First-best collusion without communication

I study a 2-bidder infinitely repeated IPV first-price auction without transfers, communication, or public randomization, where each bidder’s valuation can assume, in each of the (statistically independent) stage games, one of three possible values. If the probability of the high valuation is sufficiently large and the medium valuation is between one third and one half of the high valuation, then the following is true: for every > 0 there is a nondegenerate interval ∆( ) ⊂ (0, 1), such that if the bidders’ discount factor belongs to ∆( ), then there exists a Perfect Public Equilibrium with payoffs -close to the first-best payoffs.