I will present a model of language evolution, based on population game dynamics with learning. Specifically, we consider the case of two genetic vari- ants of universal grammar (UG), the heart of the human language faculty, assuming each admits two possible grammars. The dynamics are driven by a communication game. We prove using dynamical systems techniques that if the payoff matrix obeys certain constraints, then the two UGs are stable against invasion by each other, that is, they are evolutionarily stable. These constraints are independent of the learning process. Intuitively, if a muta- tion in UG results in grammars that are incompatible with the established languages, then it will die out because individuals with the mutation will be unable to communicate and therefore unable to realize any potential benefit of the mutation. An example for which the proofs do not apply shows that compatible mutations may or may not be able to invade, depending on the population's history and the learning process. These results suggest that the genetic history of language is constrained by the need for compatibility and that mutations in the language faculty may have died out or taken over depending more on historical accident than on any simple notion of relative fitness. Further results in language game dynamics will require more detailed knowledge of the acquisition process. Following that line of thought, I will also sketch some recent work on a more detailed simulation for studying lan- guage change on historical (rather than geological) timescales. Preliminary results suggest that the human race may have only explored a small part of the space of possible languages.