## Utility functions

Formal analysis of choice relies upon a conception of well-being known as utility. This notion provides a real valued space in which each member of the choice set corresponds to a numerical value. A mapping from the choice set X to the real line is known as a utility function.

Utility Function: a function $u:X\rightarrow \mathbb {R}$  is a utility function representing the preference relation $\succeq$  if:
$\forall x,y\in \mathbf {X} ,\quad u(x)\geq u(y)\iff x\succsim y$ .

The utility function plays a foundational role in economics. However, as used in economics, the concept of utility has no measurable counterpart in the real world. Utility functions provide ordinal rankings of choices, thus the numerical value assigned to each utility level is completely arbitrary.
To explicate, consider the set of possible choices $\mathbf {X} ={x,y,z}$  with preferences ${x\succ y,y\succ z,x\succ z}$  Now consider two candidate utility functions
$u_{1}(c)={\begin{cases}100&{\mbox{if }}c=x\\50&{\mbox{if }}c=y\\10&{\mbox{if }}c=z\\\end{cases}}u_{2}(c)={\begin{cases}3&{\mbox{if }}c=x\\2&{\mbox{if }}c=y\\1&{\mbox{if }}c=z\\\end{cases}}$

Notice, the functions $u_{1}(c),\quad u_{2}(c)$  each provide valid representations of the preferences defined over the choice set. However, a mental-state interpretation of utility may interpret the two functions differently. Using some crude notion of 'happiness' or 'satisfaction', $u_{1}$  suggests choice x leaves our decision maker 'twice as happy' as choice y, an interpretation $u_2(c)$ fails to support. In fact, any monotone transform of $u_{1},\quad g(u_{1}(c))$  represents the original preference relation but may not (usually will not) preserve cardinal relationships between utility levels. Thus, attaching a psychological interpretation to utility creates information not present in the underlying preference relation.

### Properties of Utility Functions

1. Monotonicity: Utility functions exhibit weak monotonicity iff $x\geq y\Rightarrow x\succsim y$  and strong monotonicity iff $x\geq y\land x\neq y\Rightarrow x\succ y$