Natural Language Equilibrium I: Off-Path Conventions

We incorporate natural language into games, focusing here on the class of signaling games. The sender, using a commonly understood language, can make cheap-talk statements about the strategy that he is using. Because the sender knows his strategy, any statement that he makes is either true according to its literal meaning or is intentionally false or deceptive. It is shown that if the receiver interprets any off-path statement by the sender as true unless it may be seen as a rational attempt to deceive, then the only outcomes of the game without language that survive the introduction of language are, generically, those that are stable in the sense of Kohlberg and Mertens (1986). Incorporating language into game theory can thus reap significant benefits, with the potential to significantly refine equilibrium predictions in ways that are more intuitive and more easily justified than when language is absent.