# Conflicting Definitions of Weak Monotnocity (preferences)

Strong Montonicity my sources seem to agree on Strong monotonicity, i state equivalent definitions below. But weak montonicity i keep finding what appear to be conflicting definitions. In the following i refer to Wikipedia, Hal R. Varian, "Microeconomic Analysis," 3rd Edition, and my own Mathematical Economics coursepack.

Wikepedia: An agent's preferences are said to be strongly monotonic if, given a consumption bundle $$x$$ the agent prefers all consumption bundles $$y$$ that have more of at least one good, and not less in any other good.

• Wikipedia: $$y \ge x$$ and $$y ≠ x$$ $$\implies y\succ x$$
• Varian: $$y \ge x$$ and $$y ≠ x$$ $$\implies y > x$$
• Course pack: Calls this "Nonsatiation": Vectors $$x > y$$ implies $$x_i \ge y_i , \forall \; i$$ and $$x_i > y_i$$ for some $$i$$ then $$x\succ y$$

I understand these are all saying the same thing.

Weak Montonicity:

• Wikipedia: "In economics, an agent's preferences are said to be weakly monotonic if, given a consumption bundle $$x$$ the agent prefers all consumption bundles $$y$$ that have more of all goods. That is, $$y ≫ x$$ implies $$y\succ x$$"

• Course pack: My course pack does not mention monotnocity but also has this definition and refers to it as "Strict preferences" and that it satisfies nonsatiation: And I believe it means $$x_i > y_i, \forall \;i$$

• Varian p.96: Weak Montonicity: $$x \ge y$$ then $$x\succeq y$$

This is confusing because Varian's definition seems be the complete opposite of Wikipedia's definition, and my course packs definition of Strict preferences.

1. Could someone confirm the most standard definitions of weak and strong montonoicity, and weak and strict preferences. Thanks!

There is no universal terminology for monotonicity conditions. For example, the most widely used textbook for such things, Microeconomic Theory by Mas-Colell, Whinston, and Green, uses somewhat different terminology. If we take $$\geq$$ to be the coordinatewise ordering, $$>$$ the asymmetric part ($$x>y$$ if $$x\geq y$$ but not $$y\geq x$$), and $$\gg$$ for being strictly larger in every coordinate, they call preference monotone if $$x\gg y$$ implies $$x\succ y$$ (what Wikipedia seems to call weakly monotonic), call them strongly monotone if $$x> y$$ implies $$x\succ y$$. There is probably more agreement on what strong (sometimes strict) monotonicity means. I think the Varian version of weak monotonicity makes more sense than the Wikipedia one.

Nonsatiation is pretty much universally used for something different, namely that for each $$x$$ there is some $$y$$ such that $$y\succ x$$. Often this is strengthened to local nonsatiation: For every $$x$$ and every neighborhood $$V$$ of $$x$$, there exists some $$y\in V$$ satisfying $$y\succ x$$.

The term strict preference is usually used for the asymmetric part $$\succ$$ of a weak (reflexive) preference relation $$\succeq$$ or just for an irreflexive or asymmetric relation, depending on the desired generality.

Let's look at the relationship between the different monotonicity notions, here neutrally named (all preference relations $$\succeq$$ are assumed to be complete and transitive, $$\succ$$ is their asymmetric part, and the domain is $$\mathbb{R}^l_+$$):

M1: $$x\geq x$$ implies $$x\succeq y$$.

M2: $$x\gg y$$ implies $$x\succ y$$.

M3: $$x> y$$ implies $$x\succ y$$.

Then M3 implies both M1 and M2. No other implications need to hold in general. If $$\succeq$$ is continuous, then M2 implies M1. If $$\succeq$$ is locally nonsatiated, then M1 implies M2. M2 (and therefore also M3) imply local nonsatiation.

• This is a great answer, thank you very much! I'm glad the ambiguity is real and not just me being clueless. What's your intuition as to why you prefer the Vairan Definition? Apr 28 at 10:15
• Usually, continuity is assumed. And then it is actually weaker. The concept is also useful, so it is good not to have to come up with some notion of ultra weakness. Apr 28 at 10:45
• Hi Michael; I'm just taking some Time to go through this again as the topic is still causing frustration. Can you clarify your point from Green that x > y implies x >>y. This is confusing? x > y is that not a contradiction based on the previous definiotns of > and >> you mentioned? May 10 at 8:48
• Sorry, that was a typo. The $\gg$ should be $\succ$. I corrected it now. May 10 at 9:30
• That makes alot more sense!! Thank you! Given that this field is a minefield of different interpretations, im going to write out one more question focussing specifically on Varian which my course pack follows. I will post the link here, and if you have timed love your feedback! May 10 at 9:42