<ol>
<li>Most of what you write is correct, but the definitions of the $F_i$ sets is imprecise. The problem is that in the core $A$ and $B$ may get goods that do not match their initial endowments. In this case it is not true that the core allocation $x$ is Pareto-efficient in the restricted 2-person economy of $A$ and $B$, because $x$ is not even an allocation in that game.</li>
</ol>

<hr>

<p>Edit: (An example)</p>

<p>Consider the initial endowments
$$
\omega_{A} = (1,1),  \omega_{B} = (1,1), \omega_{C} = (2,2)
$$
and an allocation $x$
$$
x_A = (2,2), x_B = (2,2), x_C = (0,0).
$$
$A$ and $B$ cannot Pareto-improve on $x$. But $x$ is not a Pareto-efficient allocation in their 2-person economy, because it is not a feasible allocation of their economy:
$$
x_A + x_B \neq \omega_{A} + \omega_{B}
$$</p>

<hr>

<p>A better definition for the sets $F_i$  would be something like:</p>

<p>Let us denote the set of feasible allocations of the 2-person economy of $A$ and $B$ by $Y_{A,B} \subset \mathbb{R}^{2}$. Then
$$
F_{1} =\{ x \in \mathbb{R}^{2} : \nexists y \in Y_{A,B} \text{ such that } u_{A}(y_{A}) \geq u_{A}(x_{A}), u_{B}(y_{B}) &gt; u_{B}(x_{B}) \}
$$
There are still some issues with cases when $A$ is better off and $B$ is not worse off, but if the utility functions are continuous then this should not cause a problem.</p>

<p>You can define $F_{2}$ and $F_{3}$ in a similar manner.</p>

<p>A remark: $E$ is not 'special', it is the set of allocations that cannot be improved upon by the three player coalition. This is equivalent to Pareto-efficiency.</p>

<ol start="2">
<li><p>Yes. Why not? This is exactly what the core is.</p></li>
<li><p>I would not call it solving, because usually you do not get a unique solution, and in extreme cases you may get no solution. But yes, every economy ($n$ players, $m$ goods) has a core, and it is defined in this way. (As indicated, unless some conditions are met the core may be empty. A competitive equilibrium is always an element of the core, so if that exists, the core is non-empty.)</p></li>
</ol>

CC BY-SA 3.0
2015-12-18T18:46:29.873Z
First of all, thanks for your answer! However, I am not clear on the difference between the way you have defined $F_{1}$ and the way I have defined it as the pareto efficient outcome in a two person game between $A$ and $B$. It seems to me that the way you have defined $F_{1}$ is equivalent to "there does not exist a feasible allocation at least as good for $A$ and $B$, but strictly preferred by $A$ or $B$". Isn't this the definition of pareto efficiency? I am confused with this.
möbius
CC BY-SA 3.0
2015-12-18T19:09:41.197Z
@möbius I said that for allocation $x$ there is no Pareto-improvement that $A$ and $B$ can do. You said that allocation is $x$ is Pareto-efficient. The two are only the same if $A$ and $B$ can make allocation $x$ happen. I edited my answer to include an example of the difference.
Giskard
CC BY-SA 3.0
2015-12-19T15:32:02.540Z
I see what you mean. So I needed to change "pareto efficient" in $F_{1}$ to "pareto efficient and feasible"?
möbius
CC BY-SA 3.0
2015-12-19T18:15:54.737Z
@möbius No, the allocation $x$ does not have to feasible for any coalition other than the grand coalitions. Only the other allocation with which coalition $S$ blocks $x$ has to be feasible for $S$. I am afraid there is no very simple way of putting it. Perhaps the best way you could phrase it is: $x$ is such that no coalition can Pareto-improve upon their share using only their own resources.
Giskard
CC BY-SA 3.0
2015-12-18T18:35:25.540Z
2015-12-18T19:21:29.217Z
2015-12-18T19:21:29.217Z
Giskard
<p>I have been reviewing general equilibrium models and was trying to find an efficient method for computing the core of a cooperative game. I was taught this topic in a very poor way so I believe I still have some conceptual errors. </p>

