AD - Moran process

Duration: 25 minutes

Objectives

  • Compute the probability of fixation in the Hawk Dove game using the formula.

Notes

For:

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

Ask students to work in pairs, using the formula to compute \(x_1\).

This is given by:

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

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:

\[x_1 = \frac{1}{1 + 5/9 + 5/9\times2/3 +5/9\times2/3\times1}=\frac{1}{62/27}=\frac{27}{62}\approx.44\]