<p><strong>tl;dr:</strong> Short answer is that cardinal utility of rational person is in large part of a literature is derived from the von Neumann and Morgenstern expected cardinal utility framework. In such framework utility must be bounded by our definition of what rationality is. So in such framework the short answer is simply that utility has to be bounded for person to be considered rational.</p>
<p>There are also expected utility frameworks that allow expected utility to be unbounded (see review by <a href="https://www.jstor.org/stable/1886093?seq=1#metadata_info_tab_contents" rel="nofollow noreferrer">Fishburn, 1976)</a>. However, these are not very widely used as they can often lead to paradoxes and do not seem to offer any special insight. I suppose this is due to the strong influence of instrumentalism on economic thinking. A researcher would prefer a utility framework that is able to provide some testable predictions rather one that just results in a paradox and hence offers no useful testable prediction. So even if someone could perhaps consider concept of unbounded utility more elegant its instrumental value in doing research would be close to none (if in particular case it leads to unresolvable paradox).</p>
<p><strong>Full Answer:</strong></p>
<p>The utility needs to be bounded in order to avoid paradoxes such as the <a href="https://en.wikipedia.org/wiki/St._Petersburg_paradox" rel="nofollow noreferrer">St. Petersburg paradox</a> (for more nuanced overview see this entry in <a href="https://plato.stanford.edu/entries/paradox-stpetersburg/" rel="nofollow noreferrer">Stanford encyclopedia of philosophy</a>). As a matter of fact a rational person's utility should be bounded as suggested by <a href="https://www.worldcat.org/title/essays-in-the-theory-of-risk-bearing/oclc/152690" rel="nofollow noreferrer">Arrow (1970)</a> precisely in reference to the paradox above.</p>
<p>More general actually cardinal utility will be bounded by the axioms that were used to derive the expected cardinal utility in the first place. Following Neumann and Morgenstern (1947) Theory of Games and Economic Behavior,  an expected utility from gamble can be described by the so called von Neuman-Morgenstern equation:</p>
<p><span class="math-container">$$E[u(g_i)] = \sum_j u(X_{ij})p_{ij}$$</span></p>
<p>where <span class="math-container">$u$</span> is utility <span class="math-container">$g_i$</span> is gamble <span class="math-container">$X$</span> is an outcome and <span class="math-container">$p$</span> is a probability. Furthermore, for the above to be utility we must have some continuum of gambles for which:</p>
<p><span class="math-container">$$g_i,g_j \in \mathbf{G}: g_j \succeq g_i \implies E[u(g_j)]\geq E[u(g_i)] $$</span></p>
<p>Now given this we can ask (as the author of the paper did) what would be properties of such utility function?</p>
<p>Now it turns out that basic rationality constraint where preferences satisfy transitivity, completeness, continuity and independence imply that the utility has to be bounded.</p>
<p>The completeness axiom states:</p>
<p><span class="math-container">$$ \forall x,y \in \mathbf{X}: x \succeq y \vee y\succeq x \vee y \thicksim x $$</span></p>
<p>that is we can order all our options in terms of preference in some way.</p>
<p>The transitivity axiom states:</p>
<p><span class="math-container">$$ \forall x,y,z \in \mathbf{X}, \text{ if } x \succeq y \wedge y \succeq z \implies x \succeq z$$</span></p>
<p>So if someone likes <span class="math-container">$x$</span> more than <span class="math-container">$y$</span> and <span class="math-container">$y$</span> more than <span class="math-container">$z$</span> then <span class="math-container">$x$</span> must be preferable to <span class="math-container">$z$</span></p>
<p>The continuity  <span class="math-container">$z \succeq y \succeq x$</span>, then there must be some probability <span class="math-container">$p$</span> such:</p>
<p><span class="math-container">$${px,(1-p)z} \thicksim y $$</span></p>
<p>This implies that no outcome <span class="math-container">$x$</span> is so terrible that you would not take up some gamble involving <span class="math-container">$x$</span>.</p>
<p>by independence axiom if <span class="math-container">$y \succeq x$</span> then for <span class="math-container">$z$</span> and some probability <span class="math-container">$p$</span></p>
<p><span class="math-container">$${px,(1-p)z} \preceq {py,(1-p)z} $$</span></p>
<p>This axiom states that if two outcomes have the same probability, we should evaluate the two alternatives independently of what we think that the outcome is.</p>
<p>The above axioms are how we define rationality in expected cardinal utility. There are of course different possible specifications of utility functions but most modern research relies on the von Neuman-Morgenstern type (or related utility functions).</p>
<p>Now these rationality requirements - which are <em>axioms</em> so they are by definition what rationality is in this context, simply demand that no outcome can produce infinite utility. In order to see this we can try to do proof by contradiction - suppose that there is a gamble wehere: <span class="math-container">$x = 1€$</span>, <span class="math-container">$y=100€$</span> and <span class="math-container">$z= \infty €$</span> and that <span class="math-container">$u(X)=x$</span>. In this case clearly <span class="math-container">$z \succ y \succ x$</span> but there is no <span class="math-container">$p$</span> for which the continuity axiom holds. Since the continuity axiom would be violated our basic axioms of what rationality is would not hold and a person with unbounded utility would cease to be rational (within the context of von Neumann and Morgenstern framework).</p>

