<p>You're on the right track in understanding the difference between weakly monotonic and strictly monotonic preferences.</p>
<p><strong>1. Weakly Monotonic Preferences:</strong>
Preferences are weakly monotonic if for any two consumption bundles x and y, if y strictly greater than x (meaning that every element in y is strictly greater than the corresponding element in x), then y &gt; x (you strictly prefer y over x).</p>
<p><strong>What it doesn't guarantee:</strong>
Weakly monotonic preferences do not require a clear preference between two bundles if one is only greater than or equal to the other, but not strictly greater (e.g., y &gt;= x, such as y = (1,1,2) and x = (1,1,1)).
In these cases, the preference might not be clear, and it's possible under weakly monotonic
preferences that you might actually prefer x over y (i.e., y &lt;= x), or be indifferent. Weak monotonicity doesn't tell us anything about this scenario.</p>
<p><strong>2. Strictly Monotonic Preferences:</strong>
Preferences are strictly monotonic if: - For y &gt;= x, you at least weakly prefer y over x (i.e., y &lt;= x). This means if y is greater than or equal to x in all dimensions,you cannot prefer x over y, though you might be indifferent. - For y strictly greater than x, you strictly prefer y over x (i.e., y &gt; x).</p>
<p><strong>Key Difference:</strong></p>
<p>Weakly monotonic: Only ensures strict preference if y strictly greater than x. It is silent on what happens when y &gt;= x but not y strictly greater than x, so it's possible to have y &lt;= x in this case.
Strictly monotonic: Always ensures that y &gt;= x means you at least weakly prefer y, and if y strictly greater than x, you strictly prefer y over x.</p>
<p>In your example, if preferences are weakly monotonic, it's possible for y = (1,1,2) to be strictly worse than x = (1,1,1), since weak monotonicity doesn't provide guidance when y &gt;= x but not y strictly greater than x. However, in the case of strictly monotonic preferences,y = (1,1,2) would be at least weakly preferred to x = (1,1,1).</p>

CC BY-SA 4.0
2024-09-28T10:05:55.493Z
2024-09-28T10:05:55.493Z
sajib sarkar dhrubo
<p>Recently I am reviewing preference theory and I found that I am confused about the difference between weakly monotonic preference and strictly monotonic preference.</p>
<p>Now, let us suppose <span class="math-container">$X$</span> is the consumption set. If my understanding is correct, then:</p>
<ul>
<li><p>Preference is said to be <em>weakly</em> monotonic if for <span class="math-container">$x,y \in X$</span>, <span class="math-container">$y \gg x$</span> implies <span class="math-container">$y \succ x$</span>.</p>
</li>
<li><p>Preference is said to be <em>strictly</em> monotonic if for <span class="math-container">$x,y \in X$</span>, <span class="math-container">$y \geq x$</span> implies <span class="math-container">$y \succeq x$</span>, and <span class="math-container">$y \gg x$</span> implies <span class="math-container">$y \succ x$</span>.</p>
</li>
</ul>
<p>So it seems that weakly monotonic preference says nothing when <span class="math-container">$y \geq x$</span> but not <span class="math-container">$y \gg x$</span> (for example, <span class="math-container">$y=(1,1,2)$</span> and <span class="math-container">$x=(1,1,1)$</span>)? Maybe we can have <span class="math-container">$y \prec x$</span> if the preference is weakly monotonic?</p>

CC BY-SA 4.0
2024-09-28T00:47:03.277Z
Weak monotonicity, according to your definition, is just being *agnostic* about cases where $x\ge y$. It is "weak" because it imposes one fewer condition relative to the "strict" version. Also note that different authors (e.g. Kreps (2013)) define monotonicity differently (see [related question](https://economics.stackexchange.com/q/55185/42)).
Herr K.
CC BY-SA 4.0
2024-09-28T03:11:44.897Z
For continuous preferences, these are equivalent. Usually, "strictly monotone" refers to the stronger condition that $y\geq x$ and $y\neq x$ implies $y\succ x$.
Michael Greinecker
CC BY-SA 4.0
2024-09-28T09:48:11.020Z
Thank all of you very much. The info you provided is really helpful.
Shujun Tan
CC BY-SA 4.0
2024-09-28T20:48:50.373Z
@MichaelGreinecker Hi, in the question https://economics.stackexchange.com/questions/55185/conflicting-definitions-of-weak-monotnocity-preferences you mentioned that if $\succeq$ is locally nonsatiated, then M1 implies M2. However I failed to prove it, could you explain how can we approach this conclusion?
Shujun Tan
CC BY-SA 4.0
2024-09-29T01:28:29.810Z
@ShujunTan Let $x\succ y$. There is a neighborhood of $y$ such that for every point $z$ in the neighborhood, $x\gg z$. The neighborhood must also contain a point $z$ such that $z\succ y$ by local nonsatiation. Then, $x\geq z\succ y$. By M1, $x\succeq z\succ y$. Hence, $x\succ y$.
Michael Greinecker
CC BY-SA 4.0
2024-09-29T06:54:44.027Z
@MichaelGreinecker. Thanks. However I am struggling to understand why if $x \succ y$, then there is a neighborhood of $y$ such that for every point $z$ in the neighborhood, $x \gg z$. 

