The following is an adaptation of part of my (unsuccessful) application to the Winter 2022 cohort of SERI MATS, as prompted by Victoria Krakovna’s “Explain the power-seeking theorems in the reinforcement learning setting in your own words”
I realise that nowadays, high-level summaries of this kind provide little value as summarizing bots have reached a very acceptable level. I am publishing to get into the habit of posting more.
So, Epistemic Status: Rendered obsolete by ChatGPT et al.
An action or behaviour is instrumental to an objective when it helps achieve that objective. When this action or behaviour is instrumental to a range of objectives, it said to be convergently instrumental.
The power seeking theorem outlined in  formalises the notion that power seeking is convergently instrumental. That is, it mathematically shows that power seeking is helpful in achieving a wide range of objectives under certain conditions, by considering the power-seeking tendencies of optimal policies in finite MDPs.
Here, power is formalised as the ability to achieve a wide variety of goals. To “seek” power, an action leads an agent to a state with greater power. Mathematically, the authors express power as a modified version of average optimal value. That is, the optimal value averaged over a range of goals. Value is simply the expected return, or the expected sum of rewards when acting under a given policy. The optimal value is then the maximum achievable value, which is obtained when acting under an optimal policy. Note that a goal is formalised as the maximisation of a reward function.
The authors’ formalisation rests on the use of optimal policies as the representation of intelligent agents, and considers the setting of a Markov Decision Process (MDP), i.e. assuming the environment is fully observable. Under these premises, the authors show that certain graphical symmetries in MDPs cause optimal policies to tend to seek power. They note that optimal policies will avoid visiting terminal states when possible, and prefer states with higher optionality.
 A. Turner, L. Smith, R. Shah, A. Critch, and P. Tadepalli, ‘Optimal Policies Tend To Seek Power’, in Advances in Neural Information Processing Systems, 2021, vol. 34, pp. 23063–23074. [Online]. Available: https://proceedings.neurips.cc/paper/2021/hash/c26820b8a4c1b3c2aa868d6d57e14a79-Abstract.html