I have a background in mathematics rather than economics, and currently reading Choices, Values, and Frames[1]. The paper defines a "hypothetical value function" (the s-shape that is concave for gains, convex for losses).
Given the definition of value to be descriptive, and utility to be normative, should it not be a "hypothetical utility function" instead?
The expected value of a risky decision is defined by the sum of all of weighted possible outcomes, from Wikipedia e.g.:
$${\displaystyle \operatorname {E} [X]=\sum _{i=1}^{k}x_{i}\,p_{i}=x_{1}p_{1}+x_{2}p_{2}+\cdots +x_{k}p_{k}}$$
The corresponding expected utility is then:
$${\displaystyle \operatorname {E} [u(x)]=\sum _{i=1}^{k}u(x_{i})\,p_{i}=u(x_{1})p_{1}+u(x_{2})p_{2}+\cdots +u(x_{k})p_{k}}$$
I understand (incorrectly?) one key difference between expected utility theory and prospect theory is in the way that $u$ is constructed - the former dependent on the total wealth, and the latter dependent on the gain/loss of the change itself. Nevertheless, they are both dealing with the "satisfaction" of a gamble, and not the expected value, which does not change - hence my question.
I feel that I'm missing a trick here. Any illumination would be greatly appreciated!
[1] Kahneman, D., Tversky, A., 1984. Choices, Values, and Frames. American Psychologist, Choices, Values, and Frames 39, 341–350.