Johannes Treutlein and Rubi Hudson worked on this post while participating in SERI MATS, under Evan Hubinger's and Leo Gao's mentorship respectively. We are grateful to Marius Hobbahn, Erik Jenner, and Adam Jermyn for useful discussions and feedback, and to Bastian Stern for pointing us to relevant related work. Update 30 May 2023: We have now published a paper based on this post. In this paper, we also discuss in detail the relationship to the related literature on performative prediction. Introduction One issue with oracle AIs is that they might be able to influence the world with their predictions. For example, an AI predicting stock market prices might be able to influence whether people buy or sell stocks, and thus influence the outcome of its prediction. In such a situation, there is not one fixed ground truth distribution against which the AI's predictions may be evaluated. Instead, the chosen prediction can influence what the model believes about the world. We say that a prediction is a self-fulfilling prophecy or a fixed point if it is equal to the model's beliefs about the world, after the model makes that prediction. If an AI has a fixed belief about the world, then optimizing a strictly proper scoring rule incentivizes it to output this belief (assuming the AI is inner aligned to this objective). In contrast, if the AI can influence the world with its predictions, this opens up the possibility for it to manipulate the world to receive a higher score. For instance, if the AI optimizes the world to make it more predictable, this would be dangerous, since the most predictable worlds are lower entropy ones in which humans are more likely dead or controlled by a misaligned AI. Optimizing in the other direction and making the world as unpredictable as possible would presumably also not be desirable. If, instead, the AI selects one fixed point (of potentially many) at random, this would still involve some non-aligned optimization to find a fixed point, but