# Core vs. Strong Core in Housing Allocation Games

I am presently reviewing the course notes for my Game Theory course, and I'm struggling with the concepts of the core vs. the strong core. In the notes, we have three players, with preferences regarding houses (ranked from most to least preferred):

-$P_{1}: h_{3}, h_{2}, h_{1}$
-$P_{2}: h_{1}, h_{2}, h_{3}$
-$P_{3}: h_{2}, h_{3}, h_{1}$

The course notes tell us that the allocation $(h_{2}, h_{1}, h_{3})$ is in the core. However, couldn't $(P_{1}, P_{3})$ form a blocking coalition? $P_{1}$ and $P_{3}$ could swap amongst themselves and both strictly improve, which would violate the definition of being in the core.

The definition for the core with which I'm working is (for $A$ our set of players): The set of matchings $\mu$ such that there exists no coalition $B \subset A$ and a matching $v$ such that:
(a) For any $a \in B$, $v(a) = h_{l}$ for some $a_{l} \in B$. (That is, members of the coalition can only trade amongst themselves).

(b) For any $a \in B$, $v(a) \succeq_{a} \mu(a)$ and for some $b \in B$, $v(b) \succ_{b} \mu(b)$.

Also, if anyone can clarify between the core and strong core, that would be greatly appreciated! Thank you in advance for any help.

## 1 Answer

A Course on Cooperative Game Theory by Chakravarty et al. gives the simplest textual definitions I've seen on this. See page 164-165.

Core: An assignment of houses to owners is in the core of the house exchange market if there does not exist a coalition $S$ of house owners that can redistribute the houses they own such that they all prefer the houses resulting from the reallocation to the houses they are getting under the assignment. Then the core is defined as the set of all such assignments.

Strong Core: An assignment of houses to owners is in the strong core if there exists no coalition that could make all its members at least as good as and at least one member better off.

Therefore, any allocation in the strong core is also in the core.

Regarding the example you give, you're correct that $(h_2,h_1,h_3)$ is neither in the core nor the strong core. I'm guessing there was a typo somewhere in those preferences. If you're assigning three agents to three houses and they each prefer different houses, then the assignment problem is pretty trivial. The core (and strong core) consists only of $(h_3,h_1,h_2)$. There's really no scarcity here, as each agent can get her first choice. Without scarcity, there's no economics!

• That makes a lot of sense! Thank you for your help. – ml0105 Apr 2 '15 at 19:56