“Utilities as Legendre duals of probabilities” by Fernando Rosas

TLDR: In recent work, Roy Fox proposes to understand an agent's capabilities in terms of the set of environment dynamics it can bring about. This leads to an intriguing duality between probabilities and utilities via the Legendre-Fenchel transform.

Introduction

Some agents are more powerful than others. Indeed, some can yield a wider range of outcomes, maybe because they are capable long-term planners or because they have built rich world models. Being able to clearly delineate the capabilities of agents is an important challenge for AI alignment.

A natural place to start thinking about how to describe the capabilities of an agent is reinforcement learning (RL), or more generally, approaches that see behaviour as arising from the maximisation of expected utility. By taking this view, one can describe "capability" as the range of reward/utility functions that an agent can successfully maximise — as done e.g. in classic work by Legg & Hutter and also in more recent work.

Such a perspective is very useful, but I am not a big fan of rewards/utilities. Rewards are great in games and other settings where they come naturally, but real life often does not handle rewards on a silver plate. When absent [...]

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Outline:

(00:27) Introduction

(03:31) Defining capability space

(05:37) The Legendre-Fenchel transform

(07:58) Utilities as Legendre duals of probabilities

(10:08) Conclusion

The original text contained 9 footnotes which were omitted from this narration.

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First published:
August 23rd, 2026

Source:
https://www.lesswrong.com/posts/ALmBydH53DE3dSzCh/utilities-as-legendre-duals-of-probabilities

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Narrated by TYPE III AUDIO.

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