In class we spoke about the replicator dynamics equation: a differential equation that is a building block of evolutionary game theory.

You can see a recording of this here.

I asked you all to suggest examples of social conventions. I then picked the idea of asking "How are you?" and responding "Fine" as a socially conventionally greeting to explore from an evolutionary game theoretic point of view.

We used the following game to model this using Replicator Dynamics:

$$ A = \begin{pmatrix} 2 & -1 \\ 0 & 2\\ \end{pmatrix} $$

This assumes a population of two types of individuals:

• The first type $H$: individuals who say "How are you?" and expect "Fine" to complete the greeting. These individuals get a utility of 2 when meeting each other and a utility of -1 when meeting someone who does not follow the convention.
• The second type $\bar H$: individuals who use a different type of greeting. These individuals get a utility of 0 when meeting someone of type $H$ and 2 when meeting someone of their own type.

We then model a given population using a vector $x=(x_1, x_2)$ where $x_1$ corresponds to the proportion of individuals of the type $H$ and $x_2=1-x_1$ is the proportion of the type $\bar H$.

Let us use sympy to create all these variables:

import sympy as sym
import nashpy as nash
import numpy as np
import matplotlib.pyplot as plt

x1 = sym.Symbol("x_1")
x2 = sym.Symbol("x_2")

We can then compute the average fitness of individuals of each type:

$$ f_1 = 2x_1 - x_2 $$

The average fitness of the individuals of the second type are:

$$ f_2 = 0\times x_1 + 2x_2 = 2x_2 $$

f1 = 2 * x1 - x2
f1

$\displaystyle 2 x_{1} - x_{2}$

f2 = 0 * x1 + 2 * x2

f2

$\displaystyle 2 x_{2}$

The average utility over the entire population is then given by:

$$\phi=x_1f_1+x_2f_2$$

phi = x1 * f1 + x2 * f2
phi

$\displaystyle x_{1} \left(2 x_{1} - x_{2}\right) + 2 x_{2}^{2}$

In the notes on Replicator Dynamics you can find linear algebraic expressions of these quantities $f$ and $\phi$ that extend naturally to populations with more than just 2 types.

The actual Replicator Dynamics equation is then given by:

$$ \frac{dx_i}{dt} = x_i(f_i-\phi)\text{ for all}i $$

In the case of our game this corresponds to:

$$ \begin{align} \frac{dx_1}{dt} =& x_1 (2x_1 - x_2 - x_1(2x_1 - x_2)-2x_2)\\ \frac{dx_2}{dt} =& x_2 (2x_2 - x_2 - x_1(2x_1 - x_2)-2x_2) \end{align} $$

rhs_of_dx1dt = x1 * (f1 - phi)
rhs_of_dx1dt

$\displaystyle x_{1} \left(- x_{1} \left(2 x_{1} - x_{2}\right) + 2 x_{1} - 2 x_{2}^{2} - x_{2}\right)$

rhs_of_dx2dt = x2 * (f2 - phi)
rhs_of_dx2dt

$\displaystyle x_{2} \left(- x_{1} \left(2 x_{1} - x_{2}\right) - 2 x_{2}^{2} + 2 x_{2}\right)$

Let us substitute $x_2=1-x_1$ and factorise:

sym.factor(rhs_of_dx1dt.subs({x2 : 1 - x1}))

$\displaystyle - x_{1} \left(x_{1} - 1\right) \left(5 x_{1} - 3\right)$

sym.factor(rhs_of_dx2dt.subs({x2 : 1 - x1}))

$\displaystyle x_{1} \left(x_{1} - 1\right) \left(5 x_{1} - 3\right)$

This gives:

$$ \begin{align} \frac{dx_1}{dt} =& - x_{1} \left(x_{1} - 1\right) \left(5 x_{1} - 3\right)\\ \frac{dx_2}{dt} =& x_{1} \left(x_{1} - 1\right) \left(5 x_{1} - 3\right) \end{align} $$

We note that $\frac{dx_1}{dt}=-\frac{dx_2}{dt}$ which is expected as we have $x_1+x_2=1$.

