Did you know that the concept of utility was originally used to describe preferences over choices, not states of the world? Might be worth reading up on. One important idea is that you can write preferences entirely in a language of what decisions you make, not how much you value different amounts of X. This is the language that circular preferences can be written in. Pop quiz: if your utility was the angle of a wheel (so, if it turns twice, your utility is 4 pi), could you prove that writing utility in terms of the physical state of the wheel breaks down, but that you can still always have preferences over choices?
Did you know that the concept of utility was originally used to describe preferences over choices, not states of the world? Might be worth reading up on. One important idea is that you can write preferences entirely in a language of what decisions you make, not how much you value different amounts of X. This is the language that circular preferences can be written in. Pop quiz: if your utility was the angle of a wheel (so, if it turns twice, your utility is 4 pi), could you prove that writing utility in terms of the physical state of the wheel breaks down, but that you can still always have preferences over choices?