<p>A relation <span class="math-container">$P$</span> is <em>negative transitive</em> if <span class="math-container">$\neg(xPy)$</span> and <span class="math-container">$\neg(yPz)$</span> imply <span class="math-container">$\neg(xPz)$</span>.</p>
<p>If <span class="math-container">$\succeq$</span> is a transitive and complete relation, then the relation <span class="math-container">$\succ$</span> defined by <span class="math-container">$x\succ y\iff(x\succeq y)\wedge\neg(y\succeq x)$</span> is irreflexive, transitive, and negative transitive. The proof is a simple exercise.</p>
<p>Conversely, if <span class="math-container">$\succ$</span> is irreflexive, transitive, and negative transitive, then the relation <span class="math-container">$\succeq$</span> defined by <span class="math-container">$x\succeq y\iff\neg(y\succ x)$</span> is transitive and complete. Transitivity of <span class="math-container">$\succeq$</span> follows directly from the negative transitivity of <span class="math-container">$\succ$</span>. To see that <span class="math-container">$\succeq$</span> is complete, we argue by contradiction. Suppose that for some <span class="math-container">$x,y$</span>, neither <span class="math-container">$y\succeq x$</span> nor <span class="math-container">$x\succeq y$</span> holds. Then we have, by definition, both <span class="math-container">$x\succ y$</span> and <span class="math-container">$y\succ x$</span>. Since <span class="math-container">$\succ$</span> is transitive, we have <span class="math-container">$x\succ x$</span>, in contradiction to <span class="math-container">$\succ$</span> being irreflexive.</p>
<p>Note that the argument also works if we replace transitivity and irreflexivity with asymmetry.</p>

CC BY-SA 4.0
2024-05-16T07:47:50.597Z
Thank you very much for your answer. Does the argument also work if transitivity and reflexivity are replaced by **antisymmetry**?
EoDmnFOr3q
CC BY-SA 4.0
2024-05-16T07:57:09.577Z
No, it does not.
Michael Greinecker
CC BY-SA 4.0
2024-05-16T08:52:21.483Z
Thank you for your clarification. Then, if I understand it correctly, a strict preference relation satisfies **antisymmetry** but there exists no collection of properties on binary relations that uniquely identifies strict preference relations AND includes **antisymmetry**. Is that right=
EoDmnFOr3q
CC BY-SA 4.0
2024-05-16T09:15:09.303Z
@EoDmnFOr3q A strict preference relation trivially satisfies antisymmetry (the LHS of the definition is always false), so the collection of properties {asymmetric, negative transitive, antisymmetric} does "uniquely identify a strict preference relation", but this collection is not minimal as the antisymmetry property is redundant in it.
VARulle
CC BY-SA 4.0
2024-05-16T09:24:17.883Z
@VARulle Thank you for your comment. But then, if asymmetry is redundant in it, why does {negative transitivity, antisymmetry} not uniquely identify a strict preference relation?
EoDmnFOr3q
CC BY-SA 4.0
2024-05-16T09:26:39.673Z
@EoDmnFOr3q Asymmetry is not redundant, antisymmetry is!
VARulle
CC BY-SA 4.0
2024-05-16T09:28:38.937Z
My apologies—I didn’t read your comment carefully enough. Thank you: I think I begin to understand why I was confused.
EoDmnFOr3q
CC BY-SA 4.0
2024-05-16T07:35:40.703Z
2024-05-16T08:57:12.040Z
2024-05-16T08:57:12.040Z
Michael Greinecker
<p>I am wondering which are the standard axioms with which indifference and (particularly) strict preference are usually defined. Normally, a weak preference relation for agent <span class="math-container">$i$</span> on a set <span class="math-container">$X$</span>, denoted <span class="math-container">$R_i$</span>, is a binary relation satisfying:</p>
<ol>
<li><strong>Transitivity</strong>: for all <span class="math-container">$a,b,c\in X$</span>, <span class="math-container">$aR_ib$</span> and <span class="math-container">$bR_ic$</span> imply <span class="math-container">$aR_ic$</span>.</li>
<li><strong>Completeness</strong>: for all <span class="math-container">$a,b\in X$</span>, <span class="math-container">$aR_ib$</span> or <span class="math-container">$bR_ia$</span>.</li>
<li><strong>Reflexivity</strong>: for all <span class="math-container">$a\in X$</span>, <span class="math-container">$aR_ia$</span> (<em>implied by completeness</em>).</li>
</ol>
<p>It is my understanding that an indifference relation, denoted <span class="math-container">$I_i$</span>, is a binary relation satisfying:</p>
<ol>
<li><strong>Transitivity</strong>: for all <span class="math-container">$a,b,c\in X$</span>, <span class="math-container">$aI_ib$</span> and <span class="math-container">$bI_ic$</span> imply <span class="math-container">$aI_ic$</span>.</li>
<li><strong>Reflexivity</strong>: for all <span class="math-container">$a\in X$</span>, <span class="math-container">$aI_ia$</span>.</li>
<li><strong>Symmetry</strong>: for all <span class="math-container">$a,b\in X$</span>, <span class="math-container">$aI_ib$</span> if and only if <span class="math-container">$bI_ia$</span>.</li>
</ol>
<p>Finally, a strict preference relation, denoted <span class="math-container">$P_i$</span>, is a binary relation satisfying:</p>
<ol>
<li><strong>Transitivity</strong>: for all <span class="math-container">$a,b,c\in X$</span>, <span class="math-container">$aP_ib$</span> and <span class="math-container">$bP_ic$</span> imply <span class="math-container">$aP_ic$</span>.</li>
<li><strong>Irreflexivity</strong>: for all <span class="math-container">$a\in X$</span>, <span class="math-container">$\neg aP_ia$</span>.</li>
</ol>
<p>However,  a strict preference relation also seems to satisfy:</p>
<ol>
<li><strong>Asymmetry</strong>: for all <span class="math-container">$a,b\in X$</span>, <span class="math-container">$aP_ib$</span> implies <span class="math-container">$\neg bP_ia$</span>.</li>
<li><strong>Antisymmetry</strong>: for all <span class="math-container">$a,b\in X$</span>, <span class="math-container">$aP_ib$</span> and <span class="math-container">$bP_ia$</span> imply <span class="math-container">$a=b$</span>.</li>
</ol>
<p>I have a few closely related questions:</p>
<ol>
<li>What are the most common axioms to define a strict preference relation?</li>
<li>Do there exist different yet correct collections of axioms to define strict preference? If so, which ones are those?</li>
<li>Are my axioms for weak preference and indifference correct?</li>
</ol>

