Complete Pre-Order Preference Structure

Complete Pre-Order

This is a preference structure for Modeling Preferences in MCDA

Also called the traditional representation of preferences

$\forall a,b \in A:$

• $a \ P \ b \iff g(a) > g(b)$ (The complete order: $R$ relation)
• $a \ I \ b \iff g(a) = g(b)$ (The complete pre-oder: $I$ relation)

$g$ is some global aggregation function:

• Weighted Sum Model
• Ideal Point Model

This way we cannot model incomparability ($>$ can always compare things)

• $J$ is always empty - everything is comparable
• $P$ is transitive
• $I$ also becomes transitive

Example 1

Suppose there are three sport teams: $a$, $b$, and $c$.

• If $a$ beats $b$, $a$ receives 3 points and the loser receives 0 points
• If they draw, both receives 1 point.
• The three teams will play with each other and at the end they will have total points.
• If all total scores are different, there will be a ‘‘complete order’’
• If there is a tie, the order will be a ‘‘complete preorder’’

Example 2

Expected gains of different actions:

| $a$ | $b$ | $c$ | $d$ | $e$ | $f$ | $g$ | 100 | 100 | 120 | 130 | 130 | 130 | 131 |

Sources

• Decision Engineering (ULB)
• http://web.itu.edu.tr/~topcuil/ya/MDM03ConstructingModel.pptx