CC BY-SA 4.0
2020-10-11T22:27:16.927Z
Many thanks for the literature! Two questions. First, in your paragraph 2, what are the "paradoxes" besides St Petersburg paradox? Second, in you last paragraph you proved that $z\neq\infty$. But by "unbounded" I guess people mean that $z\in (-\infty,+\infty)$ and $u\in(-\infty,+\infty)$? That is, the utility cannot be infinity but it can arbitrarily approaches infinity.
High GPA
CC BY-SA 4.0
2020-10-11T22:29:51.277Z
I think even if the utility is unbounded, the St. Petersburg paradox might not be a problem, as realizing the final payoffs in St. Petersburg paradox take infinite amount of time, and agents can have a time discount factor. The experimenter can never convince the agents that they can pay arbitrarily high amount of money, or they can compute the reward instantly.
High GPA
CC BY-SA 4.0
2020-10-11T22:35:34.630Z
@HighGPA 1. Paradox is any logical inconsistency most aren't named. 2. I could have used the same example with just making one of the options $-\infty$ I was not attempting to make a rigorous proof but rather just an example, but it works both ways - any infinity at either end just messes up with the continuity axiom. 3. You are right there are other ways of resolving St. Petersburg paradox - but in a sense they are often unsatisfactory. For example, one solution to St. Petersburg paradox is that we can just assume agents ignore small probabilities which might even be true but ...
1muflon1
CC BY-SA 4.0
2020-10-11T22:36:57.097Z
when you want to construct a rigorous model it would be unsatisfactory to just 'handwave' and say that agents ignore some arbitrary small probabilities. Also, just because this particular paradox would be resolved that does not mean other logical inconsistencies would not pop-up somewhere else. It again comes down back to instrumental value of a model.
1muflon1
CC BY-SA 4.0
2020-10-11T22:59:19.710Z
Many thanks for your teaching! I guess in the last paragraph, you proved that $z\neq +\infty$ and $z\neq -\infty$. However, "unbounded" usually mean that $z\to +\infty$, which is different from $z= +\infty$, to my limited knowledge.
High GPA
CC BY-SA 4.0
2020-10-11T23:22:06.253Z
@HighGPA in mathematics bounded function is a function for which there is some number A such that $|f(x)| \leq A$. Hence, if utility cannot be $\infty$ there will be some number A for which the above holds and function will be bounded. Furthermore, the continuity axiom would be violated even if there is a limit because limit of $\lim_{x\rightarrow \infty} x = \infty$
1muflon1
CC BY-SA 4.0
2024-09-03T21:42:47.487Z
@1muflon1 I still think this doesn't show "unbounded utility" implies there is an event with ∞ utility. Suppose for all z in R, there is an event with that utility. No matter which z you take, you can find a gamble to satisfy continuity.
martinkunev
CC BY-SA 4.0
2020-10-11T14:19:59.607Z
2020-10-11T14:46:58.830Z
2020-10-11T14:46:58.830Z
1muflon1
<p>For Expected Utility and SEU, people make axioms to ensure that the utility is bounded. However, I personally believe that the utility function must be unbounded, especially if we are considering cardinal utility. I'll explain my point in a though experiment.</p>
<p>Even if I have a diminishing marginal utility, there will be always some increase in money that will improve my utility by one unit. For example, it is possible that doubling money means one unit of utility for me, or it is possible that tripling money means one unit of utility for me. Either way, the range of my utility function must be unbounded. In a more rigorous language, if a utility function of money is bounded, then, when the money is approaching infinity, the utility function becomes arbitrarily close to a constant function, which violates non-satiation.</p>
<p>My question is, which economist argue that utility in real life must be unbounded, and who argue that a bounded utility is more normative, rational, or natural? Did economists reached a consensus that utility function must be bounded in real life? Any literature will be very helpful.</p>
<hr />
<p>Sources for bounded utility: EU: Bounded expected utility, PC Fishburn (1967)</p>

