Games with Noisy Signals About Emotions

We formalize a novel framework allowing for the observation of noisy signals about co-players' emotions, or states of mind. Insofar as the latter are belief-dependent, such feedback allows players to draw inferences informing their strategic thinking. First, we give a definition of players' rationality: we require that rational players hold coherent beliefs, that they update their beliefs consistently with the evidence and according to Bayes rule, that they plan optimally, and that they implement their plans. Under mild and reasonable technical assumptions, we prove that the set of states of the world in which a player is rational is an event (a Borel set). Second, we analyze players' strategic reasoning adapting the strong rationalizability solution concept, and we give its epistemic justification in terms of players' rationality and interactive beliefs. The "forward-induction" reasoning entailed by such solution allows players to make inferences about their co-players' beliefs, private information, and future, or past and unobserved behavior based on the behavioral and emotional feedback they obtain as the game unfolds. We illustrate our framework with a signaling-like example, showing that the possibility of betraying lies, e.g., by blushing, may incentivize truth-telling.