# 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

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

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* \(y\not\succ x\).