CC BY-SA 4.0
2020-10-11T13:32:16.643Z
"*For Expected Utility and SEU, people make axioms to ensure that the utility is bounded.*" Can you please post a source for this claim? It seems contrary to my experience.
Giskard
CC BY-SA 4.0
2020-10-11T13:33:29.503Z
"*Even if I have a diminishing marginal utility, there will be always some increase in money that will improve my utility by one unit.*" This is false, e.g., $u(x) = -1/x$ is a counterexample.
Giskard
CC BY-SA 4.0
2020-10-11T13:34:04.603Z
"*which economist believes that utility must be unbounded and who believes the other way?*" I am not sure what you mean by this, but I think if you leave it like this the question will be closed.
Giskard
CC BY-SA 4.0
2020-10-11T13:36:39.460Z
@Giskard My argument is, the bounded utility like $u(x)=-1/x$ does not always make sense. For example, if you already have 10k dollars, do you believe that receiving 1 billion dollars will not increase your utility by one unit?
High GPA
CC BY-SA 4.0
2020-10-11T13:37:15.387Z
What is a utility unit?
Giskard
CC BY-SA 4.0
2020-10-11T13:39:05.077Z
@Giskard utility function usually have a codomain of $\mathbb R$, so one unit of utility can be elicited provided that we are using cardinal utility like EU.
High GPA
CC BY-SA 4.0
2020-10-11T13:39:24.177Z
So, what exactly is a utility unit?
Giskard
CC BY-SA 4.0
2020-10-11T13:41:50.520Z
$u(x)-u(y)=1$ means that replacing $y$ with $x$ will increase my utility by one unit. If you have better definition please do share it; it will be very helpful
High GPA
CC BY-SA 4.0
2020-10-11T13:43:29.270Z
Sorry, but I am going to vote to close instead. Seems like your confusion is just about the basic nature of how utility functions work, or at least I don't understand how any of this is related to the other things in your question.
Giskard
CC BY-SA 4.0
2020-10-11T13:44:51.077Z
It is more like philosophy than measurement. For sure you can define that $1B$ only worth 0.9 unit of utility, this is totally fine. But for any reasonable person, if $1B$ worth 0.9 unit of utility, then $10000B$ must worth quite a bit more than 0.9 unit. Repeating this way you could add to infinite and thus get unbounded utility
High GPA
CC BY-SA 4.0
2020-10-11T13:47:00.440Z
I am afraid I fall outside of your definition of a reasonable person, or perhaps I misunderstand your point, or perhaps you are not very clear about what it is that you are asking.
Giskard
CC BY-SA 4.0
2020-10-11T13:53:35.800Z
@Giskard Let me make another example. One common definition of utility is enjoyment (for example see Loomes and Sugden 1987). As a very simple model, say if something will make you very happy, then it worth one unit of utility (aka enjoyment). If something will make you slightly happy, then it worth 0.1 unit of utility. If we agree that there is always something that make me happy, regardless of how much I already own, then my utility could go to infinity.
High GPA
CC BY-SA 4.0
2020-10-11T13:59:43.123Z
@Giskard By the way, it is totally fine if you believe that utility must be bounded; I never say that it is wrong. I am curious to learn your arguments and references (if any), like how did economists reached a consensus that utility function must be bounded.
High GPA
<macroeconomics><game-theory><reference-request><utility><decision-theory>
Why utility should be bounded (or unbounded)?
High GPA
# Why utility should be bounded (or unbounded)?

(Asked by: username_0 on 2020-10-11)

For Expected Utility and SEU, people make axioms to ensure that the utility is bounded. However, I personally believe that the utility function must be unbounded, especially if we are considering cardinal utility. I'll explain my point in a though experiment.

Even if I have a diminishing marginal utility, there will be always some increase in money that will improve my utility by one unit. For example, it is possible that doubling money means one unit of utility for me, or it is possible that tripling money means one unit of utility for me. Either way, the range of my utility function must be unbounded. In a more rigorous language, if a utility function of money is bounded, then, when the money is approaching infinity, the utility function becomes arbitrarily close to a constant function, which violates non-satiation.

My question is, which economist argue that utility in real life must be unbounded, and who argue that a bounded utility is more normative, rational, or natural? Did economists reached a consensus that utility function must be bounded in real life? Any literature will be very helpful.

