  Various social dilemma games that follow different strategy updating rules
have been studied on many networks.The reported results span the entire
spectrum, from significantly boosting,to marginally affecting,to seriously
decreasing the level of cooperation.Experimental results that are qualitatively
different from theoretical prediction have also been reported.It is widely
believed that the results are largely determined by three elements,including
payoff matrices of the underlying 2*2 games,the way that the strategic states
of the players are updated and the structure of the networks.Here we discuss
the impact of a seemly non-essential mechanism -- what we refer to as a "payoff
scheme". Specifically, in each round after the states of all of the players are
determined,the payoff scheme is how each player's payoff is calculated.In
addition to the two conventions in which either the accumulated or the averaged
payoff is calculated from playing with all of the neighboring players,we here
study the effects of calculating the payoff from pairing up with one random
player from among the neighboring players. Based on probability theory, in a
situation of uncorrelated events, the average payoff that involves all of the
neighbors should,in principal,be equivalent to the payoff from pairing up with
one neighbor.However,our simulation of games on graphs shows that, in many
cases,the two payoff schemes lead to qualitatively different levels of
cooperation.This finding appears to provide a possible explanation for a wide
spectrum of observed behaviors in the literature.We have also observed that
results from the randomly-pairing-one mechanism are more robust than the
involving-all-neighbours mechanism because,in the former case, neither the
other three main elements nor the initial states of the players have a large
impact on the final level of cooperation compared with in the latter case.
