When a compromise gets compromised by another compromise

In many situations it is difficult to avoid a conflict and cooperate with others, even when all parties would like to do so. We show that while having one coordination device tends to be beneficial, having more of them may actually make things worse. Our game theoretic analysis examines the relationship between two widely used coordination devices, namely the focal point and leadership. However, instead of conventional Stackelberg leadership we use a more general Stochastic leadership framework that allows for probabilistic revisions of the players' decisions. Upon observing the initial simultaneous actions, each player $i$ can (simultaneously) change her action with some probability $p_i$, known to all players in advance. In this framework, the more rigid player with lower $p_i$ assumes the role of a Stochastic leader, and the more flexible player becomes the Stochastic follower. We demonstrate how focal point and leadership may clash in coordination and anti-coordination games such as the Battle of the sexes, Hawk and dove and Stag hunt. Because the focal point is based on symmetry whereas leadership on asymmetry, having these two coordination devices jointly makes Pareto-inferior outcomes more likely. Two distinct effects are behind this phenomenon, we call them multiplicity-broadening and multiplicity-deepening. Our theoretic findings are then linked to many instances from business, sport as well as politics. We also discuss the implications for the design of welfare-improving managerial processes and public policies that might help to avoid costly mis-coordination.