CC BY-SA 4.0
2024-05-15T21:01:44.990Z
What does it mean for axioms to be "correct?"
Michael Greinecker
CC BY-SA 4.0
2024-05-15T21:17:36.800Z
Btw: transitivity+irreflexivity implies asymmetry, and asymmetry implies antisymmetry.
Michael Greinecker
CC BY-SA 4.0
2024-05-16T05:20:31.153Z
Thank you for your comments. By correct, I mean “*uniquely identifying axioms*” (i.e., *if and only if*). I understand from your comment that a binary relation is a *strict preference relation* if and only if it is *transitive* and *irreflexive*. However, I don’t see how **asymmetry** and **antisymmetry** can be used to define a strict preference relation. Any help?
EoDmnFOr3q
CC BY-SA 4.0
2024-05-16T07:15:28.747Z
I'm even more confused now. My comment says exactly what it says. Do you want to have conditions on $P$ such that for some transitive, complete, reflexive relation $R$, one has $xPy$ if and only if $xRy$ but not $yRx$?
Michael Greinecker
CC BY-SA 4.0
2024-05-16T07:19:27.137Z
Thank you for your new comment and apologies for the confusion I created. As you point out, I want to have conditions on $P_i$ such that for some transitive, complete and reflexive binary relation $R_i$, one has $xP_i$ if and only if $xR_iy$ but not $yR_ix$. I understand *transitivity* and *irreflexivity* constitute one such set of conditions. I am wondering whether one can have a different set of conditions which include **assymetry** and/or **antisymmetry**.
EoDmnFOr3q
CC BY-SA 4.0
2024-05-16T07:23:31.787Z
Transitivity and irreflexivity are not enough. But I can answer your question now.
Michael Greinecker
CC BY-SA 4.0
2024-05-16T07:32:28.963Z
Thank you again for your time. After reading Proposition 1.B.1 in Mas-Colell, Whinston and Green (1995), I thought transitivity and reflexivity were enough. Said Proposition states that a strict preference derived from a transitive, reflexive and complete binary relation is irreflexive and transitive. Since there is no mention of either asymmetry or antisymmetry, I thought transitivity and irreflexivity were enough, which left me a but confused…
EoDmnFOr3q
<microeconomics><preferences><theory>
What are the most common axioms to define a strict preference relation?
EoDmnFOr3q
# What are the most common axioms to define a strict preference relation?

(Asked by: username_0 on 2024-05-15)

I am wondering which are the standard axioms with which indifference and (particularly) strict preference are usually defined. Normally, a weak preference relation for agent $i$ on a set $X$, denoted $R\_i$, is a binary relation satisfying:

1. **Transitivity**: for all $a,b,c\in X$, $aR\_ib$ and $bR\_ic$ imply $aR\_ic$.
2. **Completeness**: for all $a,b\in X$, $aR\_ib$ or $bR\_ia$.
3. **Reflexivity**: for all $a\in X$, $aR\_ia$ (*implied by completeness*).

It is my understanding that an indifference relation, denoted $I\_i$, is a binary relation satisfying:

1. **Transitivity**: for all $a,b,c\in X$, $aI\_ib$ and $bI\_ic$ imply $aI\_ic$.
2. **Reflexivity**: for all $a\in X$, $aI\_ia$.
3. **Symmetry**: for all $a,b\in X$, $aI\_ib$ if and only if $bI\_ia$.

