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First, let's start with the definition of a monotone binary relation:

Definition: Let $C \subset \mathbb{R}^n$ be nonempty and convex. A binary relation $\succeq$ on $C^m$ is monotonic if for any $(x_1, \dots, x_m)$ and $(y_1, \dots, y_m)$ in $C^m$ if $(x_i, \dots, x_i) \succeq (y_i, \dots, y_i)$ for $i = 1, \dots, m$ imply $(x_1,\dots, x_m) \succeq (y_1, ... y_m)$

My question: Why does $C$ need to be convex? Couldn't we just choose any non-empty subset of $\mathbb{R}^n$?

My guess: Because we eventually want to assume continuity, then any convex combination between $x$ and $y$ must be in $C$, hence $C$ must be convex.

Rodrigo de Azevedo
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MrCorote
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1 Answers1

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As mentioned in the comments, convexity on $C$ is assumed to use the fact that any convex subset of $\mathbb{R}^n$ is connected. (proof can be found anywhere really, but in Math Exchange, we have this post.)

Connectedness is used in the proof of the following proposition:

Proposition: Let $\succeq$ be a complete, monotonic, transitive and continuous binary relation on $C^m$. For any $(x_1, \dots, x_m) \in C^m$, there is $z \in C$, such that:

$$(x_1, \dots, x_m) \sim (z,\dots, z)$$

MrCorote
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