Repeated Sequential Prisoner's Dilemma: The Stackleberg Variant

Abstract

We study the Stackleberg variant of the repeated Sequential Prisoner’s Dilemma (SPD for short). The game goes in two stages, and the two players, the leader and the follower, are asymmetric in both stages. In the first stage of the game, the leader chooses a strategy (for the repeated SPD of the second stage), which is immediately known to the follower. In the second stage, they play repeated SPD: in each round the follower moves after observing the leader’s action.

Assuming complete rationality, we find some extraordinary properties of this model. (i) The (sub-game perfect) equilibrium payoff profile is unique, which lies on the corner of the region predicted by classical folk theorems: it is best for the leader and at the same time worst for the follower. (ii) The leader has simple optimal strategies that are one-step memory and stationary. These features are in great contrast with classical results, where either uniqueness cannot be guaranteed and equilibrium strategies are often quite complicated, or bounded rationality is required. Although full cooperation, i.e. the outcome is always (cooperate, cooperate), is not attainable in our model, at least a half of the optimal social welfare can be guaranteed. We also do a non-equilibrium analysis which makes the usual equilibrium analysis more convincing.

Keywords: Prisoner's Dilemma, sequential repeated game, folk theorem, Markov decision process

JEL Classification: C72