SOME SIMPLE UTILITY FUNCTIONS
As it follows, x denotes the amount of one’s assets (for example, one’s money minus one’s debts). A common but strong assumption shall be used:
x ≥ 0
Linear utility function
The present section assumes that the possible outcomes are finite and known in a given decision problem.
Linear transformations
x → (ax + b)
do not change the preference ordering of monetary expectations (expected values). Changing the money scale or changing the origin of the money axis neither influences that preference order. It is comfortable to make such a transformation that x = 0 corresponds to the minimal possible outcome. Then, it is possible to choose such a utility function
u(x) = ax + b
that
u(0) = 0; and u = 1 at the maximal value of x.
If the maximal possible value is 20, then the corresponding linear utility function is shown in the Figure below.
All monetary expectations preferences remain the same if to use linear utility graphs like the one shown in the Figure.
That utility can be interpreted as a probability. The utility u(x) is the win probability in a utility-fair gamble, in which one can lose all one’s money x and can achieve, as a result of the win, the final assets equal to 20.
For example, u(12) = 0.6. Thus, if one’s initial assets are x = 12, then the following gamble would be monetary fair: with probability p = 0.6 one wins 20 ‒ 12 = 8 units and with probability 1 ‒ p = 0.4 one loses 12 units of money.
However, the linear utility has some shortages.
First, if there is no upper bound of the possible wealth, then the linear utility meets with mathematical obstacles. If the point A on the Figure moves to the right, then the angle α decreases. In the limit, it diminishes to zero.
Second, the linear utility does not take into account the effect of risk aversion.
Generally, the decision-makers tend to avoid monetary fair bets. Losing 1 unit of money is considered more serious change than winning the same amount of money. Also, every next unit of money earned is considered less important than the previous unit earned.
Both of these obstacles can be overcome if to assume that the utility function is concave — if to assume the diminishing marginal utility.
Logarithmic utility function
In 1738, Daniel Bernoulli [Bernoulli 1738] (for the English translation see [Bernoulli 1954]) introduced the logarithmic utility function (also known as the log utility):
u(x) = ln(x)
This function is concave, and it yields to the risk aversion. More importantly, it also yields to the diminishing risk aversion. According to Bernoulli, it is important, that than richer one is, the lesser is one’s reluctance against tossing a coin in a fair gamble with the same fixed bet. An infinitely rich man should behave as if one’s utility function is linear.
It is crucial to understand that risk aversion is the result of the concaveness of the utility function while diminishing risk aversion is not.
Unfortunately, Bernoulli’s utility function is without lower and upper bounds, which creates shortages. It is possible to use the improved Bernoulli’s function
u(x) = ln(x + 1)
Everything remains the same, except that the utility function has a lower bound (see the Figure). Unfortunately, this improved utility still lacks the upper bound.
The risk aversion indicator or the Arrow-Pratt measure [Pratt 1964] can be defined as follows:
λ(x) = — u’’(x)/u’(x)
(In some sources, this formula's right side has been multiplied with a positive constant.)
If λ > 0, a local risk aversion appears. In the case of logarithmic utility u = ln(x + 1), the risk aversion is decreasing in x, and it is diminishing to zero:
λ(x) = 1/(x + 1)
See the Figure.
Exponential utility function
Often, the following simple exponential utility is presented:
u(x) = 1 — 1/exp(α · x)
where α > 0 is a parameter.
This utility function is bounded:
0 ≤ u(x) < 1
and it can be interpreted as a probability. If x is the value of the assets of the decision-maker, and u(x) above is one’s utility function, then one regards one’s assets x as equivalent to the gamble, in which with the probability u(x) one wins an infinite amount of money, while otherwise, one loses everything.
To prove this claim, let p be the win probability in such a utility-fair gamble. Then
p · u(∞) + (1 — p) · u(0) = u(x)
therefore
p · 1 + (1 — p) · 0 = u(x)
therefore
p = u(x)
which concludes the proof.
In the case
α = ln(2)
which can be achieved by changing the money unit, that utility function has a particularly simple form (see the Figure):
u(x) = 1–1/2˟
in which case the disutility
w(x) = 1 — u(x) = 1/2˟
is bisected after every additional unit of money obtained.
Such a utility is also called as a utility with constant risk aversion. The indicator of risk aversion (the Arrow-Pratt measure) λ is constant:
λ(x) = α = const.
This result shows that while the utility function's concavity produces the effect of risk aversion, it is not enough to secure that this risk aversion decreases in x and that it is diminishing to zero in the infinity of x. However, this is a serious obstacle when trying to model mutually motivated contracts, for example, mutually motivated insurance contracts.
Utility function u = x/(x + 1)
Already in 1738, Daniel Bernoulli [Bernoulli 1738], who introduced the notion of the utility function, regarded it as important to be able to explain the insurance contracts. Why is a rich man willing to provide the insurance and a poor man is not? The answer consists not in the existence of the risk aversion but decreasing risk aversion. Incidentally, Bernoulli’s logarithmic utility u = ln(x) is not only concave, providing the risk aversion, but also with decreasing risk aversion, as was shown above. This risk aversion is diminishing in the infinity. However, Bernoulli’s utility function is unbounded.
Since then, it has often been asked about whether there was some simple utility function that was bounded and had a decreasing risk aversion, moreover, a risk aversion diminishing to zero in the infinity (see, for example, [Pratt 1964]). It has been asked about “everyman’s utility function”, but usually only the logarithmic or exponential utilities have been mentioned.
However, there is a simple utility function
u(x) = x/(x + b)
where b > 0 is a parameter.
This utility function is bounded:
0 ≤ u(x) < 1
and it can be interpreted as a probability.
