Tight social welfare approximation of probabilistic serial

We study the problem of approximate social welfare maximization (without money) in assignment problems for agents who have cardinal preferences over a finite set of items. Probabilistic Serial (PS) mechanism is known to be ordinally efficient and envy-free. We prove that its social welfare approximation ratio is while no envy-free mechanism can achieve an approximation ratio better than . In addition, our proof also implies a rank approximation of PS, answering an open question posted in .