  Destructive agents, who opt out of the game and indiscriminately harm others,
paradoxically foster cooperation, representing an intriguing variant of the
voluntary participation strategy. Yet, their impact on cooperation remains
inadequately understood, particularly in the context of pairwise social dilemma
games and in comparison to their counterparts, constructive agents, who opt out
of the game but indiscriminately benefit others. Furthermore, little is known
about the combined effects of both agent types on cooperation dynamics. Using
replicator dynamics in infinite and well-mixed populations, we find that,
contrary to their role in facilitating cooperation in multiplayer games,
destructive agents fail to encourage cooperation in pairwise social dilemmas.
Instead, they destabilize and may even replace defection in the prisoners'
dilemma and stag-hunt games. Similarly, in the chicken game, they can
destabilize or replace the mixed equilibrium of cooperation and defection, and
they undermine cooperation in the harmony game. Conversely, constructive
agents, when their payoffs exceed their contributions to opponents, can exhibit
effects similar to destructive agents. However, if their payoffs are lower,
while they destabilize defection in prisoners' dilemma and stag-hunt games,
they do not disrupt the cooperation equilibrium in harmony games and have a
negligible impact on the coexistence of cooperation in chicken games. The
combination of destructive and constructive agents does not facilitate
cooperation but instead generates complex evolutionary dynamics, including
bi-stable, tri-stable, and quad-stable states, with outcomes contingent on
their relative payoffs and game types. These results, taken together, enhance
our understanding of the impact of the voluntary participation mechanism on
cooperation, contributing to a more comprehensive understanding of its
influence.
