# Existence of maximal element in a compact choice set

I have a compact subset of $$\mathbb{R}$$, $$X$$. An agent has a continuous, transitive and complete preference relation $$\succsim$$ over $$X$$. I am wondering whether there exists a $$y\in X$$ such that $$y\succsim x$$ for all $$x\in X$$. I have the following so far:

If $$X$$ were simply closed, then the answer would be no. This is because we could define the preference relation: $$x\succsim y$$ iff $$x\geq y$$ on $$\mathbb{R}$$ (which is a closed set). Clearly, there is no maximal element.

But I am not sure about the case in which $$X$$ is compact.

Thank you

• Hint: Use extreme value theorem and the result that a continuous, transitive and complete preference relation can be represented by a continuous utility function. Sep 20, 2021 at 15:36

Yes, compactness is sufficient and no need to restrict yourself to subsets of $$\mathbb{R}$$.

The proof is by contradiction. Assume that $$\succeq$$ has no greatest element. Consider the sets: $$V_x = \{y \in X| x \succ y\}$$ These are open sets as $$\succeq$$ is continuous. If $$(V_x)_{x \in X}$$ do not cover $$X$$ then there is an $$y$$ that is no set $$V_x$$ for all $$x$$. This means that for all $$x \in X$$, $$x \not \succ y$$, so $$y \succeq x$$ for all $$x$$, which means that $$y$$ is the greatest element a contradiction.

From this, it follows that $$(V_x)_{x \in X}$$ is an open cover of $$X$$. By compactness, it has a finite subcover, say $$V_{x_1},\ldots, V_{x_n}$$. By completeness and transitivity, we can find $$x_i \in \{x_1, \ldots, x_n\}$$ such that $$x_i \succeq x_j$$ for all $$j \in \{1,\ldots, n\}$$.

Now for any $$y \in X$$ either $$y = x_j$$ for some $$j$$ which means $$x_i \succeq y = x_j$$ or $$y \in V_{x_j}$$ for some $$j$$ which means that: $$x_i \succeq x_j \succ y.$$ So $$x_i$$ is a greatest element, which gives the desired contradiction.