# 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 $$y\not\succ x$$.