Preference Relations

Let the set of choice alternatives being analysed be denoted by \(X\).

Weak Preferences

A binary relation \(\succsim\) on \(X\) is a weak preference relation if it satisfies

Reflexivity

For all \(x\in X\), \(x\succsim x\).

The relation of strict preference that is derived from \(\succsim\) is defined by

\[x\succ y\;\; \text{if}\;\; x\succsim y\;\; \text{and}\;\; y\not\succsim x\]

The relation of indifference that is derived from \(\succsim\) is defined by

\[x\sim y\;\; \text{if}\;\; x\succsim y\;\; \text{and}\;\; y\succsim x\]

Additional properties that a weak preference relation may have are:

Completeness

For all \(x,y\in X\), either \(x\succsim y\) or \(y\succsim x\).

Transitivity

For all \(x,y,z\in X\), \(x\succsim y\succsim z\) implies \(x\succsim z\).

Preorder

\(\succsim\) is reflexive and transitive.

Weak Order

\(\succsim\) is complete and transitive.

Incomplete Preorder

\(\succsim\) is reflexive and transitive and there exist \(x,y\in X\) such that \(x\not\succsim y\) and \(y\not\succsim x\).

Strict Preferences

If it is assumed that no two distinct alternatives are related by indifference, then a strict preference relation \(\succ\) on \(X\) is taken as primitive. Such a relation \(\succ\) satisfies:

Asymmetry

For all \(x,y\in X\), \(x\succ y\) implies \(y\not\succ x\).

Additional properties that a strict preference relation \(\succ\) may have are:

Totality

For all distinct \(x,y\in X\), either \(x\succ y\) or \(y\succ x\).

Transitivity

For all \(x,y,z\in X\), \(x\succ y\succ z\) implies \(x\succ z\).

Strict Linear Order

\(\succ\) is asymmetric, total and transitive.

Strict Partial Order

\(\succ\) is asymmetric and transitive and there exist distinct \(x,y\in X\) such that \(x\not\succ y\) and \(y\not\succ x\).