Finally, a strict preference relation, denoted $P\_i$, is a binary relation satisfying:

1. **Transitivity**: for all $a,b,c\in X$, $aP\_ib$ and $bP\_ic$ imply $aP\_ic$.
2. **Irreflexivity**: for all $a\in X$, $\neg aP\_ia$.

However, a strict preference relation also seems to satisfy:

1. **Asymmetry**: for all $a,b\in X$, $aP\_ib$ implies $\neg bP\_ia$.
2. **Antisymmetry**: for all $a,b\in X$, $aP\_ib$ and $bP\_ia$ imply $a=b$.

I have a few closely related questions:

1. What are the most common axioms to define a strict preference relation?
2. Do there exist different yet correct collections of axioms to define strict preference? If so, which ones are those?
3. Are my axioms for weak preference and indifference correct?

**Tags:** microeconomics, preferences, theory

> **Comments:**
> **username_1**: What does it mean for axioms to be "correct?"  
> **username_1**: Btw: transitivity+irreflexivity implies asymmetry, and asymmetry implies antisymmetry.  
> **username_0**: Thank you for your comments. By correct, I mean “*uniquely identifying axioms*” (i.e., *if and only if*). I understand from your comment that a binary relation is a *strict preference relation* if and only if it is *transitive* and *irreflexive*. However, I don’t see how **asymmetry** and **antisymmetry** can be used to define a strict preference relation. Any help?  
> **username_1**: I'm even more confused now. My comment says exactly what it says. Do you want to have conditions on $P$ such that for some transitive, complete, reflexive relation $R$, one has $xPy$ if and only if $xRy$ but not $yRx$?  
> **username_0**: Thank you for your new comment and apologies for the confusion I created. As you point out, I want to have conditions on $P_i$ such that for some transitive, complete and reflexive binary relation $R_i$, one has $xP_i$ if and only if $xR_iy$ but not $yR_ix$. I understand *transitivity* and *irreflexivity* constitute one such set of conditions. I am wondering whether one can have a different set of conditions which include **assymetry** and/or **antisymmetry**.  
> **username_1**: Transitivity and irreflexivity are not enough. But I can answer your question now.  
> **username_0**: Thank you again for your time. After reading Proposition 1.B.1 in Mas-Colell, Whinston and Green (1995), I thought transitivity and reflexivity were enough. Said Proposition states that a strict preference derived from a transitive, reflexive and complete binary relation is irreflexive and transitive. Since there is no mention of either asymmetry or antisymmetry, I thought transitivity and irreflexivity were enough, which left me a but confused…  

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## 1 answers

### Answer by username_1 on 2024-05-16. Score: 1 | Accepted answer

A relation $P$ is *negative transitive* if $\neg(xPy)$ and $\neg(yPz)$ imply $\neg(xPz)$.

If $\succeq$ is a transitive and complete relation, then the relation $\succ$ defined by $x\succ y\iff(x\succeq y)\wedge\neg(y\succeq x)$ is irreflexive, transitive, and negative transitive. The proof is a simple exercise.

Conversely, if $\succ$ is irreflexive, transitive, and negative transitive, then the relation $\succeq$ defined by $x\succeq y\iff\neg(y\succ x)$ is transitive and complete. Transitivity of $\succeq$ follows directly from the negative transitivity of $\succ$. To see that $\succeq$ is complete, we argue by contradiction. Suppose that for some $x,y$, neither $y\succeq x$ nor $x\succeq y$ holds. Then we have, by definition, both $x\succ y$ and $y\succ x$. Since $\succ$ is transitive, we have $x\succ x$, in contradiction to $\succ$ being irreflexive.

Note that the argument also works if we replace transitivity and irreflexivity with asymmetry.

> **Comments:**
> **username_0**: Thank you very much for your answer. Does the argument also work if transitivity and reflexivity are replaced by **antisymmetry**?  
> **username_1**: No, it does not.  
> **username_0**: Thank you for your clarification. Then, if I understand it correctly, a strict preference relation satisfies **antisymmetry** but there exists no collection of properties on binary relations that uniquely identifies strict preference relations AND includes **antisymmetry**. Is that right=  
> **username_2**: @username_0 A strict preference relation trivially satisfies antisymmetry (the LHS of the definition is always false), so the collection of properties {asymmetric, negative transitive, antisymmetric} does "uniquely identify a strict preference relation", but this collection is not minimal as the antisymmetry property is redundant in it.  
> **username_0**: @username_2 Thank you for your comment. But then, if asymmetry is redundant in it, why does {negative transitivity, antisymmetry} not uniquely identify a strict preference relation?  
> **username_2**: @username_0 Asymmetry is not redundant, antisymmetry is!  
> **username_0**: My apologies—I didn’t read your comment carefully enough. Thank you: I think I begin to understand why I was confused.  

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