  We construct a family of Markov decision processes for which the policy
iteration algorithm needs an exponential number of improving switches with
Dantzig's rule, with Bland's rule, and with the Largest Increase pivot rule.
This immediately translates to a family of linear programs for which the
simplex algorithm needs an exponential number of pivot steps with the same
three pivot rules. Our results yield a unified construction that simultaneously
reproduces well-known lower bounds for these classical pivot rules, and we are
able to infer that any (deterministic or randomized) combination of them cannot
avoid an exponential worst-case behavior. Regarding the policy iteration
algorithm, pivot rules typically switch multiple edges simultaneously and our
lower bound for Dantzig's rule and the Largest Increase rule, which perform
only single switches, seem novel. Regarding the simplex algorithm, the
individual lower bounds were previously obtained separately via deformed
hypercube constructions. In contrast to previous bounds for the simplex
algorithm via Markov decision processes, our rigorous analysis is reasonably
concise.

Discrete Mathematics;Computational Complexity;Data Structures and Algorithms;Combinatorics