Some researchers speculate that intelligent reinforcement learning (RL) agents would be incentivized to seek resources and power in pursuit of their objectives. Other researchers are skeptical, because human-like power-seeking instincts need not be present in RL agents. To clarify this debate, we develop the first formal theory of the statistical tendencies of optimal policies in reinforcement learning. In the context of Markov decision processes, we prove that certain environmental symmetries are sufficient for optimal policies to tend to seek power over the environment. These symmetries exist in many environments in which the agent can be shut down or destroyed. We prove that for most prior beliefs one might have about the agent’s reward function (including as a special case the situations where the reward function is known), one should expect optimal policies to seek power in these environments. These policies seek power by keeping a range of options available and, when the discount rate is sufficiently close to 1, by navigating towards larger sets of potential terminal states.