Ordinal utility

Suppose the set of all states of the world is ${\displaystyle X}$ and an agent has a preference relation on ${\displaystyle X}$. It is common to mark the weak preference relation by ${\displaystyle \preceq }$, so that ${\displaystyle A\preceq B}$ reads "the agent wants B at least as much as A".

The symbol ${\displaystyle \sim }$ is used as a shorthand to the indifference relation: ${\displaystyle A\sim B\iff (A\preceq B\land B\preceq A)}$, which reads "The agent is indifferent between B and A".

The symbol ${\displaystyle \prec }$ is used as a shorthand to the strong preference relation: ${\displaystyle A\prec B\iff (A\preceq B\land B\not \preceq A)}$, which reads "The agent strictly prefers B to A".

A function ${\displaystyle u:X\to \mathbb {R} }$ is said to represent the relation ${\displaystyle \preceq }$ if:

${\displaystyle A\preceq B\iff u(A)\leq u(B)}$

Some conditions on ${\displaystyle \preceq }$ are necessary to guarantee the existence of a representing function:

• Transitivity: if ${\displaystyle A\preceq B}$ and ${\displaystyle B\preceq C}$ then ${\displaystyle A\preceq C}$.
• Completeness: for all bundles ${\displaystyle A,B\in X}$: either ${\displaystyle A\preceq B}$ or ${\displaystyle B\preceq A}$ or both.
  • Completeness also implies reflexivity: for every ${\displaystyle A\in X}$: ${\displaystyle A\preceq A}$.

When these conditions are met and the set ${\displaystyle X}$ is finite, it is easy to create a function ${\displaystyle u}$ which represents ${\displaystyle \prec }$ by just assigning an appropriate number to each element of ${\displaystyle X}$, as exemplified in the opening paragraph. The same is true when X is countably infinite. Moreover, it is possible to inductively construct a representing utility function whose values are in the range ${\displaystyle (-1,1)}$.[4]

When ${\displaystyle X}$ is infinite, these conditions are insufficient. For example, lexicographic preferences are transitive and complete, but they cannot be represented by any utility function.[4] The additional condition required is continuity.

A preference relation is called continuous if, whenever B is preferred to A, small deviations from B or A will not reverse the ordering between them. Formally, a preference relation on a set X is called continuous if it satisfies one of the following equivalent conditions:

1. For every ${\displaystyle A\in X}$, the set ${\displaystyle \{(A,B)|A\preceq B\}}$ is topologically closed in ${\displaystyle X\times X}$ with the product topology (this definition requires ${\displaystyle X}$ to be a topological space).
2. For every sequence ${\displaystyle (A_{i},B_{i})}$, if for all i ${\displaystyle A_{i}\preceq B_{i}}$ and ${\displaystyle A_{i}\to A}$ and ${\displaystyle B_{i}\to B}$, then ${\displaystyle A\preceq B}$.
3. For every ${\displaystyle A,B\in X}$ such that ${\displaystyle A\prec B}$, there exists a ball around ${\displaystyle A}$ and a ball around ${\displaystyle B}$ such that, for every ${\displaystyle a}$ in the ball around ${\displaystyle A}$ and every ${\displaystyle b}$ in the ball around ${\displaystyle B}$, ${\displaystyle a\prec b}$ (this definition requires ${\displaystyle X}$ to be a metric space).

If a preference relation is represented by a continuous utility function, then it is clearly continuous. By the theorems of Debreu (1954), the opposite is also true:

Every continuous complete preference relation can be represented by a continuous ordinal utility function.

Note that the lexicographic preferences are not continuous. For example, ${\displaystyle (5,0)\prec (5,1)}$, but in every ball around (5,1) there are points with ${\displaystyle x<5}$ and these points are inferior to ${\displaystyle (5,0)}$. This is in accordance with the fact, stated above, that these preferences cannot be represented by a utility function.

For every utility function v, there is a unique preference relation represented by v. However, the opposite is not true: a preference relation may be represented by many different utility functions. The same preferences could be expressed as any utility function that is a monotonically increasing transformation of v. E.g., if

${\displaystyle v(A)\equiv f(v(A))}$

where ${\displaystyle f:\mathbb {R} \to \mathbb {R} }$ is any monotonically increasing function, then the functions v and v give rise to identical indifference curve mappings.

This equivalence is succinctly described in the following way:

An ordinal utility function is unique up to increasing monotone transformation.

In contrast, a cardinal utility function is only unique up to increasing affine transformation. Every affine transformation is monotone; hence, if two functions are cardinally equivalent they are also ordinally equivalent, but not vice versa.

Suppose, from now on, that the set ${\displaystyle X}$ is the set of all non-negative real two-dimensional vectors. So an element of ${\displaystyle X}$ is a pair ${\displaystyle (x,y)}$ that represents the amounts consumed from two products, e.g., apples and bananas.

