We continued to examine the Abreu-Pearce-Stacchetti (APS) operator, particularly focusing on the following two theorems.

Theorem1 (Necessity)
V* = LV*

Theorem2 (Sufficiency)
If V = LV (and V is bounded), then V is a subset of V*

where L is APS operator and V* is the set of SPE payoffs of the repeated game.

The proof of Theorem 1 is not difficult. We used "unimprovability" to prove Theorem 2. APS operator also establishes following results.

1. V* is compact
2. V* is increasing in the discount factor
3. APS operator is monotone

Using the third result with two theorems mentioned above, we can derive the algorithm to compute SPE payoffs. That is, starting with a large set of candidate equilibrium payoffs (say, a convex hull of the set of feasible payoffs), we just need to apply the APS operator iteratively until the sequence of sets will converge. Then, the limit must coincide with V*.