Decomposed games, focal points, and the framing of collective and individual interests

The literature attributes high coordination rates in pure coordination games with focal points to team reasoning and low coordination rates in related battle of the sexes games to level k reasoning. We investigate whether coordination success changes in these games when they are decomposed in two component games. Among others, we decompose a pure coordination game into two battle of the sexes components and a battle of the sexes game in one pure coordination component and one battle of the sexes component. In line with narrow bracketing, we observe that the game decompositions are behaviourally relevant. We find that coordination success increases and decreases depending on the type of decomposition and order of component games.