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Replicator dynamics as the large population limit of a discrete Moran process in the weak selection regime: A proof via Eulerian specification

Marco Morandotti, Gianluca Orlando

TL;DR

The paper rigorously derives the replicator dynamics as the large-population, weak-selection limit of a multi-strategy Moran process by recasting the discrete stochastic dynamics in an Eulerian framework on the strategy simplex. A velocity field $b(\lambda)$ is identified from the payoff matrix $A$, and the evolution of the law of proportions is shown to satisfy an approximate continuity equation in the $1$-Wasserstein sense; compactness is established to pass to a limit. Under the scaling $N_k\sim\tau_k^{-\alpha}$, $w_k\sim\tau_k^{\beta}$ with $\alpha+\beta=1$ and $\alpha>1/2$, the sequence of laws converges to a Lipschitz curve $\Lambda_t$ solving $\partial_t\Lambda_t+\mathrm{div}(b\Lambda_t)=0$, with $\Lambda_t=\Psi(t,\cdot)_\#\Lambda_0$ representing the pushforward by the replicator flow. Consequently, the replicator ODE for strategy proportions emerges as the mean-field limit of the discrete Moran dynamics, with the Eulerian-Lagrangian duality formalized via the superposition principle. This provides a robust bridge between finite stochastic evolutionary processes and continuous-time replicator dynamics, with precise parameter scaling governing the limit.

Abstract

We study the large population limit of a multi-strategy discrete-time Moran process in the weak selection regime. We show that the replicator dynamics is interpreted as the large-population limit of the Moran process. This result is obtained by interpreting the discrete process in its Eulerian specification, proving a compactness result in the Wasserstein space of probability measures for the law of the proportions of strategies, and passing to the limit in the continuity equation that describes the evolution of the proportions.

Replicator dynamics as the large population limit of a discrete Moran process in the weak selection regime: A proof via Eulerian specification

TL;DR

The paper rigorously derives the replicator dynamics as the large-population, weak-selection limit of a multi-strategy Moran process by recasting the discrete stochastic dynamics in an Eulerian framework on the strategy simplex. A velocity field is identified from the payoff matrix , and the evolution of the law of proportions is shown to satisfy an approximate continuity equation in the -Wasserstein sense; compactness is established to pass to a limit. Under the scaling , with and , the sequence of laws converges to a Lipschitz curve solving , with representing the pushforward by the replicator flow. Consequently, the replicator ODE for strategy proportions emerges as the mean-field limit of the discrete Moran dynamics, with the Eulerian-Lagrangian duality formalized via the superposition principle. This provides a robust bridge between finite stochastic evolutionary processes and continuous-time replicator dynamics, with precise parameter scaling governing the limit.

Abstract

We study the large population limit of a multi-strategy discrete-time Moran process in the weak selection regime. We show that the replicator dynamics is interpreted as the large-population limit of the Moran process. This result is obtained by interpreting the discrete process in its Eulerian specification, proving a compactness result in the Wasserstein space of probability measures for the law of the proportions of strategies, and passing to the limit in the continuity equation that describes the evolution of the proportions.
Paper Structure (6 sections, 7 theorems, 126 equations, 1 figure, 1 table)

This paper contains 6 sections, 7 theorems, 126 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Notation and basic definitions
  3. The discrete stochastic process
  4. Eulerian specification of the discrete stochastic process
  5. Continuous-time large population limit
  6. Replicator dynamics

Key Result

Lemma 4.2

Let $\Lambda^k_t$ be as in eq:Lambda_k_t and $\overline \Lambda^k_t$ be as in eq:Lambda_bar_k_t. We have that $\mathcal{W}_1(\Lambda^k_t,\overline \Lambda^k_t) \to 0$ as $k \to +\infty$ for every $t \in [0,T]$.

Figures (1)

  • Figure 1: Graphical representation of one time step of a Moran process. In the picture, we have $N = 8$ agents and $M = 3$ strategies, $u_1$ (white circle), $u_2$ (grey circle), $u_3$ (black circle). At time $t_h$, for each agent a fitness is calculated in terms of the expected payoff due to the interaction with an agent sampled randomly uniformly in the population. The fitness is used to determine the probability of reproduction. In the figure, an agent with strategy $u_1$ has been chosen for reproduction (biggest circle), and an agent with strategy $u_2$ has been chosen to die (smallest circle). The generation is updated at time $t_{h+1}$ accordingly.

Theorems & Definitions (17)

  • Remark 4.1
  • Lemma 4.2
  • proof
  • Remark 4.3
  • Proposition 4.4
  • Lemma 4.5
  • proof
  • proof : Proof of Proposition \ref{['prop:approximate_distributional_solution']}
  • Remark 5.1
  • Remark 5.2
  • ...and 7 more