Writing the core as the intersection of pareto efficient outcomes of all coalitions

Question

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!

Answer 1

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.)