If only M1 is given, then M1 keeps silent on what relation between $x$ and $y$ must make $x \succ y$. So, for example, if $x=(1,2)$ and $y=(2,1)$, then it is possible that we have $x \succ y$ under M1. However, _any_ neighborhood of $y$ has at least one point $z$ such that $x \not\gg z$.
Shujun Tan
CC BY-SA 4.0
2024-09-29T08:09:29.187Z
@ShujunTan Sorry, that's a typo. The first $x\succ y$ should be $x\gg y$.
Michael Greinecker
<consumer-theory><preferences><consumption>
Are weakly monotonic preference and strictly monotomic preference the same?
Shujun Tan
# Are weakly monotonic preference and strictly monotomic preference the same?

(Asked by: username_0 on 2024-09-27)

Recently I am reviewing preference theory and I found that I am confused about the difference between weakly monotonic preference and strictly monotonic preference.

Now, let us suppose $X$ is the consumption set. If my understanding is correct, then:

* Preference is said to be *weakly* monotonic if for $x,y \in X$, $y \gg x$ implies $y \succ x$.
* Preference is said to be *strictly* monotonic if for $x,y \in X$, $y \geq x$ implies $y \succeq x$, and $y \gg x$ implies $y \succ x$.

So it seems that weakly monotonic preference says nothing when $y \geq x$ but not $y \gg x$ (for example, $y=(1,1,2)$ and $x=(1,1,1)$)? Maybe we can have $y \prec x$ if the preference is weakly monotonic?

**Tags:** consumer-theory, preferences, consumption
> **Comments:**
> **username_1**: Weak monotonicity, according to your definition, is just being *agnostic* about cases where $x\ge y$. It is "weak" because it imposes one fewer condition relative to the "strict" version. Also note that different authors (e.g. Kreps (2013)) define monotonicity differently (see [related question](https://economics.stackexchange.com/q/55185/42)).  
> **username_2**: For continuous preferences, these are equivalent. Usually, "strictly monotone" refers to the stronger condition that $y\geq x$ and $y\neq x$ implies $y\succ x$.  
> **username_0**: Thank all of you very much. The info you provided is really helpful.  
> **username_0**: @MichaelGreinecker Hi, in the question https://economics.stackexchange.com/questions/55185/conflicting-definitions-of-weak-monotnocity-preferences you mentioned that if $\succeq$ is locally nonsatiated, then M1 implies M2. However I failed to prove it, could you explain how can we approach this conclusion?  
> **username_2**: @ShujunTan Let $x\succ y$. There is a neighborhood of $y$ such that for every point $z$ in the neighborhood, $x\gg z$. The neighborhood must also contain a point $z$ such that $z\succ y$ by local nonsatiation. Then, $x\geq z\succ y$. By M1, $x\succeq z\succ y$. Hence, $x\succ y$.  
> **username_0**: @MichaelGreinecker. Thanks. However I am struggling to understand why if $x \succ y$, then there is a neighborhood of $y$ such that for every point $z$ in the neighborhood, $x \gg z$.   If only M1 is given, then M1 keeps silent on what relation between $x$ and $y$ must make $x \succ y$. So, for example, if $x=(1,2)$ and $y=(2,1)$, then it is possible that we have $x \succ y$ under M1. However, _any_ neighborhood of $y$ has at least one point $z$ such that $x \not\gg z$.  
> **username_2**: @ShujunTan Sorry, that's a typo. The first $x\succ y$ should be $x\gg y$.  

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## 1 answers

### Answer by username_3 on 2024-09-28. Score: 1

You're on the right track in understanding the difference between weakly monotonic and strictly monotonic preferences.

**1. Weakly Monotonic Preferences:**
Preferences are weakly monotonic if for any two consumption bundles x and y, if y strictly greater than x (meaning that every element in y is strictly greater than the corresponding element in x), then y > x (you strictly prefer y over x).

**What it doesn't guarantee:**
Weakly monotonic preferences do not require a clear preference between two bundles if one is only greater than or equal to the other, but not strictly greater (e.g., y >= x, such as y = (1,1,2) and x = (1,1,1)).
In these cases, the preference might not be clear, and it's possible under weakly monotonic
preferences that you might actually prefer x over y (i.e., y <= x), or be indifferent. Weak monotonicity doesn't tell us anything about this scenario.

**2. Strictly Monotonic Preferences:**
Preferences are strictly monotonic if: - For y >= x, you at least weakly prefer y over x (i.e., y <= x). This means if y is greater than or equal to x in all dimensions,you cannot prefer x over y, though you might be indifferent. - For y strictly greater than x, you strictly prefer y over x (i.e., y > x).

**Key Difference:**

Weakly monotonic: Only ensures strict preference if y strictly greater than x. It is silent on what happens when y >= x but not y strictly greater than x, so it's possible to have y <= x in this case.
Strictly monotonic: Always ensures that y >= x means you at least weakly prefer y, and if y strictly greater than x, you strictly prefer y over x.

In your example, if preferences are weakly monotonic, it's possible for y = (1,1,2) to be strictly worse than x = (1,1,1), since weak monotonicity doesn't provide guidance when y >= x but not y strictly greater than x. However, in the case of strictly monotonic preferences,y = (1,1,2) would be at least weakly preferred to x = (1,1,1).


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