A couple of random thoughts. From the point of view on prior+utility as vectors in probability-shouldness coordinates, it's easy to see that the ability to rescale and shift utilities without changing preference corresponds to transformations to the shouldness component. These transformations don't change the order on vectors' (events') angles, and so even if we allow shouldness to go negative, expected utility will still work as preference. Similarly, if the shouldness is fixed positive, one could allow rescaling and shifting probability, so that it, too, can go negative.

Another transformation: if we swap the roles of probability and shouldness, the resulting prior+utility will have shouldness of the original system as prior and inverse utility of the original system as utility. In this system, expected utility minimization will describe the same optimization as the expected utility maximization in the original system. The same effect could be achieved by flipping the sign on utility (another symmetry), which can also be easily seen from the probability-shouldness diagram.

Applying both transformations, we get the same preference, but with shouldness of the original system as prior. Utility of the transformed system is negated inverted utility of the original representation. This shows that conceptually, probability distribution and shouldness distribution are interchangeable.

## Comments (35)

Best*0 points [-]A couple of random thoughts. From the point of view on prior+utility as vectors in probability-shouldness coordinates, it's easy to see that the ability to rescale and shift utilities without changing preference corresponds to transformations to the shouldness component. These transformations don't change the order on vectors' (events') angles, and so even if we allow shouldness to go negative, expected utility will still work as preference. Similarly, if the shouldness is fixed positive, one could allow rescaling and shifting

probability, so that it, too, can go negative.Another transformation: if we swap the roles of probability and shouldness, the resulting prior+utility will have shouldness of the original system as prior and inverse utility of the original system as utility. In this system, expected utility

minimizationwill describe the same optimization as the expected utility maximization in the original system. The same effect could be achieved by flipping the sign on utility (another symmetry), which can also be easily seen from the probability-shouldness diagram.Applying both transformations, we get the same preference, but with shouldness of the original system as prior. Utility of the transformed system is negated inverted utility of the original representation. This shows that conceptually, probability distribution and shouldness distribution are interchangeable.