---

Sources for bounded utility: EU: Bounded expected utility, PC Fishburn (1967)

**Tags:** macroeconomics, game-theory, reference-request, utility, decision-theory

> **Comments:**
> **username_1**: "*For Expected Utility and SEU, people make axioms to ensure that the utility is bounded.*" Can you please post a source for this claim? It seems contrary to my experience.  
> **username_1**: "*Even if I have a diminishing marginal utility, there will be always some increase in money that will improve my utility by one unit.*" This is false, e.g., $u(x) = -1/x$ is a counterexample.  
> **username_1**: "*which economist believes that utility must be unbounded and who believes the other way?*" I am not sure what you mean by this, but I think if you leave it like this the question will be closed.  
> **username_0**: @username_1 My argument is, the bounded utility like $u(x)=-1/x$ does not always make sense. For example, if you already have 10k dollars, do you believe that receiving 1 billion dollars will not increase your utility by one unit?  
> **username_1**: What is a utility unit?  
> **username_0**: @username_1 utility function usually have a codomain of $\mathbb R$, so one unit of utility can be elicited provided that we are using cardinal utility like EU.  
> **username_1**: So, what exactly is a utility unit?  
> **username_0**: $u(x)-u(y)=1$ means that replacing $y$ with $x$ will increase my utility by one unit. If you have better definition please do share it; it will be very helpful  
> **username_1**: Sorry, but I am going to vote to close instead. Seems like your confusion is just about the basic nature of how utility functions work, or at least I don't understand how any of this is related to the other things in your question.  
> **username_0**: It is more like philosophy than measurement. For sure you can define that $1B$ only worth 0.9 unit of utility, this is totally fine. But for any reasonable person, if $1B$ worth 0.9 unit of utility, then $10000B$ must worth quite a bit more than 0.9 unit. Repeating this way you could add to infinite and thus get unbounded utility  
> **username_1**: I am afraid I fall outside of your definition of a reasonable person, or perhaps I misunderstand your point, or perhaps you are not very clear about what it is that you are asking.  
> **username_0**: @username_1 Let me make another example. One common definition of utility is enjoyment (for example see Loomes and Sugden 1987). As a very simple model, say if something will make you very happy, then it worth one unit of utility (aka enjoyment). If something will make you slightly happy, then it worth 0.1 unit of utility. If we agree that there is always something that make me happy, regardless of how much I already own, then my utility could go to infinity.  
> **username_0**: @username_1 By the way, it is totally fine if you believe that utility must be bounded; I never say that it is wrong. I am curious to learn your arguments and references (if any), like how did economists reached a consensus that utility function must be bounded.  

---

## 1 answers

### Answer by username_2 on 2020-10-11. Score: 4 | Accepted answer

**tl;dr:** Short answer is that cardinal utility of rational person is in large part of a literature is derived from the von Neumann and Morgenstern expected cardinal utility framework. In such framework utility must be bounded by our definition of what rationality is. So in such framework the short answer is simply that utility has to be bounded for person to be considered rational.

There are also expected utility frameworks that allow expected utility to be unbounded (see review by [Fishburn, 1976)](https://www.jstor.org/stable/1886093?seq=1#metadata_info_tab_contents). However, these are not very widely used as they can often lead to paradoxes and do not seem to offer any special insight. I suppose this is due to the strong influence of instrumentalism on economic thinking. A researcher would prefer a utility framework that is able to provide some testable predictions rather one that just results in a paradox and hence offers no useful testable prediction. So even if someone could perhaps consider concept of unbounded utility more elegant its instrumental value in doing research would be close to none (if in particular case it leads to unresolvable paradox).

**Full Answer:**

The utility needs to be bounded in order to avoid paradoxes such as the [St. Petersburg paradox](https://en.wikipedia.org/wiki/St._Petersburg_paradox) (for more nuanced overview see this entry in [Stanford encyclopedia of philosophy](https://plato.stanford.edu/entries/paradox-stpetersburg/)). As a matter of fact a rational person's utility should be bounded as suggested by [Arrow (1970)](https://www.worldcat.org/title/essays-in-the-theory-of-risk-bearing/oclc/152690) precisely in reference to the paradox above.

More general actually cardinal utility will be bounded by the axioms that were used to derive the expected cardinal utility in the first place. Following Neumann and Morgenstern (1947) Theory of Games and Economic Behavior, an expected utility from gamble can be described by the so called von Neuman-Morgenstern equation:

$$E[u(g\_i)] = \sum\_j u(X\_{ij})p\_{ij}$$

where $u$ is utility $g\_i$ is gamble $X$ is an outcome and $p$ is a probability. Furthermore, for the above to be utility we must have some continuum of gambles for which:

$$g\_i,g\_j \in \mathbf{G}: g\_j \succeq g\_i \implies E[u(g\_j)]\geq E[u(g\_i)] $$

Now given this we can ask (as the author of the paper did) what would be properties of such utility function?

