The dual problems of coordination and anti-coordination on random bipartite graphs

TL;DR

This paper investigates the difficulty of achieving coordination versus anti-coordination in bipartite graphs, revealing their fundamental equivalence through Markov chain isomorphisms under specific decision rules.

Contribution

It establishes the duality between coordination and anti-coordination problems on bipartite graphs and constructs an isomorphism between their Markov chains under certain stochastic decision rules.

Findings

Coordination and anti-coordination problems are equally difficult on bipartite graphs.

An isomorphism between the Markov chains of the two problems is constructed.

Results provide new insights into collective action problems on networks.

Abstract

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.