  In some scenarios ("anti-coordination games"), individuals are better off
choosing different actions than their neighbors while in other scenarios
("coordination games"), it is beneficial for individuals to choose the same
strategy as their neighbors. Despite having different incentives and resulting
population dynamics, it is largely unknown which collective outcome,
anti-coordination or coordination, is easier to achieve. To address this issue,
we focus on the distributed graph coloring problem on bipartite graphs. We show
that with only two strategies, anti-coordination games (2-colorings) and
coordination games (uniform colorings) are dual problems that are equally
difficult to solve. To prove this, we construct an isomorphism between the
Markov chains arising from the corresponding anti-coordination and coordination
games under certain specific individual stochastic decision-making rules. Our
results provide novel insights into solving collective action problems on
networks.
