10 Moran Processes

Corresponding chapters

Objectives

  • Play a class activity of a Moran process;
  • Define a Moran process;
  • Prove theorem for formula of fixation probabilities;
  • Numeric calculations.

Activity

Ask students to form groups of 4: this is a population.

Explain that players/individuals are either Hawks or Doves and that they will play the Hawk-Dove game with row matrix:

\[\begin{split}A = \begin{pmatrix} 0 & 3\\ 1 & 2 \end{pmatrix}\end{split}\]

Recall, this corresponds to sharing of 4 resources:

  • Two hawks both get nothing;
  • A hawk meeting a dove gets 3 out of 4.
  • A dove meeting a hawk gets 1 out of 4.
  • A dove meeting a dove gets 2 out of 4.

Hand out a 10 and 20 sided dice to each group.

Explain that one student is to be a Hawk and the others Doves.

Every student players each other and write down their total scores:

  • The Hawk gets: 9 (\(3\times 3\)).
  • The Doves all get: 5 (\(1\times 1 + 2 \times 2=5\)).

One of the individuals is chosen to reproduce:

\[P(H) = \frac{9}{24}=3/8 \qquad P(D) = \frac{9}{24}=5/24\]

Use the dice (invite the students to figure out a way to do this, various different ways - ignore none throws etc…).

Then randomly choose a player to eliminate: this can be the same player chosen to reproduce (life can suck).

Now recalculate the fitness (every player plays everyone else).

Repeat until all players are of the same type.

Count from groups to obtain mean fixation rate of a Hawk.

Depending on time, potentially repeat this using Doves.

Now work through the notes: culminating in the proof of the theorem for the absorption probabilities of a birth death process.

Discuss and use code from chapter to show the fixation with the Hawk Dove game:

>>> import numpy as np
>>> A = np.array([[0, 3], [1, 2]])

Calculate theoretic value using formula from theorem:

\[\begin{split}\begin{align} f_{1i} &= \frac{3(N-i)}{N - 1}=3\frac{N-i}{N-1}\\ f_{2i} &= \frac{i+2(N - i -1)}{N - 1}=\frac{2N-2-i}{N - 1}\\ \end{align}\end{split}\]

This gives (for \(N=4\)):

  \(i=1\) \(i=2\) \(i=3\)
\(f_{1i}\) 3 2 1
\(f_{2i}\) 5/3 4/3 1
\(\gamma_i\) 5/9 2/3 1

Thus:

\[x_1 = \frac{1}{1 + 5/9 + 5/9\times2/3 +5/9\times2/3\times1}=\frac{1}{62/27}=\frac{27}{62}\approx.44\]
  • Discuss work of Maynard smith but that this actually used Hawk Dove game in infinite population games.
  • Discussion possibility for using a utility model on top of fitness.
  • A lot of current work looks at Moran processes: a good model of invasion of a specifies etc…
  • The Prisoners dilemma can also be included, there is documentation about simulating this with Axelrod is here: http://axelrod.readthedocs.io/en/stable/tutorials/getting_started/moran.html