Then under certain circumstances a preference relation ${\displaystyle \preceq }$ is represented by a utility function ${\displaystyle v(x,y)}$.

Suppose the preference relation is monotonically increasing, which means that "more is always better":

${\displaystyle x<x'\implies (x,y)\prec (x',y)}$

${\displaystyle y<y'\implies (x,y')\prec (x,y')}$

Then, both partial derivatives, if they exist, of v are positive. In short:

If a utility function represents a monotonically increasing preference relation, then the utility function is monotonically increasing.

Suppose a person has a bundle ${\displaystyle (x_{0},y_{0})}$ and claims that he is indifferent between this bundle and the bundle ${\displaystyle (x_{0}-\lambda \cdot \delta ,y_{0}+\delta )}$. This means that he is willing to give ${\displaystyle \lambda \cdot \delta }$ units of x to get ${\displaystyle \delta }$ units of y. If this ratio is kept as ${\displaystyle \delta \to 0}$, we say that ${\displaystyle \lambda }$ is the marginal rate of substitution (MRS) between x and y at the point ${\displaystyle (x_{0},y_{0})}$.[5]^:82

This definition of the MRS is based only on the ordinal preference relation – it does not depend on a numeric utility function. If the preference relation is represented by a utility function and the function is differentiable, then the MRS can be calculated from the derivatives of that function:

${\displaystyle MRS={\frac {v'_{x}}{v'_{y}}}.}$

For example, if the preference relation is represented by ${\displaystyle v(x,y)=x^{a}\cdot y^{b}}$ then ${\displaystyle MRS={\frac {a\cdot x^{a-1}\cdot y^{b}}{b\cdot y^{b-1}\cdot x^{a}}}={\frac {ay}{bx}}}$. The MRS is the same for the function ${\displaystyle v(x,y)=a\cdot \log {x}+b\cdot \log {y}}$. This is not a coincidence as these two functions represent the same preference relation – each one is an increasing monotone transformation of the other.

In general, the MRS may be different at different points ${\displaystyle (x_{0},y_{0})}$. For example, it is possible that at ${\displaystyle (9,1)}$ the MRS is low because the person has a lot of x and only one y, but at ${\displaystyle (9,9)}$ or ${\displaystyle (1,1)}$ the MRS is higher. Some special cases are described below.

When the MRS of a certain preference relation does not depend on the bundle, i.e., the MRS is the same for all ${\displaystyle (x_{0},y_{0})}$, the indifference curves are linear and of the form:

${\displaystyle x+\lambda y={\text{const}},}$

and the preference relation can be represented by a linear function:

${\displaystyle v(x,y)=x+\lambda y.}$

(Of course, the same relation can be represented by many other non-linear functions, such as ${\displaystyle {\sqrt {x+\lambda y}}}$ or ${\displaystyle (x+\lambda y)^{2}}$, but the linear function is simplest.)[5]^:85

When the MRS depends on ${\displaystyle y_{0}}$ but not on ${\displaystyle x_{0}}$, the preference relation can be represented by a quasilinear utility function, of the form

${\displaystyle v(x,y)=x+\gamma v_{Y}(y)}$

where ${\displaystyle v_{Y}}$ is a certain monotonically increasing function. Because the MRS is a function ${\displaystyle \lambda (y)}$, a possible function ${\displaystyle v_{Y}}$ can be calculated as an integral of ${\displaystyle \lambda (y)}$:[6][5]^:87

${\displaystyle v_{Y}(y)=\int _{0}^{y}{\lambda (y')dy'}}$

In this case, all the indifference curves are parallel – they are horizontal transfers of each other.

A more general type of utility function is an additive function:

${\displaystyle v(x,y)=v_{X}(x)+v_{Y}(y)}$

There are several ways to check whether given preferences are representable by an additive utility function.

When there are three or more commodities, the condition for the additivity of the utility function is surprisingly simpler than for two commodities. This is an outcome of Theorem 3 of Debreu (1960). The condition required for additivity is preferential independence.[5]^:104

A subset A of commodities is said to be preferentially independent of a subset B of commodities, if the preference relation in subset A, given constant values for subset B, is independent of these constant values. For example, suppose there are three commodities: x y and z. The subset {x,y} is preferentially-independent of the subset {z}, if for all ${\displaystyle x_{i},y_{i},z,z'}$:

${\displaystyle (x_{1},y_{1},z)\preceq (x_{2},y_{2},z)\iff (x_{1},y_{1},z')\preceq (x_{2},y_{2},z')}$.

In this case, we can simply say that:

${\displaystyle (x_{1},y_{1})\preceq (x_{2},y_{2})}$ for constant z.

Preferential independence makes sense in case of independent goods. For example, the preferences between bundles of apples and bananas are probably independent of the number of shoes and socks that an agent has, and vice versa.

