Penance strategy game theory

Question

I am having some difficulty in understanding the following use of the penance strategy as a strategy to support compliance in the repeated game presented in Asheim et al. (2006). I understand penance in the following way:

Begin in the cooperative phase where both states choose compliance; if State A then defects, the other State B immediately switches to defect for all subsequent periods until and including the first period that State A reverts to compliance. After that period, State B also reverts to compliance

Now, in their work Asheim et al. state the following:

We assume that compliance in the global agreement is sustained by way of the "Penance" strategy. This strategy specifies that a participating country plays cooperate except if another participating country has been the sole deviator from Penance in the previous period, in which case defect is played.

My doubt has to do with the use of the word "sole". Specifically, following the penance strategy in a game with 3 or more players, is the punishment the same whether a single player deviates or more than one does?

Answer 1

I see two ways to produce histories with multi-state deviations (not including the degenerate case in which all states defect):

1. Multiple states defect in different periods. This possibility is ruled out so long as Equation 3 (the necessary and sufficient condition for unilateral defection at time $t-1$ to be punished by all other states at time $t$) is satisfied - in that case, only one state can ever defect at a time because all other states have credibly committed to Penance.
2. Multiple states defect in the same period. There are two obvious ways this can happen: 1) regional agreements are in effect and one of the regions is punishing a defection. Equation 7 gives the conditions required for both regions to revert to Regional Penance in the following period, so this case isn't interesting.

The other way to have multiple states defect in the same period is the interesting case. It implies collusion between a subset of states. This reduces to the Regional Penance case if the subset is equal to half the states. If it isn't, Footnote 10 explains that such an arrangement is not weakly renegotiation-proof. This makes sense, as collusion implies that (Regional) Penance is not, in fact, subgame perfect.