Should the "value function" be "utility function" in prospect theory?

Question

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.

Answer 1

The terminology in microeconomics is not completely unified but typically differs slightly from the mathematical one. For a real-valued random (outcome) variable $X$, the mathematical expected value $\mathbb{E}(X)$ would rather be called the expectation of $X$. The utility $u(x)$ of an outcome $x$ is understood to be given by a (Bernoulli) utility function in the sense of EUT, i.e. a function $u(.)$ such that the expected utility $\mathbb{E}(u(.))$, sometimes written as a (von-Neumann-Morgenstern) utility function $U(.)$ on lotteries on outcomes, represents the preferences over lotteries on outcomes, while in behavioral economics the term value or valuation $v(x)$ typically represents some kind of subjective valuation of $x$, as e.g. in prospect theory. (So here, value is not to be understood as the mere numerical value of $x$, as in the usual mathematical definition of a "function value" or an "expected value".)

Confusingly, however, $v(.)$ is sometimes also used to denote an alternative (Bernoulli) utility function. For example when explaining that a positive linear transformation of a utility function is again a utility function representing the same preferences, one of them is usually denoted by $u$ and the other by $v$. Moreover, in a utilitarian framework, $v(.)$ is often used to denote the willingness to pay for $x$, i.e. a monetary valuation of $x$.