same 50 each to two other people has resulted in a net loss of 25 utils. If a class of utility functions is cardinal, intrapersonal comparisons of utility differences are allowed. If, in addition, some comparisons of utility are meaningful interpersonally, the linear transformations used to produce the class of utility functions must be restricted across people. An example is cardinal unit comparability. In that information environment, admissible transformations are increasing affine functions and, in addition, the scaling factor must be the same for everyone. This information assumption allows for interpersonal comparisons of utility differences, but utility levels cannot be compared interpersonally because the intercept of the affine transformations may differ across people. Marginalism Under cardinal utility theory, the sign of the marginal utility of a good is the same for all the numerical representations of a particular preference structure. The magnitude of the marginal utility is not the same for all cardinal utility indices representing the same specific preference structure. The sign of the second derivative of a differentiable utility function that is cardinal, is the same for all the numerical representations of a particular preference structure. Given that this is usually a negative sign, there is room for a law of diminishing marginal utility in cardinal utility theory. The magnitude of the second derivative of a differentiable utility function is not the same for all cardinal utility indices representing the same specific preference structure. Expected utility theory This type of indices involves choices under risk. In this case, , and , are lotteries associated with outcomes. Unlike cardinal utility theory under certainty, in which the possibility of moving from preferences to quantified utility was almost trivial, here it is paramount to be able to map preferences into the set of real numbers, so that the operation of mathematical expectation can be executed. Once the mapping is done, the introduction of additional assumptions would result in a consistent behavior of people regarding fair bets. But fair bets are, by definition, the result of comparing a gamble with an expected value of zero to some other gamble. Although it is impossible to model attitudes toward risk if one doesn't quantify utility, the theory should not be interpreted as measuring strength of preference under certainty. Construction of the utility function Suppose that certain outcomes are associated with three states of nature, so that x3 is preferred over x2 which in turn