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!


1 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.

  1. Yes. Why not? This is exactly what the core is.

  2. 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.)

  • $\begingroup$ 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. $\endgroup$
    – möbius
    Commented Dec 18, 2015 at 18:46
  • $\begingroup$ @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. $\endgroup$
    – Giskard
    Commented Dec 18, 2015 at 19:09
  • $\begingroup$ I see what you mean. So I needed to change "pareto efficient" in $F_{1}$ to "pareto efficient and feasible"? $\endgroup$
    – möbius
    Commented Dec 19, 2015 at 15:32
  • $\begingroup$ @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. $\endgroup$
    – Giskard
    Commented Dec 19, 2015 at 18:15

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.