Secretary Selection When Ties Are Possible

Three items arrive in uniformly random order. Their qualities are NOT all distinct: two of them have quality 2 (tied for best) and one has quality 1. After each item you observe its quality relative to those seen so far, reported as 'higher', 'tied', or 'lower' (so a tie is visible). You accept irrevocably or reject (the last is forced). You want to maximize the expected quality of the item you accept. Find the optimal policy and the maximum expected quality, and explain how the possibility of an observed tie changes what you can guarantee.