We study a society inside which two official languages, the majority language A and the minority language B, are in contact and compete for the same social functions. We propose a non-cooperative game to capture some features of this competitive situation. In the game, there are two types of players: the bilingual one who speaks both A and B and the monolingual one who speaks only A. The information about which type is each player is private. A real life situation captured by the game is that in many interactions bilingual players must decide under incomplete information about which language to use. One implication of this information structure is that while A satisfies the main properties of a public good, B does not. Another implication is that it may have dangerous consequences on the language diversity of the society. We show that in many equilibria bilingual players fail to coordinate in their preferred language and end up using the majority language A.