= Average-reward optimal policy
{title2=$g=\liminf_{n\to\infty}n^{-1}\mathbb E\sum_{j=1}^nr_j$}
An average-reward optimal policy maximizes long-run expected reward per time step. In a finite recurrent <Markov chain>, a stationary policy's gain is its expected one-step reward under the stationary distribution. Initial-state effects are encoded by a bias function rather than by the gain. In a communicating finite control problem, the <average-reward Bellman equation> certifies a policy that is optimal from every initial state.
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