Table 1.

Fixation probabilities for a single random mutant in graph occupied by individuals of the other type: The first column describes the graph type, and the second and third columns represent the fixation probability of a single cooperator and defector, respectively

	${\rho }_C^n$	${\rho }_D^n$	${\rho }_C^n\bowtie {\rho }_D^n$	$A_n=\frac{{\rho }_C^n}{{\rho }_D^n}$	${\displaystyle \underset{n\to \infty }{lim}}{A}_n$	$B_n=\frac{{\rho }_C^n}{{\rho }_n}$	${\displaystyle \underset{n\to \infty }{lim}}{B}_n$
Complete graph	$2^{-c_{1}n}$	c 2	$<<$	$2^{-c_{1}n}/c_2$	$0$	$n{2}^{-c_{1}n}$	$0$
Grid	$2^{-c_{3}n}$	c 4	$<<$	$2^{-c_{3}n}/c_4$	$0$	$n{2}^{-c_{3}n}$	$0$
Star	$c_5{n}^{-1}$	$c_6{n}^{-1}$	≈	$c_5/c_6$	$c_5/c_6$	c 5	c 5
${\mathcal{A}}^r$	c 7	$2^{-c_{8}n}$	$>>$	$c_7{2}^{c_{8}n}$	∞	$c_{7}n$	∞

In the fourth column, we compare the two fixation probabilities where ⋈ is the comparison operator, which can be very small <<, comparable ≈, or very large >>. The fifth and sixth columns consider the ratio A_n of the two fixation probabilities and the respective large-population limit. The seventh and eighth columns consider the ratio B_n of the fixation probability of a cooperator to the fixation probability of a neutral mutant, denoted ρ_n, which is $1/n$, and the respective large-population limit. In the table, c₁, c₂, ⋯, c₈ denote constants that are independent of n. The table summarizes the following: (a) for complete graph and grid, the fixation probability of cooperators is exponentially small in n, whereas for defectors it is constant; (b) for star, the fixation probabilities are proportional to $1/n$; and (c) for ${\mathcal{A}}^r$ the fixation probability of cooperators is constant, whereas for defectors it is exponentially small. The large-population limit of the desired comparison ratios vanishes for the complete graph and grid, is constant for the star, and goes to ∞ for ${\mathcal{A}}^r$. Observe that only ${\mathcal{A}}^r$ yields that the desired ratio goes to ∞ in the large-population limit.