# Understanding the definition of monotone

In Microeconomic Theory by Mas-Colell, Whinston, and Green, the definition of monotone preference relations is given as follows:

Definition 3.B.2$$\quad$$ The preference relation $$\succsim$$ on $$X$$ is monotone if $$x \in X$$ and $$y \gg x$$ implies $$y \succ x$$.

So suppose that $$\succsim$$ is monotone. I was wondering if it is possible for two bundles $$x$$ and $$y$$ to be such that $$y \geq x$$ but $$x \succ y$$?

I'm not sure about this, but I think that the above statement is not possible; that is, I think:

If $$\succsim$$ is monotone, then $$y \succsim x$$ whenever $$y \geq x$$.

Note that $$x \geq y$$ means $$x_n \geq y_n$$ for all $$n=1,\dots,N$$ and that $$x \gg y$$ means $$x_n > y_n$$ for all $$n=1,\dots,N$$.

• I think this works. Why can't my preferences be defined by a preference for less? That is, $U(x) > U(y)$ wherever $x<y$ such that $x\succ y$ whenever $y > x$. I was never under the impression that it was necessarily an increasing relation. Preference relations can be monotonically increasing or decreasing. Not sure how to go about directly proving this though, I would just suppose the preference relation I defined and then prove it by contradiction. Commented Apr 6 at 1:31
• @Brennan But that's bizarre. If you take a look at my first question in this post "economics.stackexchange.com/questions/58053/… =", it should indicate that "$y \succsim x$ whenever $y \geq x$". Commented Apr 6 at 2:51
• I don't think it's that bizarre. My preference relation is monotonically increasing in less suffering. Or it's monotonically decreasing in suffering. I dont see how monotonicity is violated if I just say I prefer less of something in a monotonic way. Commented Apr 6 at 16:00

Throughout my answer, I assume that $$x \succ y$$ is read as "$$x$$ is strictly preferred to $$y$$". Refer to footnote (a) if that is not the case.

I was wondering if it is possible for two bundles $$x$$ and $$y$$ to be such that $$y \geq x$$ but $$x \succ y$$?

1. Counterexample: $$X = \{(1,1), (2,2), (3,1)\}$$ such that $$(2,2) \succ (1,1) \succ (3,1)$$. Completeness, transitivity and monotonicity are satisfied. Now pick $$x = (3,1)$$ and $$y = (1,1)$$.

2. With the assumption that $$X \cong \mathbb{R}^m \ (m \in \mathbb{N})$$ and continuity of $$\succeq_X$$, we have $$y \geq x \implies y \succeq x$$.

Proof. Suppose $$y \geq x$$ for some $$x,y \in X$$ and assume for the sake of contradiction that $$x \succ y$$. Define $$Z := \{z \in X : z >> y\}$$. Consider any two open balls $$B_x$$ and $$B_y$$ and observe that $$z_y \in B_y \cap Z$$ must satisfy $$x \succ z_y$$ (due to continuity and our assumption that $$x \succ y$$) and $$z_y \succ x$$ (due to monotonicity) simultaneously. Contradiction!

(a) If $$x \succ y$$ is read as "$$x$$ is weakly preferred to $$y$$", consider $$U(X) = 1$$ and see that $$(y \geq x) \land (x \succ y)$$ holds for all $$x,y \in X$$ satisfying $$y \geq x$$.

(b) Definition of continuity of preferences: A preference relation $$\succeq$$ on $$X$$ is continuous if whenever $$a \succ b$$, $$\exists$$ open balls $$B_a$$ and $$B_b$$ such that for all $$x \in B_a$$ and $$y \in B_b$$, $$x \succ y$$.

(c) Note: I would appreciate an answer that has a weaker condition on $$X$$ compared to $$X$$ being isomorphic to $$\mathbb{R}^m$$ $$(m \in \mathbb{N})$$. Having said that, the above proof works for $$X = \mathbb{R}^m_{\geq 0}$$ $$(m \in \mathbb{N})$$ which makes more sense from the economic point.

• Correctme if I'm wrong. For your definition of continuity, I feel it should be: $a \succ b$ whenever there are open balls $B_a$ and $B_b$ such that $x \succ y$ for all $x \in B_a$ and $y \in B_b$. Commented Apr 6 at 14:51
• @Beerus No, your definition is incorrect. Counterexample: Pick $X = \mathbb{R}^2, a = (2,2), b = (1,1)$. For $\epsilon = 0.1$, there are open balls $B(a,\epsilon)$ and $B(b,\epsilon)$ such that $x \succ y$ for all $x \in B(a,\epsilon)$ and $y \in B(y,\epsilon)$, and also $a \succ b$. However, lexicographic preferences not continuous. I took the definition from Ariel Rubinstein's Lecture Notes in Micro Theory, Lecture 2 (Utility), Page 16, Definition C1. Commented Apr 6 at 16:58
• Oh, I see. Yes you are right! Commented Apr 6 at 17:34