To be able to find the best solution for MCDA problems we need to know *subjective preferences*:

- these are binary relations provided by a decision maker
- defined analogously to Voting Theory Relations

So given two alternatives $a$ and $b$ a decision maker can say if

- $a \ P \ b$ or $b \ P \ a$: $a$ is preferred to $b$ -
*the preference relation* - $a \ I \ b$ - the indifference relation
- $a \ J \ b$ - the incomparability relation, when you cannot compare things

Main properties:

- $P$ is asymmetric
- $a \ P \ b \equiv b \ \overline{P} \ a$

- $I$ is reflexive and symmetric:
- $a \ I \ a$ and $a \ I \ b \equiv b \ I \ a$

- $J$ is irreflexible and symmetric
- $a \ \overline{J} \ a$ and $a \ J \ b \equiv b \ J \ a$

Transitivity

- $P$ is transitive
- but $I$ is not! by Luce's Coffee Cups (In contract to Voting Theory - there we assumed it's transitive)

We can show the preferences of a decision maker with a graph:

How to build a mathematical model from statements of a decision maker?

There are several preference structures that can do that:

- Complete Pre-Order Preference Structure ($I$ is not transitive) - The Traditional Model
- SemiOrder Preference Structure ($I$ is transitive, no $J$) - The Threshold Model
- Partial Order Preference Structure (with $J$)
- Valued Preference

- Main Article:
*Preferential Independence*

This is an important condition between preferences and criteria: the criteria should be preferentially independent.