Purification and Perturbations of Communication and Repeated Games

I prove that it is irrational for agents with even slightly private preferences to condition their strategy on private information that is payoff-irrelevant to them, contrary to powerful techniques for analyzing communication and repeated games. In repeated games with public+private monitoring, this means all pure equilibria are perfect public equilibria, and non-trivial belief free equilibria do not exist. In a wide class of communication games (up to allowing receiver commitment), this means persuasion is impossible with state-independent preferences. Nevertheless, these analytic techniques may be made compatible with private preferences through perturbation approaches: considering either payoff-relevance of the private information or correlation between parties' information. An example of the latter occurs by introducing `atonement' to repeated games equilibria.