  The probabilistic serial (PS) rule is one of the most prominent randomized
rules for the assignment problem. It is well-known for its superior fairness
and welfare properties. However, PS is not immune to manipulative behaviour by
the agents. We examine computational and non-computational aspects of
strategising under the PS rule. Firstly, we study the computational complexity
of an agent manipulating the PS rule. We present polynomial-time algorithms for
optimal manipulation. Secondly, we show that expected utility best responses
can cycle. Thirdly, we examine the existence and computation of Nash
equilibrium profiles under the PS rule. We show that a pure Nash equilibrium is
guaranteed to exist under the PS rule. For two agents, we identify two
different types of preference profiles that are not only in Nash equilibrium
but can also be computed in linear time. Finally, we conduct experiments to
check the frequency of manipulability of the PS rule under different
combinations of the number of agents, objects, and utility functions.
