Justifiable punishments in repeated games

- We introduce justifiable punishments in infinitely repeated games.
- We characterize the set of payoffs that can be supported by subgame perfect equilibria with justifiable punishments.
- Justifiable punishments may significantly restrict what can be supported by equilibria.
- Any efficient perfect equilibrium payoff can also be supported by equilibrium with justifiable punishments.
- Any weakly renegotiation-proof equilibrium payoff can also be supported by equilibrium with justifiable punishments.

In repeated games, subgame perfection requires all continuation strategy profiles must be effective to enforce the equilibrium; they serve as punishments should deviations occur. It does not require whether a punishment can be justified for the deviation, which creates a great deal of freedom in constructing equilibrium strategies, resulting the well-known folk theorem. We introduce justifiable punishments in repeated games. After one player deviates, the corresponding continuation or punishment is justifiable if either the deviation is bad to the other player or the continuation itself is good to the other player. We characterize the set of payoff vectors that can be supported by subgame perfect equilibria with justifiable punishments, as the discount factor goes to one. This limiting set of equilibrium payoffs can be quite different from the set of subgame perfect equilibrium payoffs. Any efficient, feasible, and strictly individually rational payoff can be supported by equilibrium with justifiable punishments.