# Why utility should be bounded (or unbounded)?

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)

• "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 Oct 11 '20 at 13:32
• "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 Oct 11 '20 at 13:33
• "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 Oct 11 '20 at 13:34
• @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 Oct 11 '20 at 13:36
• What is a utility unit? – Giskard Oct 11 '20 at 13:37

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). 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).

The utility needs to be bounded in order to avoid paradoxes such as the St. Petersburg paradox (for more nuanced overview see this entry in Stanford encyclopedia of philosophy). As a matter of fact a rational person's utility should be bounded as suggested by Arrow (1970) 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).

• 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 Oct 11 '20 at 22:27
• 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 Oct 11 '20 at 22:29
• @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 Oct 11 '20 at 22:35
• 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 Oct 11 '20 at 22:36
• 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 Oct 11 '20 at 22:59