[Paper Review] Sojourn times and fixation dynamics in multi-player games with fluctuating environments

Joseph W. Baron, Tobias Galla|arXiv (Cornell University)|Dec 16, 2016

Game Theory and Applications14 references3 citations

TL;DR

This paper develops an analytical framework to compute sojourn times and fixation dynamics in multi-player evolutionary games under fluctuating environmental conditions. It reveals that environmental switching rates can non-monotonically affect fixation times and probabilities, with optimal switching rates enabling fastest fixation despite complex nonlinear fitness landscapes in three-player games.

ABSTRACT

We study evolutionary multi-player games in finite populations, subject to fluctuating environments. The population undergoes a birth-death process with absorbing states, and the environment follows a Markovian process, resulting in a fluctuating payoff matrix for the evolutionary game. Our focus is on the fixation or extinction of a single mutant in a population of wildtypes. We show that the nonlinear nature of fitnesses in multi-player games gives rise to an intricate interplay of selection, genetic drift and environmental fluctuations. This generates effects not seen in simpler two-player games. To analyse trajectories towards fixation we analytically calculate sojourn times for general birth-death processes in populations of two types of individuals and in fluctuating environments.

Motivation & Objective

• To understand how environmental fluctuations interact with nonlinear fitness landscapes in multi-player evolutionary games.
• To analyze fixation dynamics of a single mutant in finite populations under time-varying payoff matrices.
• To develop a general analytical method for sojourn times in birth-death processes with Markovian environmental switching.
• To investigate the interplay between selection, genetic drift, and environmental noise in shaping fixation outcomes.
• To extend prior two-player game models to include higher-order, nonlinear interactions typical of real-world biological and social systems.

Proposed method

• Model a finite population of N individuals with two types (A and B) undergoing a birth-death process with absorbing states.
• Introduce a discrete-time, finite-state Markov process for environmental switching that alters the payoff matrix of the multi-player game.
• Use a backward-Fokker-Planck-like approach and backward-master equations to analytically compute sojourn times in each population state.
• Apply the framework to three-player games with nonlinear fitness functions and two internal fixed points in the replicator dynamics.
• Calculate fixation probabilities and conditional fixation times as functions of environmental switching parameters (e.g., switching rate T and transition rates δ⁺).
• Derive analytical expressions for sojourn times in each state i (number of type A individuals) under different environmental states, using the full time-averaged dynamics.

Experimental results

Research questions

• RQ1 How do environmental fluctuations affect the sojourn times and fixation dynamics in multi-player evolutionary games?
• RQ2 What is the role of switching rate and environmental persistence in determining the likelihood and speed of mutant fixation?
• RQ3 How does the nonlinearity of fitness functions in multi-player games alter fixation dynamics compared to two-player games?
• RQ4 Can optimal environmental switching rates be identified that minimize conditional fixation time while maximizing fixation probability?
• RQ5 How do multiple internal fixed points in the replicator dynamics influence the path and timing of fixation under environmental noise?

Key findings

• Conditional fixation times exhibit a local minimum at intermediate switching rates, indicating that mutants can reach fixation most quickly under specific environmental dynamics.
• Fixation probability shows monotonous dependence on the switching time scale T, suggesting a trade-off between rapid fixation and long-term success.
• The system displays non-trivial interplay between environmental switching and nonlinear fitness landscapes, leading to complex dynamics not observed in two-player games.
• Sojourn times in intermediate population states (e.g., i = 1 to i = N−1) are analytically calculable and depend on both the environmental state and the current population composition.
• The presence of two well-separated internal fixed points in the replicator dynamics leads to complex transient behavior, with environmental switching modulating the likelihood of escaping or being trapped near these points.
• Numerical simulations confirm that fixation likelihood and timing are sensitive to environmental switching parameters, with T = 500 yielding higher early fixation rates than T = 50 or T = 5000 at t ≈ 1000–5000.