The interaction between learning at the individual level and the trajectory of a population over time is fundamental to our understanding of linguistic change. Here we use game theory as a mathematical framework for formulating and testing hypotheses about this interaction. We formalize two hypotheses regarding the spread of vowel mergers and use them to derive the proportion of merged in-migrants that would precipitate a merger in a previously non-merged speech community. We test these predictions against the documented spread of the low-back merger. In light of these results, we consider the impact of social network structures on both models.