# Can we define the core as the intersection of Pareto efficient allocations and upper-contour sets?

Suppose that an economy has $$n$$ agents, each with endowment $$\omega_i$$. Their preferences are represented by the quasiconcave and increasing function $$u_i$$. Let the set of Pareto efficient allocations be $$E$$ and let $$C_i$$ be the upper-contour set for $$u_i(\omega_i)$$ for all $$i$$.

My question is: can we define the core as the intersection of $$E$$ and all $$C_i$$, so $$core = E \cap \left(\bigcap_{i = 1}^n C_i\right)$$? I think this definition works when $$n = 2$$, but what about when $$n > 2$$?

Assume there are three individuals $$A, B, C$$ and three goods $$x, y, z$$. Endowment for $$A$$ is $$(1,0,0)$$ for $$B$$ is $$(0,1,0)$$ and for $$C$$ is $$(0,0,1)$$.
Assume the utility function for the individuals are: $$u^A(x,y,z) = xy + z, u^B(x,y,z) = x + yz, U^C(x,y,z) = xz + y$$ The utilities at the endowments are $$u^A(1,0,0) = 0, u^B(0,1,0) = 0, u^C(0,0,1) = 0.$$ Now consider the allocation where $$A$$ gets $$(0,0,0)$$ for $$B$$ receives $$(1,1,1)$$ and for $$C$$ receives $$(0,0,0)$$. This gives utilities: $$u^A(0,0,0) = 0, u^B(1,1,1) = 2, u^C(0,0,0) = 0,$$ So these are individual rational. It is also clear that this is Pareto efficient as any reallocation would decrease the utility of $$B$$.
Note however that the coalition $$\{A, C\}$$ has a profitable deviation. For example, they could switch their endowments, so $$A$$ consumes $$(0, 0, 1)$$ and $$C$$ consumes $$(1, 0,0)$$, giving both a utility equal to 1. So the above Pareto efficient and individual rational allocation is not in the core.