<p>Here is a thought I had:</p>

<p>Suppose we are in an economy with three consumers, $A$, $B$, and $C$, with utility $u_{i}(x)$ defined over bundles $x \in \mathbb{R}^{2}$ and endowments $\omega_{i}$ for $i = A, B ,C$. I want to compute the core for this economy.</p>

<p>I know the core must satisfy:
\begin{align}
u_{A}(x_{A}) &amp;\geq u_{A}(\omega_{A})\\
u_{B}(x_{B}) &amp;\geq u_{B}(\omega_{B})\\
u_{C}(x_{C}) &amp;\geq u_{C}(\omega_{C})\\
\end{align}
i.e. the core must be individually rational. So let $$D =\{ x \in \mathbb{R}^{2} : x \text{ is individually rational for $A$, $B$ and $C$} \}$$ I also know that the core is a subset of the pareto efficient outcomes, so let $$E =\{ x \in \mathbb{R}^{2} : x \text{ is pareto efficient} \}$$ Now here is the part I am not sure about: I know that the core is also not blocked by any two-person coalition. I think this means that any core allocation is pareto efficient for any two-person game. Thus I define: 
\begin{align}
F_{1} =\{ x \in \mathbb{R}^{2} : x \text{ is pareto efficient in the cooperative game with only $A$ and $B$ } \}\\
F_{2} =\{ x \in \mathbb{R}^{2} : x \text{ is pareto efficient in the cooperative game with only $A$ and $C$ } \}\\
F_{3} =\{ x \in \mathbb{R}^{2} : x \text{ is pareto efficient in the cooperative game with only $B$ and $C$ } \}
\end{align} </p>

<p>Here are my questions:</p>

<ol>
<li>Is the above analysis correct?</li>
<li>Can I write the set of core allocations $\mathcal{C}$ as $$\mathcal{C} = D \cap E \cap F_{1} \cap F_{2} \cap F_{3}\text{?}$$</li>
<li>Can this method of solving be generalized to a game with $n$ players and $m$ goods?</li>
</ol>

<p>Let me know if anything is not clear!</p>

<general-equilibrium><pareto-efficiency><cooperative-game-theory>
Writing the core as the intersection of pareto efficient outcomes of all coalitions
möbius
# Writing the core as the intersection of pareto efficient outcomes of all coalitions

(Asked by: username_0 on 2015-12-18)

I have been reviewing general equilibrium models and was trying to find an efficient method for computing the core of a cooperative game. I was taught this topic in a very poor way so I believe I still have some conceptual errors.

Here is a thought I had:

Suppose we are in an economy with three consumers, $A$, $B$, and $C$, with utility $u\_{i}(x)$ defined over bundles $x \in \mathbb{R}^{2}$ and endowments $\omega\_{i}$ for $i = A, B ,C$. I want to compute the core for this economy.

I know the core must satisfy:
\begin{align}
u\_{A}(x\_{A}) &\geq u\_{A}(\omega\_{A})\\
u\_{B}(x\_{B}) &\geq u\_{B}(\omega\_{B})\\
u\_{C}(x\_{C}) &\geq u\_{C}(\omega\_{C})\\
\end{align}
i.e. the core must be individually rational. So let $$D =\{ x \in \mathbb{R}^{2} : x \text{ is individually rational for $A$, $B$ and $C$} \}$$ I also know that the core is a subset of the pareto efficient outcomes, so let $$E =\{ x \in \mathbb{R}^{2} : x \text{ is pareto efficient} \}$$ Now here is the part I am not sure about: I know that the core is also not blocked by any two-person coalition. I think this means that any core allocation is pareto efficient for any two-person game. Thus I define:
\begin{align}
F\_{1} =\{ x \in \mathbb{R}^{2} : x \text{ is pareto efficient in the cooperative game with only $A$ and $B$ } \}\\
F\_{2} =\{ x \in \mathbb{R}^{2} : x \text{ is pareto efficient in the cooperative game with only $A$ and $C$ } \}\\
F\_{3} =\{ x \in \mathbb{R}^{2} : x \text{ is pareto efficient in the cooperative game with only $B$ and $C$ } \}
\end{align}

Here are my questions:

1. Is the above analysis correct?
2. Can I write the set of core allocations $\mathcal{C}$ as $$\mathcal{C} = D \cap E \cap F\_{1} \cap F\_{2} \cap F\_{3}\text{?}$$
3. Can this method of solving be generalized to a game with $n$ players and $m$ goods?

