Replicator dynamics

In the hawk-dove game a hawk does well among doves and badly among hawks, so neither strategy can take over: whatever the start, the share of hawks settles at V / C, the mixed equilibrium of the game.

The model is created by name from the catalog with the value V = 2 and the cost C = 3 of a fight, and started with 5% hawks. The observer saves every second frame with the default blueprint, whose primary view is the time series of the shares of hawks and doves.

import matplotlib.pyplot as plt

from skcomplex.engine.runtime import Simulation
from skcomplex.engine.viz import RerunObserver
from skcomplex.models import create

MODEL = 'replicator_dynamics'

system = create(MODEL, x=[0.05, 0.95], V=2.0, C=3.0)
observer = RerunObserver(MODEL, save_path=f'{MODEL}.rrd', blueprint='default', interval=2)
simulation = Simulation(system, dt=0.1, observers=[observer]).check().prepare().run(until=30)

plot replicator dynamics

Open the recording in the Rerun viewer

The mixed equilibrium

The hawks grow when their payoff (V - C) / 2 x + V (1 - x) exceeds that of the doves V / 2 (1 - x), that is while their share x is below V / C. Started from few or many hawks, the share should converge to 2 / 3.

fig, ax = plt.subplots(figsize=(7, 4))
for start in [0.05, 0.3, 0.6, 0.95]:
    trajectory = Simulation(create(MODEL, x=[start, 1 - start]), dt=0.1).check().prepare().run(until=30).trajectory_
    ax.plot(trajectory.times, trajectory.values('population.x')[:, 0], label=f'{start:.0%} hawks at the start')
ax.axhline(2 / 3, color='black', linestyle='--', label='V / C')
ax.set_xlabel('time')
ax.set_ylabel('share of hawks')
ax.set_title('Every start converges to V / C hawks')
ax.legend()
plt.show()

Every start converges to V / C hawks

Download Python source code: plot_replicator_dynamics.py

Download Jupyter notebook: plot_replicator_dynamics.ipynb

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About the model

The replicator dynamics of a game, by default hawks and doves, which settle at a share V / C of hawks.

References

Taylor and Jonker 1978
Maynard Smith 1982
Gros 2015
Szabó and Fáth 2007