Referential games and reconstruction games are the most common game types for studying emergent languages. We investigate how the type of the language game affects the emergent language in terms of: i) language compositionality and ii) transfer of an emergent language to a task different from its origin, which we refer to as language expressivity. With empirical experiments on a handcrafted symbolic dataset, we show that languages emerged from different games have different compositionality and further different expressivity.