Let me know if anything is not clear!

**Tags:** general-equilibrium, pareto-efficiency, cooperative-game-theory


---

## 1 answers

### Answer by username_1 on 2015-12-18. Score: 2 | Accepted answer

1. Most of what you write is correct, but the definitions of the $F\_i$ sets is imprecise. The problem is that in the core $A$ and $B$ may get goods that do not match their initial endowments. In this case it is not true that the core allocation $x$ is Pareto-efficient in the restricted 2-person economy of $A$ and $B$, because $x$ is not even an allocation in that game.

---

Edit: (An example)

Consider the initial endowments
$$
\omega\_{A} = (1,1), \omega\_{B} = (1,1), \omega\_{C} = (2,2)
$$
and an allocation $x$
$$
x\_A = (2,2), x\_B = (2,2), x\_C = (0,0).
$$
$A$ and $B$ cannot Pareto-improve on $x$. But $x$ is not a Pareto-efficient allocation in their 2-person economy, because it is not a feasible allocation of their economy:
$$
x\_A + x\_B \neq \omega\_{A} + \omega\_{B}
$$

---

A better definition for the sets $F\_i$ would be something like:

Let us denote the set of feasible allocations of the 2-person economy of $A$ and $B$ by $Y\_{A,B} \subset \mathbb{R}^{2}$. Then
$$
F\_{1} =\{ x \in \mathbb{R}^{2} : \nexists y \in Y\_{A,B} \text{ such that } u\_{A}(y\_{A}) \geq u\_{A}(x\_{A}), u\_{B}(y\_{B}) > u\_{B}(x\_{B}) \}
$$
There are still some issues with cases when $A$ is better off and $B$ is not worse off, but if the utility functions are continuous then this should not cause a problem.

You can define $F\_{2}$ and $F\_{3}$ in a similar manner.

A remark: $E$ is not 'special', it is the set of allocations that cannot be improved upon by the three player coalition. This is equivalent to Pareto-efficiency.

2. Yes. Why not? This is exactly what the core is.
3. I would not call it solving, because usually you do not get a unique solution, and in extreme cases you may get no solution. But yes, every economy ($n$ players, $m$ goods) has a core, and it is defined in this way. (As indicated, unless some conditions are met the core may be empty. A competitive equilibrium is always an element of the core, so if that exists, the core is non-empty.)

> **Comments:**
> **username_0**: First of all, thanks for your answer! However, I am not clear on the difference between the way you have defined $F_{1}$ and the way I have defined it as the pareto efficient outcome in a two person game between $A$ and $B$. It seems to me that the way you have defined $F_{1}$ is equivalent to "there does not exist a feasible allocation at least as good for $A$ and $B$, but strictly preferred by $A$ or $B$". Isn't this the definition of pareto efficiency? I am confused with this.  
> **username_1**: @username_0 I said that for allocation $x$ there is no Pareto-improvement that $A$ and $B$ can do. You said that allocation is $x$ is Pareto-efficient. The two are only the same if $A$ and $B$ can make allocation $x$ happen. I edited my answer to include an example of the difference.  
> **username_0**: I see what you mean. So I needed to change "pareto efficient" in $F_{1}$ to "pareto efficient and feasible"?  
> **username_1**: @username_0 No, the allocation $x$ does not have to feasible for any coalition other than the grand coalitions. Only the other allocation with which coalition $S$ blocks $x$ has to be feasible for $S$. I am afraid there is no very simple way of putting it. Perhaps the best way you could phrase it is: $x$ is such that no coalition can Pareto-improve upon their share using only their own resources.  

---

