<p>(Weak) monotonicity captures the idea of a preference being <em>non-decreasing</em>, namely, if bundle <span class="math-container">$x$</span> contains no less quantities of each good than bundle <span class="math-container">$y$</span>, then <span class="math-container">$x$</span> is at least as good as <span class="math-container">$y$</span>. Therefore it is consistent with the second definition you have.</p>
<p>In contrast, strict (or strong) monotonicity captures the idea of a preference being <em>strictly increasing</em>; that is, if a bundle <span class="math-container">$x$</span> contains no less quantities of each good and strictly more quantities for some goods than bundle <span class="math-container">$y$</span>, then <span class="math-container">$x$</span> is strictly better than <span class="math-container">$y$</span>. This is commonly defined (e.g. in MWG or Kreps) as</p>
<blockquote>
<p><span class="math-container">$(x\ge y \wedge x\ne y)\Rightarrow x\succ y$</span>.</p>
</blockquote>
<p>Your definition (1) of monotonicity is of an intermediate &quot;strength&quot; in terms of its condition for strict preference. On the one hand, it disallows cases like <span class="math-container">$(3,3,1)\sim (1,1,0)$</span>, which is permissible under the weak version above. On the other hand, your definition (1) allows for cases like <span class="math-container">$(3,3,1)\sim(3,1,0)$</span>, which is not permissible under the strict version a la MWG/Kreps.</p>
<p>Which version to use would depend on the context rather than the general meanings conveyed by the labels &quot;weak&quot; and &quot;strong&quot;/&quot;strict&quot;.</p>

CC BY-SA 4.0
2022-11-10T20:03:02.737Z
Just one follow-up: Acc. to your definition of weak monotonicity, $(3,3,1) \sim (1,1,0)$ is allowed. [The definition on Wikipedia](https://en.wikipedia.org/wiki/Monotone_preferences) doesn't seem to allow it. This also seems to be the definition given by Rubinstein (in his lecture notes). Which definition is used more often? (Just checked that Kreps' definition of weak monotonicity is same as (2).)
monotone
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2022-11-11T03:10:09.763Z
@monotone: I updated the last part of my answer. Let me know if it's unclear.
Herr K.
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2022-11-11T04:43:06.187Z
Thank you, that clarifies my doubt.
monotone
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2022-11-10T19:53:36.653Z
2022-11-11T03:10:11.560Z
2022-11-11T03:10:11.560Z
Herr K.
<p>Which one's the correct definition of weak monotonicity?</p>
<ol>
<li><p><span class="math-container">$(x \geq y \implies x \succeq y) \land (x &gt; y \implies x \succ y)$</span></p>
</li>
<li><p><span class="math-container">$x \geq y \implies x \succeq y$</span></p>
</li>
</ol>
<p>The first definition doesn't allow <span class="math-container">$(3,3,1) \sim (1,1,0)$</span> while the second does.</p>

<microeconomics>
What is weak monotonicity?
monotone
# What is weak monotonicity?

(Asked by: username_0 on 2022-11-10)

Which one's the correct definition of weak monotonicity?

1. $(x \geq y \implies x \succeq y) \land (x > y \implies x \succ y)$
2. $x \geq y \implies x \succeq y$

The first definition doesn't allow $(3,3,1) \sim (1,1,0)$ while the second does.

**Tags:** microeconomics

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## 1 answers

### Answer by username_1 on 2022-11-10. Score: 2 | Accepted answer

(Weak) monotonicity captures the idea of a preference being *non-decreasing*, namely, if bundle $x$ contains no less quantities of each good than bundle $y$, then $x$ is at least as good as $y$. Therefore it is consistent with the second definition you have.

In contrast, strict (or strong) monotonicity captures the idea of a preference being *strictly increasing*; that is, if a bundle $x$ contains no less quantities of each good and strictly more quantities for some goods than bundle $y$, then $x$ is strictly better than $y$. This is commonly defined (e.g. in MWG or Kreps) as

> $(x\ge y \wedge x\ne y)\Rightarrow x\succ y$.

Your definition (1) of monotonicity is of an intermediate "strength" in terms of its condition for strict preference. On the one hand, it disallows cases like $(3,3,1)\sim (1,1,0)$, which is permissible under the weak version above. On the other hand, your definition (1) allows for cases like $(3,3,1)\sim(3,1,0)$, which is not permissible under the strict version a la MWG/Kreps.

Which version to use would depend on the context rather than the general meanings conveyed by the labels "weak" and "strong"/"strict".

> **Comments:**
> **username_0**: Just one follow-up: Acc. to your definition of weak monotonicity, $(3,3,1) \sim (1,1,0)$ is allowed. [The definition on Wikipedia](https://en.wikipedia.org/wiki/Monotone_preferences) doesn't seem to allow it. This also seems to be the definition given by Rubinstein (in his lecture notes). Which definition is used more often? (Just checked that Kreps' definition of weak monotonicity is same as (2).)  
> **username_1**: @monotone: I updated the last part of my answer. Let me know if it's unclear.  
> **username_0**: Thank you, that clarifies my doubt.  

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