# Utility

Utility is the tendency of an object to produce happiness or prevent unhappiness for an individual or a community.

How can we assign utilities to represent preferences?

## # Interval Scales

- Assign to each outcome $x$ a value $v(x)$ such that $v(x) \geq v(y) \iff x \geq y$ and $v(x) = v(y) \iff x \sim y$
- Transformation is linear
- Called an ordinal transformation

Ordinal Scales must satisfy the following properties:

- Completeness: $x \succ y$ or $x \sim y$ or $y \succ x$
- Asymmetry: if $x \succ y$ then it is false that $y \succ x$
- Transitivity: if $x \succ y$ and $y \succ z$ then $x \succ z$

## # Infinite Utility

An agent values A infinitely relative to B and C if we deny Continuity: $[\lambda A, (1-\lambda)C] \succ B$ for any $\lambda > 0$

The agent is willing to trade B for any gamble that offers a positive chance of A, when the ‘losing outcome’ is C.)