And we see (setting the derivatives to be equal to 0) that there are 3 stable populations:

equation = sym.Eq(lhs=rhs_of_dx1dt.subs({x2 : 1 - x1}), rhs=0)
sym.solveset(equation, x1)

$\displaystyle \left\{0, \frac{3}{5}, 1\right\}$

• $x_1=0$: No one uses "Hello how are you" as a greeting.
• $x_1=1$: Everyone uses "Hello how are you" as a greeting.
• $x_1=3/5$: 60% the population uses "Hello how are you" as a greeting.

The fact that this is stable mathematically (ie the derivatives are zero) corresponds to the game theoretic fact that in these populations every type of individual has the same fitness: so no one behaviour has an evolutionary advantage.

Let us see how the replicator dynamics equation can be solved numerically so that the populations can be observed over time.

A = np.array([[2, -1], [0, 2]])
game = nash.Game(A)

Starting with a uniformly distributed population

initial_population = np.array([.5, .5])
timepoints = np.linspace(0, 10, 1000)
populations = game.replicator_dynamics(y0=initial_population, timepoints=timepoints)
x1s, x2s = populations.T


plt.figure()
plt.plot(x1s, label="$x_1$")
plt.plot(x2s, label="$x_2$")
plt.legend()
plt.xlabel("time")
plt.ylabel("$x$");

We see that if we start with $x=(1/2, 1/2)$ the population converges to no one using "Hello how are you" as a greeting.

Starting with a small population of $\bar H$

initial_population = np.array([.9, .1])
timepoints = np.linspace(0, 10, 1000)
populations = game.replicator_dynamics(y0=initial_population, timepoints=timepoints)
x1s, x2s = populations.T


plt.figure()
plt.plot(x1s, label="$x_1$")
plt.plot(x2s, label="$x_2$")
plt.legend()
plt.xlabel("time")
plt.ylabel("$x$");

We see that with a small amount of $\bar H$ that small amount gets rejected by the population.

Confirm that $x_1=3/5$ is stable:

initial_population = np.array([3 / 5, 2 / 5])
timepoints = np.linspace(0, 10, 1000)
populations = game.replicator_dynamics(y0=initial_population, timepoints=timepoints)
x1s, x2s = populations.T


plt.figure()
plt.plot(x1s, label="$x_1$")
plt.plot(x2s, label="$x_2$")
plt.legend()
plt.xlabel("time")
plt.ylabel("$x$");

This confirms the 0 derivative. Let us see what happens if we budge slightly from that population.

First: a small increase of $H$:

epsilon = 10 ** -3

initial_population = np.array([3 / 5 + epsilon, 2 / 5 - epsilon])
timepoints = np.linspace(0, 10, 1000)
populations = game.replicator_dynamics(y0=initial_population, timepoints=timepoints)
x1s, x2s = populations.T


plt.figure()
plt.plot(x1s, label="$x_1$")
plt.plot(x2s, label="$x_2$")
plt.legend()
plt.xlabel("time")
plt.ylabel("$x$");

We see that that small nudge is enough to force the population to all adopt "Hello how are you" as a greeting.

What if the nudge was the other way?

epsilon = 10 ** -3

initial_population = np.array([3 / 5 - epsilon, 2 / 5 + epsilon])
timepoints = np.linspace(0, 10, 1000)
populations = game.replicator_dynamics(y0=initial_population, timepoints=timepoints)
x1s, x2s = populations.T


plt.figure()
plt.plot(x1s, label="$x_1$")
plt.plot(x2s, label="$x_2$")
plt.legend()
plt.xlabel("time")
plt.ylabel("$x$");

This is enough to nudge the population to stop using "Hello how are you" as a greeting.