Nonlinear perception of happiness

Epistemic status: Speculative.

tl;dr: Perception of hapiness is related to some “raw” happiness by an equivalent of a psychophysics law. The “raw” quantity should be used when aggregating. Far-reaching implications for utility calculation would follow.

A body of research seeks to understand happiness and measure it quantitatively. Often the measurements use tools such as Oxford Happiness Inventory, Subjective Happiness Scale, Panas Scale, etc. What these instruments have in common is they measure a perception, and the scales used are linear.

A proposal: let’s make a distinction between the perception of happiness, which is measured in this way, and a hypothetical raw happiness. While we cannot measure such quantity in practice, we can at least imagine how it would be measured in a thought experiment – e.g., by an outside observer who has complete access to the mental states of beings, and has some algorithmic way how to determine happiness of mental states.

Given this distinction, we may then ask, how would a human perception of happiness be related to such raw quantity?

Conjecture: Human perception of happiness has a nonlinear form, that is, with a linear increase of the raw happiness there is nonlinear increase of the perceived happiness.

One candidate for the relation can be the widely known psychophysics law, Weber–Fechner law, stating the subjective sensations are proportional to the logarithm of the stimuli intensity. This models light intensity and the perceived difference in weight. It was also proposed the logarithmic perception applies to more indirect senses, e.g. sense of time intervals. It seems plausible it would hold for the perception of quantities like wealth: if we measure the perception of wealth by asking people to rate their wealth on a scale from 0 to 10, we would get log(monetary value).

Now - what if this holds also for the sense of happiness, as used in philosophy and utilitarian calculus? Specifically, we may propose the percieved happiness related to the raw hapiness H as

where is an unknown proportionality constant.
(While I chose happiness, the argument would be the same for related or similar quantities, such as well-being.)


It may seem such logarithmic rescaling is just an irrelevant change of scale. However, when we aggregate a quantity over many people, there are significant differences between using the raw quantity and the perception.

Conjecture: the raw quantity is often the more useful when aggregating.

This can be easily seen in case of physical quantities, like weight. If we want to calculate total weight carried by a group of people, or total illumination created by a group of celestial objects, we can not simply add the perceived weights or perceived intensities, but we must first recover the raw quantity of stimuli and only latter sum or integrate. Same holds for averaging.

Per analogiam, we should neither sum nor average happiness of people as measured by various linear scales, but we should try to recover the “raw” quantity and only then do the averaging. (And possibly do the log transformation afterwards)

Implications to ethics

As various some of the utilitarian normative ethical theories suggest we should attempt to maximize quantities like happiness or well-being, the difference between counting the raw happiness (“hedons) or aggregating the perceptions leads to different results. While in the non-log-corrected happiness calculus, we would integrate over beings and time the percepts of happiness directly, in the exponentially corrected version the integral has the form

where is the percieved hapiness of a beeing in time , summed over all beings and time, and is the unknown constant.


We can see how this changes the conclusion on the example of a famous philosophical problem: the “repugnant conclusion” problem (Parfit, D., Reasons and Persons.)

In its classical formulation, the repugnant conclusions is: “For any possible population of at least ten billion people, all with a very high quality of life, there must be some much larger imaginable population whose existence, if other things are equal, would be better even though its members have lives that are barely worth living”

If we use some measure of the “raw” happiness or “quality of life”, the exponential step makes the “much larger” population size hardly feasible. Then, while the conclusion is still technically true in a sense, the paradox is resolved for all practical purposes by taking into account the resource demanded by such populations.

As an illustrative comparison with some numbers: we can imagine an open-ended subjetive quality of life scale where 1 means life of no quality, life with happiness 1.1 is just worth living, moderately happy life can be rated 5 and a life with very high quality rated 10. Then, if we take the base of the logarithmic scale to be e, the “much larger population” in the original formulation would have to be more than 240 billion people to be equal to be better than the original population. Most likely the resource cost of existence of such an immense population would be many times greater than of the original population, even if lives barely worth living are cheaper than high quality life.

Stated in other way—in the real world we are always solving a constrained optimization problem, where resources to create more souls are not exactly unlimited. In such situations relevant for reality, the question is “what is the best population given the limited resources”. Optimizing the “raw hapiness” resolves the paradox for most practical purposes.

Similarly, the use of raw happiness would affect many other questions in moral philosophy.

Experimental tests

While in presence it does not seem feasible to test whether the raw happiness is more fundamental than the perception, it at least seems possible to observe if people’s preferences are broadly consistent with the view. In a possible experimental setup one part of the participants would reveal their preferences by choosing between options like “five nice dinners, or one day of skiing in the mountains” and in the other part rate the experiences on a linear scale. From the former part we should be able to convert the joyful value of all the experiences to a single unit (“hedons”), and then compare the value of the experiences in hedons to the values assigned to them on the linear scale. Our prediction is the dependence would be approximately logarithmic.


It seems plausible the often measured perceptions of hapiness is related to a hypotethical quantity, raw hapiness, by some non-linear relation, e.g. logarithmically. Using the raw quantity when calculating aggregates and averages of hapiness over many people could be a better way of aggregation. This would have broad implications in utilitarian ethics, medical ethics, population ethics, and many other fields where aggregates of hapiness or similar quantities are used. (Similarly for aggregation over time)