  We investigate the geometry of optimal memoryless time independent decision
making in relation to the amount of information that the acting agent has about
the state of the system. We show that the expected long term reward, discounted
or per time step, is maximized by policies that randomize among at most $k$
actions whenever at most $k$ world states are consistent with the agent's
observation. Moreover, we show that the expected reward per time step can be
studied in terms of the expected discounted reward. Our main tool is a
geometric version of the policy improvement lemma, which identifies a
polyhedral cone of policy changes in which the state value function increases
for all states.
