  The commonly used accumulated payoff scheme is not invariant with respect to
shifts of payoff values when applied locally in degree-inhomogeneous population
structures. We propose a suitably modified payoff scheme and we show both
formally and by numerical simulation, that it leaves the replicator dynamics
invariant with respect to affine transformations of the game payoff matrix. We
then show empirically that, using the modified payoff scheme, an interesting
amount of cooperation can be reached in three paradigmatic non-cooperative
two-person games in populations that are structured according to graphs that
have a marked degree inhomogeneity, similar to actual graphs found in society.
The three games are the Prisoner's Dilemma, the Hawks-Doves and the Stag-Hunt.
This confirms previous important observations that, under certain conditions,
cooperation may emerge in such network-structured populations, even though
standard replicator dynamics for mixing populations prescribes equilibria in
which cooperation is totally absent in the Prisoner's Dilemma, and it is less
widespread in the other two games.
