<p>The terminology in microeconomics is not completely unified but typically differs slightly from the mathematical one. For a real-valued random (outcome) variable <span class="math-container">$X$</span>, the mathematical <em>expected value</em> <span class="math-container">$\mathbb{E}(X)$</span> would rather be called the <em>expectation</em> of <span class="math-container">$X$</span>. The <em>utility</em> <span class="math-container">$u(x)$</span> of an outcome <span class="math-container">$x$</span> is understood to be given by a (Bernoulli) utility function in the sense of EUT, i.e. a function <span class="math-container">$u(.)$</span> such that the expected utility <span class="math-container">$\mathbb{E}(u(.))$</span>, sometimes written as a (von-Neumann-Morgenstern) utility function <span class="math-container">$U(.)$</span> on lotteries on outcomes, represents the preferences over lotteries on outcomes, while in behavioral economics the term <em>value</em> or <em>valuation</em> <span class="math-container">$v(x)$</span> typically represents some kind of subjective valuation of <span class="math-container">$x$</span>, as e.g. in prospect theory. (So here, <em>value</em> is not to be understood as the mere numerical value of <span class="math-container">$x$</span>, as in the usual mathematical definition of a &quot;function value&quot; or an &quot;expected value&quot;.)</p>
<p>Confusingly, however, <span class="math-container">$v(.)$</span> 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 <span class="math-container">$u$</span> and the other by <span class="math-container">$v$</span>. Moreover, in a utilitarian framework, <span class="math-container">$v(.)$</span> is often used to denote the <em>willingness to pay</em> for <span class="math-container">$x$</span>, i.e. a monetary valuation of <span class="math-container">$x$</span>.</p>

CC BY-SA 4.0
2020-07-15T12:33:40.223Z
Thanks - that's not something that is immediately obvious. Could it also be that strictly speaking, the "hypothetical value function as proposed in prospect theory is indeed _not_ a utility function in the VNM sense? i.e. in [1], they explicitly call out the inconsistencies in VNM-rationality "invariance and dominance", which I presume corresponds to completeness/transitivity and independence VNM axoims. Therefore what is proposed does not fit in the definition accepted in EUT?
thzp
CC BY-SA 4.0
2020-07-16T10:21:33.890Z
@thzp, Yes, this "hypothetical value function" is definitely not a vNM-utility function. E.g. its argument is set relative to a reference point and can therefore be negative ("losses"). A vNM-utility function does not distinguish between gains and losses, its argument is just wealth (or some other nonnegative outcome quantity).
VARulle
CC BY-SA 4.0
2020-07-09T09:53:12.737Z
2020-07-09T09:53:12.737Z
VARulle
<p>I have a background in mathematics rather than economics, and currently reading Choices, Values, and Frames[1]. The paper defines a &quot;hypothetical value function&quot; (the s-shape that is concave for gains, convex for losses).</p>
<p>Given the definition of value to be descriptive, and utility to be normative, should it not be a &quot;hypothetical <em>utility</em> function&quot; instead?</p>
<p>The expected value of a risky decision is defined by the sum of all of weighted possible outcomes, from Wikipedia e.g.:</p>
<p><span class="math-container">$${\displaystyle \operatorname {E} [X]=\sum _{i=1}^{k}x_{i}\,p_{i}=x_{1}p_{1}+x_{2}p_{2}+\cdots +x_{k}p_{k}}$$</span></p>
<p>The corresponding expected utility is then:</p>
<p><span class="math-container">$${\displaystyle \operatorname {E} [u(x)]=\sum _{i=1}^{k}u(x_{i})\,p_{i}=u(x_{1})p_{1}+u(x_{2})p_{2}+\cdots +u(x_{k})p_{k}}$$</span></p>
<p>I understand (incorrectly?) one key difference between expected utility theory and prospect theory is in the way that <span class="math-container">$u$</span> 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 &quot;satisfaction&quot; of a gamble, and not the expected value, which does not change - hence my question.</p>
<p>I feel that I'm missing a trick here. Any illumination would be greatly appreciated!</p>
<p>[1] Kahneman, D., Tversky, A., 1984. Choices, Values, and Frames. American Psychologist, Choices, Values, and Frames 39, 341–350.</p>

CC BY-SA 4.0
2020-07-08T23:28:18.437Z
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
<behavioral-economics><expected-utility>
Should the "value function" be "utility function" in prospect theory?
thzp
# Should the "value function" be "utility function" in prospect theory?

(Asked by: username_0 on 2020-07-08)

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.

**Tags:** behavioral-economics, expected-utility

> **Comments:**
> **username_1**: 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  

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## 1 answers

### Answer by username_2 on 2020-07-09. Score: 3 | Accepted answer

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 $x$.

> **Comments:**
> **username_0**: Thanks - that's not something that is immediately obvious. Could it also be that strictly speaking, the "hypothetical value function as proposed in prospect theory is indeed _not_ a utility function in the VNM sense? i.e. in [1], they explicitly call out the inconsistencies in VNM-rationality "invariance and dominance", which I presume corresponds to completeness/transitivity and independence VNM axoims. Therefore what is proposed does not fit in the definition accepted in EUT?  
> **username_2**: @username_0, Yes, this "hypothetical value function" is definitely not a vNM-utility function. E.g. its argument is set relative to a reference point and can therefore be negative ("losses"). A vNM-utility function does not distinguish between gains and losses, its argument is just wealth (or some other nonnegative outcome quantity).  

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