<p>Consider three individuals with preferences <span class="math-container">$\succ_1, \succ_2$</span> and <span class="math-container">$\succ_3$</span>.</p>
<p>Consider the following preferences (for matchings)
<span class="math-container">$$
\begin{align*}
&amp;2 \succ_1 3 \succ_1 \emptyset\\
&amp;3 \succ_2 1 \succ_2 \emptyset\\
&amp;1 \succ_3 2 \succ_3 \emptyset.
\end{align*}
$$</span>
Here <span class="math-container">$\emptyset$</span> is the situation where the individual is not matched.</p>
<ul>
<li>If 1 is matched with 2, the couple (2,3) forms a blocking pair.</li>
<li>If 1 is matched with 3, the couple (1,2) forms a blocking pair.</li>
<li>If 2 is matched with 3, the couple (1,3) forms a blocking pair.</li>
</ul>

CC BY-SA 4.0
2024-11-12T08:36:19.597Z
2024-11-12T08:36:19.597Z
tdm
<p>I heard that the stable matching of a roomate problem might not exist.</p>
<p>My question is, what is a simple example illustrating the non-existence, and if there is simpler matching problem where stable matching does not exist?</p>

<microeconomics><game-theory><theory><matching><cooperative-game-theory>
The simplest example where stable matching does not exist?
High GPA
# The simplest example where stable matching does not exist?

(Asked by: username_0 on 2024-11-11)

I heard that the stable matching of a roomate problem might not exist.

My question is, what is a simple example illustrating the non-existence, and if there is simpler matching problem where stable matching does not exist?

**Tags:** microeconomics, game-theory, theory, matching, cooperative-game-theory

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## 1 answers

### Answer by username_1 on 2024-11-12. Score: 3 | Accepted answer

Consider three individuals with preferences $\succ\_1, \succ\_2$ and $\succ\_3$.

Consider the following preferences (for matchings)
$$
\begin{align\*}
&2 \succ\_1 3 \succ\_1 \emptyset\\
&3 \succ\_2 1 \succ\_2 \emptyset\\
&1 \succ\_3 2 \succ\_3 \emptyset.
\end{align\*}
$$
Here $\emptyset$ is the situation where the individual is not matched.

* If 1 is matched with 2, the couple (2,3) forms a blocking pair.
* If 1 is matched with 3, the couple (1,2) forms a blocking pair.
* If 2 is matched with 3, the couple (1,3) forms a blocking pair.


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