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 .. math:: 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 .. math:: 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`.