By Debreu's theorem, if all subsets of commodities are preferentially independent of their complements, then the preference relation can be represented by an additive value function. Here we provide an intuitive explanation of this result by showing how such an additive value function can be constructed.[5] The proof assumes three commodities: x, y, z. We show how to define three points for each of the three value functions ${\displaystyle v_{x},v_{y},v_{z}}$: the 0 point, the 1 point and the 2 point. Other points can be calculated in a similar way, and then continuity can be used to conclude that the functions are well-defined in their entire range.

0 point: choose arbitrary ${\displaystyle x_{0},y_{0},z_{0}}$ and assign them as the zero of the value function, i.e.:

${\displaystyle v_{x}(x_{0})=v_{y}(y_{0})=v_{z}(z_{0})=0}$

1 point: choose arbitrary ${\displaystyle x_{1}>x_{0}}$ such that ${\displaystyle (x_{1},y_{0},z_{0})\succ (x_{0},y_{0},z_{0})}$. Set it as the unit of value, i.e.:

${\displaystyle v_{x}(x_{1})=1}$

Choose ${\displaystyle y_{1}}$ and ${\displaystyle z_{1}}$ such that the following indifference relations hold:

${\displaystyle (x_{1},y_{0},z_{0})\sim (x_{0},y_{1},z_{0})\sim (x_{0},y_{0},z_{1})}$.

This indifference serves to scale the units of y and z to match those of x. The value in these three points should be 1, so we assign

${\displaystyle v_{y}(y_{1})=v_{z}(z_{1})=1}$

2 point: Now we use the preferential-independence assumption. The relation between ${\displaystyle (x_{1},y_{0})}$ and ${\displaystyle (x_{0},y_{1})}$ is independent of z, and similarly the relation between ${\displaystyle (y_{1},z_{0})}$ and ${\displaystyle (y_{0},z_{1})}$ is independent of x and the relation between ${\displaystyle (z_{1},x_{0})}$ and ${\displaystyle (z_{0},x_{1})}$ is independent of y. Hence

${\displaystyle (x_{1},y_{0},z_{1})\sim (x_{0},y_{1},z_{1})\sim (x_{1},y_{1},z_{0}).}$

This is useful because it means that the function v can have the same value – 2 – in these three points. Select ${\displaystyle x_{2},y_{2},z_{2}}$ such that

${\displaystyle (x_{2},y_{0},z_{0})\sim (x_{0},y_{2},z_{0})\sim (x_{0},y_{0},z_{2})\sim (x_{1},y_{1},z_{0})}$

and assign

${\displaystyle v_{x}(x_{2})=v_{x}(y_{2})=v_{x}(z_{2})=2.}$

3 point: To show that our assignments so far are consistent, we must show that all points that receive a total value of 3 are indifference points. Here, again, the preferential independence assumption is used, since the relation between ${\displaystyle (x_{2},y_{0})}$ and ${\displaystyle (x_{1},y_{1})}$ is independent of z (and similarly for the other pairs); hence

${\displaystyle (x_{2},y_{0},z_{1})\sim (x_{1},y_{1},z_{1})}$

and similarly for the other pairs. Hence, the 3 point is defined consistently.

We can continue like this by induction and define the per-commodity functions in all integer points, then use continuity to define it in all real points.

An implicit assumption in point 1 of the above proof is that all three commodities are essential or preference relevant.[7]^:7 This means that there exists a bundle such that, if the amount of a certain commodity is increased, the new bundle is strictly better.

The proof for more than 3 commodities is similar. In fact, we do not have to check that all subsets of points are preferentially independent; it is sufficient to check a linear number of pairs of commodities. E.g., if there are ${\displaystyle m}$ different commodities, ${\displaystyle j=1,...,m}$, then it is sufficient to check that for all ${\displaystyle j=1,...,m-1}$, the two commodities ${\displaystyle \{x_{j},x_{j+1}\}}$ are preferentially independent of the other ${\displaystyle m-2}$ commodities.[5]^:115

The mathematical foundations of most common types of utility functions — quadratic and additive — laid down by Gérard Debreu[9][10] enabled Andranik Tangian to develop methods for their construction from purely ordinal data. In particular, additive and quadratic utility functions in ${\displaystyle n}$ variables can be constructed from interviews of decision makers, where questions are aimed at tracing totally ${\displaystyle n}$ 2D-indifference curves in ${\displaystyle n-1}$ coordinate planes without referring to cardinal utility estimates.[11][12]

The following table compares the two types of utility functions common in economics:

• Lexicographic preference relation cannot be represented by a utility function. In Economics.SE
• Recognizing linear orders embeddable in R2 ordered lexicographically. In Math.SE.
• Murray N. Rothbard, "Towards a Reconstruction of Utility and Welfare Economics"