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Preliminary Notions
Let X be a set, and define a binary relation on X as a subset B of \(X \times X\)
Ex: \(X = \{\text{Mary, John, Carlos}\}\). Suppose Mary is older than Carlos, who is older than John. The binary relation "is older than" is {(Mary, Carlos), (Carlos, John), (Mary, John)}
Notation: Given a binary relation \(B\), we shall write \(x B y\) instead of \((x, y) \in B\)
Ex: \(\supset\) is a binary relation of a set \(X\). We write \(X \supset y\) instead of \((x, y) \in \supseteq\)
Let \(\geq\) be a binary relation of a set \(X\).
Def: \(\geq\) is complete s.t. \(\forall x, y \in X, \quad x \geq y\text{or} y \geq x\) (or both)
Def \(\geq\) is transitive if \(\forall x, y, z \in X, if x \geq y,\) and \(y \geq z \implies x \geq z\).
Antisymmetric if \(\forall x, y \in X,x \geq y, y \geq x \implies x = y\)
Def: The indifference part of the binary relation is defined by
\(\sim = \{(x, y) : (x, y) \in \geq\ \text{and} (y, x) \in \geq\}\)
A binary order is called a weak order if it is a complete order and a transitive order.
In this class, we call weak orders preference relations.
Interpretations: If \(\geq\) is the preference relation of an agent then \(X \geq y\) means that the agent is happy choosing \(x\) out of the set \(\{x, y\}\). And \(x > y\) means that the agent would exclusively choose \(x\) for \(\{x, y\}\).
Def: A preference relation that is also antisymmetric is called a strict preference.
Actually antisymmetry and completeness implies exclusivity (no two elements have the same place in the order)
Given a function \(u: X \to R\), we may define a binary relation \(\geq\) by \(x \geq y\) if and only if \(u(x) \geq u(y)\).
Exercise: Prove that \(\geq\) defined in this way is a preference relation.
In this case, we say that u represents \(\geq\), and that u is a utility representation of \(\geq\).
And, \(x \sim y\) iff \(u(x) = u(y)\).
First Economic Problem
Object Allocation
Model primitives:
A finite (nonempty) set \(A\) of agents.
A finite (nonempty) set \(O\) of objects.
A symbol \(\empty\) representning the outside option.
Each student \(i \in A\) is endowed with a strict prefefrence \(\succeq\) over \(O \cup\{\empty\}\).
Ineffieient if slack in system: Stable matching