  Situations in which immediate self-interest and long-term collective interest
conflict often require some form of influence to prevent them from leading to
undesirable or unsustainable outcomes. Next to sanctioning, social influence
and social structure, it is possible that strategic solutions can exist for
these social dilemmas. However, the existence of strategies that enable a
player to exert control in the long-run outcomes can be difficult to show and
different situations allow for different levels of strategic influence. Here,
we investigate the effect of threshold nonlinearities on the possibilities of
exerting unilateral control in finitely repeated n-player public goods games
and snowdrift games. These models can describe situations in which a collective
effort is necessary in order for a benefit to be created. We identify
conditions in terms of a cooperator threshold for the existence of generous,
extortionate and equalizing zero-determinant (ZD) strategies. Our results show
that, for both games, the thresholds prevent equalizing ZD strategies from
existing. In the snowdrift game, introducing a cooperator threshold has no
effect on the region of feasible extortionate ZD strategies. For extortionate
strategies in the public goods game, the threshold only restricts the region of
enforceable strategies for small values of the public goods multiplier.
Generous ZD strategies exist for both games, but introducing a cooperator
threshold forces the slope more towards the value of a fair strategy, where the
player has approximately the same payoff as the average payoff of his
opponents.
