Preference Relations¶
- \(X=\{apple,banana,orange\}\)
- \(X=\{\!\) all consumption bundles of \(n\) goods that can be bought at positive or non-negative amounts \(\!\}\)
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 (weak preferences)¶
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 (strict preferences)¶
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 \(y\not\succ x\).