By changing the money unit, the condition b = 1 can be achieved, and this utility function obtains a particularly simple form (see the Figure):
u(x) = x/(x + 1)
The Arrow-Pratt measure (or the indicator of the risk aversion) is also very simple, and it is diminishing in x (see the Figure below) [Eintalu 2019]:
λ(x) = 2/(x + 1)
One group of authors [Ikefuji, et al, 2013] has derived a set of functions they call “Pareto utilities”. The function u(x) = x/(x + b) can be considered as a special case of Pareto utilities.
There have been presented various utility functions earlier (for example, in [Pratt 1964]), having as a special case the function u(x) = x/(x + b).
The function u(x) = x/(x + 1) has one amazing feature. Technically, at every point x, the value of that function u(x) can be calculated as the value of the linear utility in the case when there is still exactly one unit of money missing from the “ultimate happiness” corresponding to the value u = 1 (see the Figure).
Therefore, that utility function can be characterized as “Always one cent is missing” [Eintalu 2019]. However, as yet, there is no deeper explanation of this feature, which might as well turn out to be a coincidence.
Written by Jüri Eintalu in 05. — 08. December 2019
The aftermath: an infowar in the Wikipedia
I do not know about the novelty. What I have seen is that when simple utility functions are presented, the list contains the linear function, the logarithmic function, the exponential function, but never the function
u(x) = x/(x + b);
b> 0
Simultaneously, several famous authors have complained that it is difficult to find a simple utility function with decreasing risk aversion.
But such a function is included in these very sets of utility functions they themselves have considered — as a special case.
So I do not understand why such a function should be missing from the textbooks, encyclopedias, etc.
Some admins of Wikipedia had actively invited me to write there something. Finally, I did so. I wrote a new sub-chapter
“Some Simple Utility Functions”
into the chapter
in the Wikipedia.
It turns out that Wikipedia has technical problems. Twice, I lost my text without making any technical mistakes of my own. It also turns out that the only possibility to save your text in the Wikipedia is to publish it — as there is no such a button visible like “Save as a draft”. And it turns out that the authors of the Wikipedia have nowhere — in some visible place — been informed that there is no possibility to save a draft. No warnings whatsoever.
At some point, I started to worry about the technically right method of citations. Finally, I found their method. Now, I had to convert my references from the section “Further reading” to the section “References”. And I had to delete unnecessary references (for example, it turned out that I did not have to refer to the source of the figures).
I was in contact with one admin, in personal contact, we were talking, he was unable to answer to all of my questions concerning the references — there are hundreds of relevant pages of instructions in the Wikipedia, in many different files.
As soon as I had converted the first citation — a reference to myself — to the right format when probably some Wikipedia’s machine discovered that I had referred to myself…
While I talked with one admin, the other one intervened on the internet and deleted my text (it can be restored, however). That admin presented a pseudo-justification that I regard as a slandering and smearing. False accusations. One deleted my text without reading it carefully. The accusation was that I had extensively referred only to myself. You can check the final version above yourself. There are 4 sections, in the last one, I am absolutely justified to refer to the book of my own. In Wikipedia, the admin should have been able to see what I have done, also, that the text was obviously not yet finished.
I should go to court against Wikipedia. But it is not my style.
Later, they deleted my text again — the final version of it — where the references were all together in the right format. Deleted without any comments. Without reacting to my explanations and comments in the Talk section.
So I decided that Wikipedia should be abandoned.
Wikipedia is not an academically sound institution.¹
They are using (actually, there is more than one such a case) their own technical mistakes and stupidity to stage and set up false and absurd accusations. And if one restores the deleted text, the other one deletes it in the next day again.
So they are writing forth-and-back, forth-and-back.
Therefore, I think that perhaps China is smart — they have prohibited the English Wikipedia altogether. Perhaps we should also be smarter.
The most curious thing about admin’s pseudo-justification was that I had used the term “us” a few times. But I did not say that “We hate…” or “We believe in God” or something like that. Instead, I said “Let us suppose that x > 0…” Still, the admin quoted this as a reason to delete the whole of my text…
However, one cannot delete Galileo Galilei.
The Earth is still moving.
¹ According to one of the founders of Wikipedia, Larry Sanger, already in 2002 there were “…trolls or borderline trolls, that ended up being in positions of authority in the Wikipedia community.” In 2019, there is already an extensive literature on the ideological biases of Wikipedia, its propaganda wars and its hidden cartels of anonymous editors.
References
- Bernoulli, Daniel (1738). “Specimen Theoriae Novae de Mensura Sortis”. Commentarii Academiae Scientiarum Imperialis Petropolitanae. 5: 175–192.
- Bernoulli, Daniel (1954). “Exposition of a New Theory on the Measurement of Risk”. Econometrica. 22: 23–36.
- Eintalu, J. (2019). Utility Function u = x/(x + 1). Berlin: Lambert Academic Publishing. ISBN 978–620–0–30723–1.
- Ikefuji, Masako, et al, (2013). “Pareto Utility”. Theory and Decision. 75 (1): 43–57.
- Jensen, Niels Erik (1967). “An Introduction to Bernoullian Utility Theory: I. Utility Functions”. The Swedish Journal of Economics. 69 (3): 163–183.
- Lindley, Dennis (1985). Making Decisions. London: Wiley.
- Pratt, John (1964). “Risk Aversion in the Small and in the Large”. Econometrica. 32 (1/2): 122–136.
- “Some Simple Utility Functions”. A text printed out from the Wikipedia, but later deleted by the Wikipedia. PDF, downloadable.
- Von Neumann, John & Morgenstern, Oskar (1944). Theory of Games and Economic Behavior. Princeton, NJ: Princeton University Press.