We consider the problem of allocating a set of indivisible goods to a set of strategic agents with arbitrary weights. While truthful mechanism is hard to guarantee any non-trivial fairness, Amanatidis et al. (WINE 2021 and MOR 2024) studied the fairness guarantees of the equilibria of the Round-Robin mechanism. They show that when all agents have the same weight, every pure Nash equilibrium of Round-Robin leads to an envy-free up to one item (EF1) allocation, with respect to agents’ true valuations. In this paper, we investigate the weighted setting where agents are asymmetric and study the weighted picking sequence protocol, which is a natural extension of Round-Robin in the weighted setting. More specifically, we show that the weighted picking sequence protocol always has pure Nash equilibria and all the corresponding allocations are weighted EF1 with respect to the true valuation functions for two agents.