  In a social dilemma game group members are allowed to decide if they
contribute to the joint venture or not. As a consequence, defectors, who do not
invest but only enjoy the mutual benefit, prevail and the system evolves onto
the tragedy of the common state. This unfortunate scenario can be avoided if
participation is not obligatory but only happens with a given probability. But
what if we also consider a player's individual wealth when to decide about
participation? To address this issue we propose a model in which the
probabilistic participation in the public goods game is combined with a
conditional investment mode that is based on individual wealth: if a player's
wealth exceeds a threshold value then it is qualified and can participate in
the joint venture. Otherwise, the participation is forbidden in the investment
interactions. We show that if only probabilistic participation is considered,
spatially structured populations cannot support cooperation better than
well-mixed populations where full defection state can also be avoided for small
participation probabilities. By adding the wealth-based criterion of
participation, however, structured populations are capable to augment network
reciprocity relevantly and allow cooperator strategy to dominate in a broader
parameter interval.
