<p>There is no universal terminology for monotonicity conditions. For example, the most widely used textbook for such things, <em>Microeconomic Theory</em> by Mas-Colell, Whinston, and Green, uses somewhat different terminology. If we take <span class="math-container">$\geq$</span> to be the coordinatewise ordering, <span class="math-container">$&gt;$</span> the asymmetric part (<span class="math-container">$x&gt;y$</span> if <span class="math-container">$x\geq y$</span> but not <span class="math-container">$y\geq x$</span>), and <span class="math-container">$\gg$</span> for being strictly larger in every coordinate, they call preference <em>monotone</em> if <span class="math-container">$x\gg y$</span> implies <span class="math-container">$x\succ y$</span> (what Wikipedia seems to call weakly monotonic), call them <em>strongly monotone</em> if <span class="math-container">$x&gt; y$</span> implies <span class="math-container">$x\succ y$</span>. 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.</p>
<p><em>Nonsatiation</em> is pretty much universally used for something different, namely that for each <span class="math-container">$x$</span> there is some <span class="math-container">$y$</span> such that <span class="math-container">$y\succ x$</span>. Often this is strengthened to <em>local nonsatiation</em>: For every <span class="math-container">$x$</span> and every neighborhood <span class="math-container">$V$</span> of <span class="math-container">$x$</span>, there exists some <span class="math-container">$y\in V$</span> satisfying <span class="math-container">$y\succ x$</span>.</p>
<p>The term <em>strict preference</em> is usually used for the asymmetric part <span class="math-container">$\succ$</span> of a weak (reflexive) preference relation <span class="math-container">$\succeq$</span> or just for an irreflexive or asymmetric relation, depending on the desired generality.</p>
<p>Let's look at the relationship between the different monotonicity notions, here neutrally named (all preference relations <span class="math-container">$\succeq$</span> are assumed to be complete and transitive, <span class="math-container">$\succ$</span> is their asymmetric part, and the domain is <span class="math-container">$\mathbb{R}^l_+$</span>):</p>
<p>M1: <span class="math-container">$x\geq y$</span> implies <span class="math-container">$x\succeq y$</span>.</p>
<p>M2: <span class="math-container">$x\gg y$</span> implies <span class="math-container">$x\succ y$</span>.</p>
<p>M3: <span class="math-container">$x&gt; y$</span> implies <span class="math-container">$x\succ y$</span>.</p>
<p>Then M3 implies both M1 and M2. No other implications need to hold in general.  If <span class="math-container">$\succeq$</span> is continuous, then M2 implies M1. If <span class="math-container">$\succeq$</span> is locally nonsatiated, then M1 implies M2. M2 (and therefore also M3) imply local nonsatiation.</p>

