# Tit-for-Tat Isn’t Subgame Perfect

The previous lecture introduced tit-for-tat as a punishment strategy that can inspire cooperation. It also explained some desirable qualities of the strategy. However, we only showed that it was not optimal for the players to deviate from the equilibrium path of play for sufficiently great patience. What we did not show is whether the punishment strategies are credible. This means that we know that tit-for-tat is a Nash equilibrium, but we do not yet know whether it is a subgame perfect equilibrium. Here, we see that it is only subgame perfect for a single value of the discount factor—an unrealistic knife-edge condition.

Takeaway Points

1. If a player deviates from tit-for-tat, it begins a sequence where the players alternate back and forth with one cooperating and one defecting in each period.
2. Imagine that the player who is supposed to defect in a period instead deviates to cooperating. This ends the sequence of flip-flopping and instead re-instills cooperation in every period.
3. To not want to do deviate from the punishment strategy in this way, the player needs to be sufficiently impatient. It just so happens that the condition necessary for this is the flipped inequality necessary for the player to want to cooperate on the equilibrium path. As such, the conditions for a subgame perfect equilibrium are knife-edge.

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