# Is a policy $\pi(s)$ on Markov decision process a random variable?

Citing Wikipedia:

The goal in a Markov decision process is to find a good "policy" for the decision maker: a function $$\pi$$ that specifies the action $$\pi(s)$$ that the decision maker will choose when in state $$s$$. Once a Markov decision process is combined with a policy in this way, this fixes the action for each state and the resulting combination behaves like a Markov chain (since the action chosen in state $$s$$ is completely determined by $$\pi(s)$$ and $$\Pr(s_{t+1}=s'\mid > s_{t}=s,a_{t}=a)$$ $$\Pr(s_{t+1}=s'\mid s_{t}=s,a_{t}=a)$$ reduces to $$Pr(s_{t+1}=s'\mid s_{t}=s)$$ $$Pr(s_{t+1}=s'\mid s_{t}=s)$$, a Markov transition matrix).

A policy that maximizes the function above is called an optimal policy and is usually denoted $$\pi^*$$. A particular MDP may have multiple distinct optimal policies. Because of the Markov property, it can be shown that the optimal policy is a function of the current state, as assumed above.

Can the policy be considered a random variable?

• The answer is in the question: "the optimal policy is a function of the current state". Oct 2, 2020 at 6:38
• A function of a random variable is a random variable? Oct 2, 2020 at 14:25