CC BY-SA 4.0
2023-04-28T10:15:25.577Z
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?
CormJack
CC BY-SA 4.0
2023-04-28T10:45:58.657Z
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.
Michael Greinecker
CC BY-SA 4.0
2023-05-10T08:48:44.173Z
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?
CormJack
CC BY-SA 4.0
2023-05-10T09:30:36.050Z
Sorry, that was a typo. The $\gg$ should be $\succ$. I corrected it now.
Michael Greinecker
CC BY-SA 4.0
2023-05-10T09:42:12.117Z
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!
CormJack
CC BY-SA 4.0
2023-05-11T07:33:30.190Z
Here is the question @Michael Greinecker. Would love your thoughts on any part of it, if you have time! https://economics.stackexchange.com/questions/55335/preference-relations-based-on-varian
CormJack
CC BY-SA 4.0
2024-05-19T12:11:26.673Z
Could you please explain the relationship between M1 and M2? I'm struggled with the proof that M1 implies M2 and M2 implies M1 given continuous. Here is the link https://economics.stackexchange.com/questions/58450/equivalence-of-two-definitions-of-monotone-preference
Nonenicht
CC BY-SA 4.0
2024-05-19T18:49:39.893Z
Continuity is not enough for M1 to imply M2. To see this, consider a consumer who is indifferent between all bundles.
Michael Greinecker
CC BY-SA 4.0
2024-05-20T11:33:24.523Z
@Nonenicht Sorry, forgot to tack you.
Michael Greinecker
CC BY-SA 4.0
2023-04-23T18:56:55.280Z
2024-05-20T08:53:10.537Z
2024-05-20T08:53:10.537Z
Michael Greinecker
<p><strong>Strong Montonicity</strong> 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, &quot;Microeconomic Analysis,&quot; 3rd Edition, and my own Mathematical Economics coursepack.</p>
<p><em><a href="https://en.wikipedia.org/wiki/Monotone_preferences" rel="nofollow noreferrer">Wikepedia</a></em>: An agent's preferences are said to be <strong>strongly monotonic</strong> if, given a consumption bundle <span class="math-container">$x$</span>  the agent prefers all consumption bundles <span class="math-container">$y$</span> that have more of at least one good, and not less in any other good.</p>
<ul>
<li><em>Wikipedia:</em> <span class="math-container">$y \ge x$</span> and <span class="math-container">$y ≠ x$</span> <span class="math-container">$\implies y\succ x$</span></li>
<li><em>Varian:</em> <span class="math-container">$y \ge x$</span> and <span class="math-container">$y ≠ x$</span> <span class="math-container">$\implies y &gt; x$</span></li>
<li><em>Course pack</em>: Calls this &quot;Nonsatiation&quot;:  Vectors <span class="math-container">$x &gt; y$</span> implies <span class="math-container">$x_i \ge y_i , \forall \; i$</span> and <span class="math-container">$x_i &gt; y_i$</span> for some <span class="math-container">$i$</span> then  <span class="math-container">$x\succ y$</span></li>
</ul>
<p>I understand these are all saying the same thing.</p>
<p><strong>Weak Montonicity:</strong></p>
<ul>
<li><p><em><a href="https://en.wikipedia.org/wiki/Monotone_preferences" rel="nofollow noreferrer">Wikipedia</a>:</em> &quot;In economics, an agent's preferences are said to be <em><strong>weakly monotonic</strong></em> if, given a consumption bundle <span class="math-container">$x$</span> the agent prefers all consumption bundles <span class="math-container">$y$</span> that have more of all goods. That is, <span class="math-container">$y ≫ x$</span> implies <span class="math-container">$y\succ x$</span>&quot;</p>
</li>
<li><p><em>Course pack</em>: My course pack does not mention monotnocity but also has this definition and refers to it as <em>&quot;<strong>Strict preferences</strong>&quot;</em> and that it satisfies <em>nonsatiation:</em> And I believe it means <span class="math-container">$x_i &gt; y_i, \forall \;i$</span></p>
</li>
<li><p><em>Varian p.96:</em> <em><strong>Weak Montonicity</strong></em>: <span class="math-container">$x \ge y$</span> <em>then</em> <span class="math-container">$x\succeq y$</span></p>
</li>
</ul>
<p>This is confusing because Varian's definition seems be the complete opposite of Wikipedia's definition, and my course packs definition of Strict preferences.</p>
<ol>
<li>Could someone confirm the most standard definitions of weak and strong montonoicity, and weak and strict preferences. Thanks!</li>
</ol>

<mathematical-economics><consumer-theory><preferences>
Conflicting Definitions of Weak Monotnocity (preferences)
CormJack
# Conflicting Definitions of Weak Monotnocity (preferences)

(Asked by: username_0 on 2023-04-23)

**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](https://en.wikipedia.org/wiki/Monotone_preferences)*: 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](https://en.wikipedia.org/wiki/Monotone_preferences):* "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!

**Tags:** mathematical-economics, consumer-theory, preferences

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## 1 answers

### Answer by username_1 on 2023-04-23. Score: 4 | Accepted answer

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: $x\geq y$ 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.

> **Comments:**
> **username_0**: 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?  
> **username_1**: 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.  
> **username_0**: 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?  
> **username_1**: Sorry, that was a typo. The $\gg$ should be $\succ$. I corrected it now.  
> **username_0**: 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!  
> **username_0**: Here is the question @Michael Greinecker. Would love your thoughts on any part of it, if you have time! https://economics.stackexchange.com/questions/55335/preference-relations-based-on-varian  
> **username_2**: Could you please explain the relationship between M1 and M2? I'm struggled with the proof that M1 implies M2 and M2 implies M1 given continuous. Here is the link https://economics.stackexchange.com/questions/58450/equivalence-of-two-definitions-of-monotone-preference  
> **username_1**: Continuity is not enough for M1 to imply M2. To see this, consider a consumer who is indifferent between all bundles.  
> **username_1**: @Nonenicht Sorry, forgot to tack you.  

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