We study evolutionary games on graphs. The individuals of a population
occupy the vertices of the graph and interact with their neighbors to receive
payoff. We consider finite population size, regular graphs, probabilistic
death-birth updating and weak selection. There are two types of strategies,
A and B, and a payoff matrix [(a, b),(c, d)]. The initial condition is given by
an arbitrary configuration where each vertex is occupied by either A or B.
The conjugate initial condition is obtained by swapping A and B. We ask:
when is the fixation probability of A for the original configuration greater
than the fixation probability of B for the conjugate configuration? The answer
is a linear condition of the form σa+b > c+σd. We calculate σ for any
initial condition. For large population size we obtain the well known result
σ = (k + 1)/(k − 1), but now this result extends to any mixed initial condition.
As a specific example we study evolution of cooperation. We calculate
the critical benefit-to-cost ratio for natural selection to favor the fixation of
cooperators for any initial condition. We obtain results that specify which
initial conditions reduce and which initial conditions increase the critical
benefit-to-cost ratio. Adding more cooperators to the initial condition does
not necessarily favor cooperation. But strategic placing of cooperators in a
network can enhance the takeover of cooperation.