AD - Moran process ================== **Duration**: 25 minutes Objectives ---------- - Compute the probability of fixation in the Hawk Dove game using the formula. Notes ----- For: .. math:: A = \begin{pmatrix} 0 & 3\\ 1 & 2 \end{pmatrix} Ask students to work in pairs, using the formula to compute :math:x_1. This is given by: .. math:: \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} This gives (for :math:N=4): +------------------+--------------+--------------+--------------+ | | :math:i=1 | :math:i=2 | :math:i=3 | +==================+==============+==============+==============+ | :math:f_{1i} | 3 | 2 | 1 | +------------------+--------------+--------------+--------------+ | :math:f_{2i} | 5/3 | 4/3 | 1 | +------------------+--------------+--------------+--------------+ | :math:\gamma_i | 5/9 | 2/3 | 1 | +------------------+--------------+--------------+--------------+ Thus: .. math:: 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