Role Swap: When the Follower Leads and the Leader Follows

The game theoretic literature has commonly explored circumstances in which the players are identical. In the real world, strategic actors such as competing firms or political parties are however heterogeneous. Most importantly, their payoffs across the various possible outcomes generally differ. We consider payoff heterogeneity within a more general ‘Stochastic leadership’ framework. It allows for probabilistic revisions of each player’s initial actions - upon observing what the others have done. The analysis shows that under Stochastic leadership it is the exact payoffs, not just their ranking, that affects the set of (subgame-perfect) equilibria. This is consistent with experimental studies that show payoff heterogeneity to hinder cooperation and aggravate conflict by moving the players away from the focal (symmetric/equitable) outcome. Furthermore, we demonstrate that if the payoffs are sufficiently asymmetric the players may essentially swap their roles in coordination and anti-coordination games. In particular, we derive circumstances within the Battle of the sexes, Stag hunt and Hawk and dove games under which the Stochastic follower (the more flexible player with a higher revision probability) starts behaving as the Stackelberg leader. Our main real-world example is from the area of climate change agreements between major countries.