The replicator coalescent

TL;DR

The paper introduces the replicator coalescent, a multi-type stochastic model generalizing Kingman's coalescent, revealing a connection to replicator equations from evolutionary game theory and describing its behavior from large initial states to stochastic effects.

Contribution

It presents the replicator coalescent as a novel multi-type coalescent model and uncovers its connection to replicator equations, bridging stochastic processes and evolutionary dynamics.

Findings

Initial behavior aligns with replicator equations.

Transition from deterministic to stochastic dynamics.

Generalizes Kingman's coalescent to multiple types.

Abstract

We consider a stochastic model, called the replicator coalescent, describing a system of blocks of $k$ different types which undergo pairwise mergers at rates depending on the block types: with rate $C_{i,j}$ blocks of type $i$ and $j$ merge, resulting in a single block of type $i$. The replicator coalescent can be seen as generalisation of Kingman's coalescent death chain in a multi-type setting, although without an underpinning exchangeable partition structure. The name is derived from a remarkable connection we uncover between the instantaneous dynamics of this multi-type coalescent when issued from an arbitrarily large number of blocks, and the so-called replicator equations from evolutionary game theory. By dilating time arbitrarily close to zero, we see that initially, on coming down from infinity, the replicator coalescent behaves like the solution to a certain replicator equation. Thereafter, stochastic effects are felt and the process evolves more in the spirit of a multi-type death chain.