# Should the “value function” be “utility function” in prospect theory?

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.

• There is a degree of rationality underlying theories based on utility challenged by prospect theory, which suggests the framing of choices affects decisions beyond mere calculations of expected utility. In particular outcomes (good or bad) known to have low probabilities are treated as being more likely to happen than they actually are, which is why the same people may buy insurance and lottery tickets even in cases when neither makes much sense – Henry Jul 8 '20 at 23:28
• If my memory serves me correctly, they never use the term value to denote the expectation of the random variable in the paper. They always call it expectation. So the term value is reserved to represent the preference of the Decision Maker (the terms utility, value, payoff are used interchangeably in different papers depending on the context). P.S - your understanding of the key difference is correct. – user28372 Jul 8 '20 at 23:46

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 $$x$$.