Thanks for this response. On notation: I want world-states, Wi, to be specific outcomes rather than random variables. As such, U(Wi) is a real number, and the expectation of a real number could only be defined as itself: E[U(Wi)]=U(Wi) in all cases. I left aside all the discussion of ‘lotteries’ in the VNM Wikipedia article, though maybe I ought not have done so.
I think your first two bullet points are wrong. We can’t reasonably interpret ~ as ‘the agent’s thinking doesn’t terminate’. ~ refers to indifference between two options, so if A>B>C and P ~ B, then A>P>C. Equating ‘unable to decide between two options’ and ‘two options are equally preferable’ will lead to a contradiction or a trivial case when combined with transitivity. I can cook up something more explicit if you’d like?
There’s a similar problem with ~ meaning ‘the agent chooses randomly’, provided the random choice isn’t prompted by equality of preferences.
This comment has sharpened my thinking, and it would be good for me to directly prove my claims above—will edit if I get there.
Thanks for this response. On notation: I want world-states, Wi, to be specific outcomes rather than random variables. As such, U(Wi) is a real number, and the expectation of a real number could only be defined as itself: E[U(Wi)]=U(Wi) in all cases. I left aside all the discussion of ‘lotteries’ in the VNM Wikipedia article, though maybe I ought not have done so.
I think your first two bullet points are wrong. We can’t reasonably interpret ~ as ‘the agent’s thinking doesn’t terminate’. ~ refers to indifference between two options, so if A>B>C and P ~ B, then A>P>C. Equating ‘unable to decide between two options’ and ‘two options are equally preferable’ will lead to a contradiction or a trivial case when combined with transitivity. I can cook up something more explicit if you’d like?
There’s a similar problem with ~ meaning ‘the agent chooses randomly’, provided the random choice isn’t prompted by equality of preferences.
This comment has sharpened my thinking, and it would be good for me to directly prove my claims above—will edit if I get there.