[3] A single-attribute utility function maps the amount of money a person has (or gains), to a number representing the subjective satisfaction he derives from it.
The motivation to define a utility function comes from the St. Petersburg paradox: the observation that people are not willing to pay much for a lottery, even if its expected monetary gain is infinite.
Bernouli himself assumed that the utility is logarithmic, that is, u(x)=log(x) where x is the amount of money; this was sufficient for solving the St. Petersburg paradox.
Gustav Fechner[4] also supplied psychophysical justification for the logarithmic function (known as the Weber–Fechner law).
But Stanley Smith Stevens[5] showed that the relation between physical stimulus and psychological perception can be better explaind by a power function, that is, u(x)=xp, with exponent p between 0.3 to 2.
[6][7][8][9][10] As a result, power functions were incorporated into psychological decision theories, such as Cumulative prospect theory, rank-affected multiplicative (RAM) weights model, and transfer of attention exchange (TAX) model.
Wakker[12] noted that power functions can have a negative exponent, but in this case their sign should change so that they remain increasing.
[13] Utility functions are usually assessed in experiments checking subjects' preferences over lotteries.
For example, the person indifferent between the lotteries [100%: $10] and [60%:$20, 40%:0] can be modeled by a linear utility function, if we assume that he underweights the probability 60% to around 50%.
Combining gain and loss domains may yield an incorrect utility function.
[6][20] He conducted experiments in which no probabilities were used; instead, he asked questions such as "how much money would you need in order to feel twice as happy as $10"?
A third problem is that most experiments compare the relative fit of different utility models to the data.
Kirby[3] presented a novel experiment design, that allowed him to get point-predictions for each model separately.
Following extensive surveys, MAU functions for health-related conditions were developed; see Quality-adjusted life year and EQ-5D#assessment.
[21] The construction is done in two steps: an online Discrete Choice Experiment (DCE) survey, and a face-to-face composite time-tradeoff interview (cTTO): In both steps, the subjects are adults, and they are asked to answer the queries from the point-of-view of a 10-years-old child.