartment model of strategy-dependent time delays in replicator dynamics

TL;DR

This paper introduces a simple compartment-based model for evolutionary game dynamics with strategy-dependent time delays, capturing complex temporal effects while remaining analytically tractable.

Contribution

It presents a novel compartment model incorporating strategy-dependent delays, bridging the gap between simple models and complex delay differential equations.

Findings

Delays negatively impact certain strategies in evolutionary games.

Including delays can alter the effective game dynamics.

Model results align qualitatively with explicit delay models.

Abstract

Real-world processes often exhibit temporal separation between actions and reactions - a characteristic frequently ignored in many modelling frameworks. Adding temporal aspects, like time delays, introduces a higher complexity of problems and leads to models that are challenging to analyse and computationally expensive to solve. In this work, we propose an intermediate solution to resolve the issue in the framework of evolutionary game theory. Our compartment-based model includes time delays while remaining relatively simple and straightforward to analyse. We show that this model yields qualitatively comparable results with models incorporating explicit delays. Particularly, we focus on the case of delays between parents' interaction and an offspring joining the population, with the magnitude of the delay depending on the parents' strategy. We analyse Stag-Hunt, Snowdrift, and the Prisoner's Dilemma game and show that strategy-dependent delays are detrimental to affected strategies. Additionally, we present how including delays may change the effective games played in the population, subsequently emphasising the importance of considering the studied systems' temporal aspects to model them accurately.