10 Evolutionary game theory

Corresponding chapters

Objectives

  • Go over the logical progression from dynamics to games;
  • Explain ESS

Notes

Recall that there are no players: just strategies in a population.

Go over the notes, explaining the mathematics.

Now consider the Matching pennies game:

\[\begin{split}A = \begin{pmatrix} 1 & -1\\ -1 & 1\\ \end{pmatrix}\end{split}\]

This gives:

\[f_1 = x_1 - x_2\qquad f_2 = -x_1 + x_2\]

which is equivalent to:

\[\begin{split}f = A x = \begin{pmatrix} x_1 - x_2\\ -x_1 + x_2 \end{pmatrix}\end{split}\]

and so:

\[phi = f x = x_1(x_1 - x_2) + x_2(x_2 - x_1) = (x_1 - x_2) ^ 2\]

thus:

\[\begin{split}\frac{dx}{dt} = \begin{pmatrix} x_1 ((x_1 - x_2) - (x_1 - x_2) ^2)\\ x_2 ((x_2 - x_1) - (x_1 - x_2) ^2) \end{pmatrix}\end{split}\]
\[\begin{split}\frac{dx}{dt} = \begin{pmatrix} x_1 (x_1 - x_2)(1 - (x_1 - x_2))\\ x_2 (x_1 - x_2)(-1 - (x_1 - x_2)) \end{pmatrix}\end{split}\]

however recall that \(x_1 + x_2 = 1\):

\[\frac{dx_1}{dt} = x_1 (2x_1 - 1)2(1-x_1))\]

Verification of above calculations:

>>> import sympy as sym
>>> x_1, x_2 = sym.symbols("x_1, x_2")
>>> f_1 = x_1 - x_2
>>> f_2 = x_2 - x_1
>>> phi = x_1 * f_1 + x_2 * f_2
>>> dx_1 = x_1 * (f_1 - phi)
>>> sym.factor(dx_1.subs({x_1: 1 - x_2}))
2*x_2*(x_2 - 1)*(2*x_2 - 1)

We have the stable points as expected:

  1. \(x_1 = 0\)
  2. \(x_1 = 1\)
  3. \(x_1 = 1 / 2\)

Relate this to the notes and discuss notion of mutated population and stable ESS.

Show output of the following:

>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> from scipy.integrate import odeint
>>> t = np.linspace(0, 10, 100)  # Obtain 100 time points
>>> def dx(x, t, A):
...     """
...     Define the derivate of x.
...     """
...     f = np.dot(A, x)
...     phi = np.dot(f, x)
...     return x * (f - phi)

Slight deviation from \(x_1=0\):

>>> A = np.array([[1, -1], [-1, 1]])
>>> epsilon = 10 ** -1
>>> xs = odeint(func=dx, y0=[epsilon, 1 - epsilon], t=t, args=(A,))
>>> plt.plot(xs);
[...

Slight deviation from \(x_1=1\):

>>> xs = odeint(func=dx, y0=[1 - epsilon, epsilon], t=t, args=(A,))
>>> plt.plot(xs);
[...

Slight deviation from \(x_1=1/2\):

>>> xs = odeint(func=dx, y0=[1 / 2 - epsilon, 1 / 2 + epsilon], t=t, args=(A,))
>>> plt.plot(xs);
[...

Look at theorem and discuss proof.

Carry out Nash equilibria computation (which corresponds to the above case):

>>> import nashpy as nash
>>> game = nash.Game(A, A.transpose())
>>> list(game.support_enumeration())
[(array([1., 0.]), array([1., 0.])), (array([0., 1.]), array([0., 1.])), (array([0.5, 0.5]), array([0.5, 0.5]))]

Now look at cases:

  1. \(x_1=0\): we are in the first case of the theorem so we have an ESS.
  2. \(x_1=1\): we are in the first case of the theorem so we have an ESS.
  3. \(x_1=1/2\): we now have the second case of the theorem \(u(x, y)=u(x, x)\) (indeed the row strategy aims to make the column strategy indifferent - according to the best response condition).

To deal with this case we need to look at the next part of the second condition:

\[u(x, y) = 0\]
\[u(y, y) = y_1 ^2 + (1 - y_1) ^2 - 2(1-y_1)y_1 = (y_1 - (1 - y_1))^2 = (2y_1 - 1) ^ 2\]

And as long as \(y_1\ne x_1=1/2\) then \(u(x, y) < u(y, y)\) thus this is not an ESS:

>>> A = sym.Matrix(A)
>>> y_1, y_2 = sym.symbols("y_1, y_2")
>>> y = sym.Matrix([y_1, y_2])
>>> sym.factor((y.transpose() * A * y)[0].subs({y_2: 1 - y_1}))
1.0*(2.0*y_1 - 1.0)**2

Discuss the work John Maynard Smith in 1973 who formalised this work. Mention that he was not actually aware of Nash equilibria at the time.