Now it turns out that basic rationality constraint where preferences satisfy transitivity, completeness, continuity and independence imply that the utility has to be bounded.

The completeness axiom states:

$$ \forall x,y \in \mathbf{X}: x \succeq y \vee y\succeq x \vee y \thicksim x $$

that is we can order all our options in terms of preference in some way.

The transitivity axiom states:

$$ \forall x,y,z \in \mathbf{X}, \text{ if } x \succeq y \wedge y \succeq z \implies x \succeq z$$

So if someone likes $x$ more than $y$ and $y$ more than $z$ then $x$ must be preferable to $z$

The continuity $z \succeq y \succeq x$, then there must be some probability $p$ such:

$${px,(1-p)z} \thicksim y $$

This implies that no outcome $x$ is so terrible that you would not take up some gamble involving $x$.

by independence axiom if $y \succeq x$ then for $z$ and some probability $p$

$${px,(1-p)z} \preceq {py,(1-p)z} $$

This axiom states that if two outcomes have the same probability, we should evaluate the two alternatives independently of what we think that the outcome is.

The above axioms are how we define rationality in expected cardinal utility. There are of course different possible specifications of utility functions but most modern research relies on the von Neuman-Morgenstern type (or related utility functions).

Now these rationality requirements - which are *axioms* so they are by definition what rationality is in this context, simply demand that no outcome can produce infinite utility. In order to see this we can try to do proof by contradiction - suppose that there is a gamble wehere: $x = 1€$, $y=100€$ and $z= \infty €$ and that $u(X)=x$. In this case clearly $z \succ y \succ x$ but there is no $p$ for which the continuity axiom holds. Since the continuity axiom would be violated our basic axioms of what rationality is would not hold and a person with unbounded utility would cease to be rational (within the context of von Neumann and Morgenstern framework).

> **Comments:**
> **username_0**: Many thanks for the literature! Two questions. First, in your paragraph 2, what are the "paradoxes" besides St Petersburg paradox? Second, in you last paragraph you proved that $z\neq\infty$. But by "unbounded" I guess people mean that $z\in (-\infty,+\infty)$ and $u\in(-\infty,+\infty)$? That is, the utility cannot be infinity but it can arbitrarily approaches infinity.  
> **username_0**: I think even if the utility is unbounded, the St. Petersburg paradox might not be a problem, as realizing the final payoffs in St. Petersburg paradox take infinite amount of time, and agents can have a time discount factor. The experimenter can never convince the agents that they can pay arbitrarily username_0 amount of money, or they can compute the reward instantly.  
> **username_2**: @username_0 1. Paradox is any logical inconsistency most aren't named. 2. I could have used the same example with just making one of the options $-\infty$ I was not attempting to make a rigorous proof but rather just an example, but it works both ways - any infinity at either end just messes up with the continuity axiom. 3. You are right there are other ways of resolving St. Petersburg paradox - but in a sense they are often unsatisfactory. For example, one solution to St. Petersburg paradox is that we can just assume agents ignore small probabilities which might even be true but ...  
> **username_2**: when you want to construct a rigorous model it would be unsatisfactory to just 'handwave' and say that agents ignore some arbitrary small probabilities. Also, just because this particular paradox would be resolved that does not mean other logical inconsistencies would not pop-up somewhere else. It again comes down back to instrumental value of a model.  
> **username_0**: Many thanks for your teaching! I guess in the last paragraph, you proved that $z\neq +\infty$ and $z\neq -\infty$. However, "unbounded" usually mean that $z\to +\infty$, which is different from $z= +\infty$, to my limited knowledge.  
> **username_2**: @username_0 in mathematics bounded function is a function for which there is some number A such that $|f(x)| \leq A$. Hence, if utility cannot be $\infty$ there will be some number A for which the above holds and function will be bounded. Furthermore, the continuity axiom would be violated even if there is a limit because limit of $\lim_{x\rightarrow \infty} x = \infty$  
> **username_3**: @username_2 I still think this doesn't show "unbounded utility" implies there is an event with ∞ utility. Suppose for all z in R, there is an event with that utility. No matter which z you take, you can find a gamble to satisfy continuity.  

---

