This note extends previous convergence results for dynamic matching and bargaining games to asymmetric information and extreme bargaining power. In a decentral market heterogeneous traders are pairwise matched and sellers make take-it-or-leave-it offers. Traders have a finite life expectancy, exiting the market with a constant hazard rate d. With vanishing d it is shown that an equilibrium exists and that the market converges to the efficient competitive outcome.