Bad apples in symmetric repeated games

We study large-population repeated games where players are symmetric but not anonymous, so player-specific rewards and punishments are feasible. Players may be commitment types who always take the same action. Even though players are not anonymous, we show that an anti-folk theorem holds when the commitment action is "population dominant," meaning that it secures a payoff greater than the population average payoff. For example, voluntary public goods provision in large populations is impossible when commitment types never contribute, even if monetary rewards can be targeted to contributors; however, provision is possible if non-contributors can be subjected to involuntary fines. A folk theorem under incomplete information provides a partial converse to our result. Along the way, we develop some general results on symmetric games with incomplete information and/or repeated play.