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 $v(x)\ge v(y)⟺x\ge y$ and $v(x)=v(y)⟺x\sim y$
• Transformation is linear
• Called an ordinal transformation

Ordinal Scales must satisfy the following properties:

1. Completeness: $x\succ y$ or $x\sim y$ or $y\succ x$
2. Asymmetry: if $x\succ y$ then it is false that $y\succ x$
3. 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.)