  Policy Iteration (PI) is a classical family of algorithms to compute an
optimal policy for any given Markov Decision Problem (MDP). The basic idea in
PI is to begin with some initial policy and to repeatedly update the policy to
one from an improving set, until an optimal policy is reached. Different
variants of PI result from the (switching) rule used for improvement. An
important theoretical question is how many iterations a specified PI variant
will take to terminate as a function of the number of states $n$ and the number
of actions $k$ in the input MDP. While there has been considerable progress
towards upper-bounding this number, there are fewer results on lower bounds. In
particular, existing lower bounds primarily focus on the special case of $k =
2$ actions. We devise lower bounds for $k \geq 3$. Our main result is that a
particular variant of PI can take $\Omega(k^{n/2})$ iterations to terminate. We
also generalise existing constructions on $2$-action MDPs to scale lower bounds
by a factor of $k$ for some common deterministic variants of PI, and by
$\log(k)$ for corresponding randomised variants.
