Lexicographic preferences

Lexicographic preferences are a relation of preferences, according to which the choice is made by sequential comparison of the number of goods included in the set. If the amount of good $i$ ${\ displaystyle i}$ $i$ in the set A is larger than in B, the agent chooses A. Otherwise, the quantities of the good are compared $i+1$ ${\ displaystyle i + 1}$ $i + 1$ . Lexicographic preferences got their name because of the similarity with the word order in the dictionary. Mathematically, such preferences are nothing more than a lexicographic order (binary relation) on many alternatives.

Formal Definition

Let many alternatives be a vector in space $\mathbb {R} ^{n},\,X\in \mathbb {R} ^{n}$ ${\ displaystyle \ mathbb {R} ^ {n}, \, X \ in \ mathbb {R} ^ {n}}$ let it go $x,y\in X$ ${\ displaystyle x, y \ in X}$ . Then $x\succsim y$ ${\ displaystyle x \ succsim y}$ , if a

$(x_{1}>y_{1})\,\lor \,(x_{1}=y_{1},\,x_{2}>y_{2})\,\lor ...\,\lor (x_{n}=y_{n})$ ${\ displaystyle (x_ {1}> y_ {1}) \, \ lor \, (x_ {1} = y_ {1}, \, x_ {2}> y_ {2}) \, \ lor ... \, \ lor (x_ {n} = y_ {n})}$

Properties

Lexicographic preferences are rational, that is, they satisfy the axioms of completeness and transitivity.
Lexicographic preferences are monotonous.
Lexicographic preferences are convex.
Lexicographic preferences are not continuous and therefore cannot be represented using the utility function.
The indifference curve for such preferences consists of a single point, and the map itself is a set of points in the space of alternatives.

Lexicographic preferences

Formal Definition

Properties

Notes

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