Unlike the previous one, the second approach does not require measuring utility in any quantity. The consumer here can compare the usefulness of individual goods or a set of them and order them according to the degree of preference. The theory of the optimal choice of the consumer proceeds from the fact that he exercises the right of comparison and free choice on a certain set of X consumer sets, each of which includes all types of products that are consumer goods for a given group of families. Without detracting from the generality, it can be considered that each such set consists of a fixed number ( n ) of elements and has the form:
x = (x1, . . . , xj, . . . , xn) ,
where the elements xj ≥ 0 because they express the quantity of output consumed.
It is further assumed that the comparative assessment of different sets by a given consumer in terms of his tastes, habits, traditions, etc., can be expressed using the so-called binary relation of weak preference.
This relation is defined on the set of consumer sets X, expressed by the formula “preferable to … or equivalent”, is written using the sign “=
” .
The formula “x =
y”, where x and y are consumer sets, means that a given consumer (group of families) in equal conditions either prefers set x to set y, or does not see the difference between them, i.e. considers them equivalent. On the basis of the ratio of weak preference, a ratio of indifference (equivalence) is introduced: two sets x and y are indifferent to the consumer if the conditions “x =
y” and “y=
are simultaneously met x». The fact that two sets are of equal value is usually written with “y ~ x”. The concept of strict (strong) preference 
is defined as follows: “x 
y” if and only if “x =
y”, and the ratio “y =
x” does not take place.
In consumption theory, it is usually assumed that the relation of weak preference satisfies important assumptions, which are called axioms of consumption theory. Thus, the basis is the use of the following axioms:
Transitivity: if the first quantity is comparable to the second, and the second to the third, then the first is comparable to the third; Complete or perfect orderliness. According to it, the consumer is able to order all kinds of goods or their sets through the relationship of preference and indifference; Unsaturations: if an additional unit of goods is added to any set of goods, the resulting set is always preferable to the previous one, since it has greater utility.
The first axiom states that the relation in question is perfect, transitive, and reflexive. The perfection of the relation means for any two sets of the set X there must necessarily be either a ratio of “x =
y”, or “y =
x”, or both together, i.e. “x ~ y”.
This means that there are no such sets that the consumer could not compare with others. The transitivity of the relation consists in the fact that from the relations “x =
y” and “y =
z”, it follows that “x =
z”, where x, y, z are consumer sets. This requirement reflects the compatibility (consistency) of consumer assessments and usually causes a lot of additional discussion. The reflexivity of the relation, i.e. the execution for any set of the ratio “x =
x”, follows from its perfection.
It should be noted that due to the fulfillment of the first axiom, the corresponding indifference relation ~ turns out to be the so-called equivalence relation. This means that the entire set of X consumer sets breaks down into pairwise non-intersecting sets – equivalence classes, each of which is called an indifference set.
Consider two examples of preference relations and corresponding sets of indifference.
1) Let n = 2 and the quantities of products in the set x = (x1, x2) be expressed in weight units (kg), and the consumer builds his comparative estimate as follows: “the set x is preferable to the set y or equivalent to it if its total weight is greater than or equal to the weight of the second set”, i.e. “x =
y”; if x1 + x2 =
y1 + y2.
It is not difficult to see that this relation satisfies the first axiom, and each class of indifference will consist of sets of the same weight.
2) lexicographic preference: the quantities of products in the set x = (x1, x2) are expressed in any units, the consumer considers the first product extremely valuable and compares the sets according to the rule “set x is preferable to the set y, if the quantity of the first product in this set is greater than its quantity in the set y, and if the quantities of the first product in both sets are equal, then the preference is determined by the quantity of the second product”.
This method of comparative evaluation is determined by the formula:
“x 
y” if x1 >y1
or if x1 = y1 and x2 >y2.
This relation also satisfies the first axiom, and each set forms its own class of indifference.
For an indifference set consisting of sets that are equivalent to some set x, the notation is used:
Cx = { y ∈ X | y ~ x }.
Let denote the set of all weakly preferred sets with respect to x through 
, and the set of all weakly non-preferred sets through 
.
The second axiom of consumption theory is that for any set x, both sets 
are closed subsets of the vector space Rn. This means that both sets contain all their limit points and the set of indifference:
,
i.e. defined as the intersection of these sets. A preference relation that has such a property is called continuous.
From the execution of these two basic axioms, it follows that there is a continuous scalar function u(x) defined on the connected set X of consumer sets and is an indicator of preference, since it has the following characteristic property:
“x =
y” if and only if u(x) 
u(y).
Thus, if the consumer weakly prefers the set x to the set y, then the value of the function u at point x will have no less value than at point y, and vice versa, if the value of the indicator for some set x is not less than for the set y, then the consumer weakly prefers the set x to the set y.
The function preference indicator – the u(x) function – is commonly referred to as the utility function of consumer sets. It is not difficult to see that any monotonic transformation of a utility function, such as a function 
, 
or 
(where a>0), are again utility functions, since they have a specified characteristic property. Thus, the utility function is not a measure of any particular “utility”, but only gives an idea of the ranking (order) of various sets, which is why it is often called an ordinal ordinal utility function.
Note that each set of Cx of indifference corresponds to its constant value of the utility function : u(x) = const.
Let’s consider from the point of view of the construction of utility functions the above examples:
(1) The “weight” preference satisfies both axioms of consumption theory, and the weight of the set itself, i.e., the weight of the set itself, can be used as a utility function.
u(x) = u(x1,x2) = x1+x2;
2) lexigraphic ordering is not continuous, since the preferred set (
) and the non-pre-irredital set (
) do not intersect with each other. Therefore, the utility function (preference indicator) does not exist here.
The ordinal approach to utility analysis is the most common. The consumer is not required to be able to measure the goods in some artificial units of measurement. It is enough that the consumer is able to order all possible commodity sets according to their “preference”. In the ordinal theory of utility, the concept of “utility” means nothing more than the order of preference. The statement, “Set A is preferable to this consumer than set B,” is the same as the statement, “Set A is more useful to this consumer than set B.” The question of how many units is more useful set A than set B is not posed. The consumer chooses the preferred set of goods from all available to him.