Blocking Coalitions

Drag the division around and watch who objects. If some group can object, the division falls.

If one 🧑 person ends up worse than going alone, that person walks out. If a group can split things better together, it falls too. Only divisions nobody can walk away from survive. That surviving set is the core.

For example, three friends split 10 🍬 candies. If one friend gets only 1 candy, that friend walks away. If two friends secretly find a better split, the original split collapses. Only a split nobody wants to leave stays standing.

x∈core  ⟺  no S blocks xx \in \text{core} \iff \text{no } S \text{ blocks } x

Examples & Exercises

With the allocation x1=(7,3)x^1 = (7, 3) (i.e., the endowment itself), determine which coalition blocks it.

MRS1(7,3)=3/7≠7/3=MRS2(3,7)MRS_1(7,3) = 3/7 \ne 7/3 = MRS_2(3,7)
Endowment as-is — blocked by the grand coalition (gains from trade).
Show proof

  1. u1=u2=21≈4.58u_1 = u_2 = \sqrt{21} \approx 4.58: individual rationality holds, so neither {1}\{1\} nor {2}\{2\} blocks.

  2. But MRS1=3/7≠7/3=MRS2MRS_1 = 3/7 \ne 7/3 = MRS_2: Pareto inefficient. Moving to x1=(5,5)x^1 = (5, 5) gives u1=u2=5>4.58u_1 = u_2 = 5 > 4.58, so the grand coalition {1,2}\{1, 2\} blocks — the lens in the picture is exactly that Pareto-improving set.

Verify that no coalition can block the allocation x1=(5,5)x^1 = (5, 5).

(5,5)(5, 5) — in the core.
Show proof

  1. u1=u2=5>4.58=ui(ei)u_1 = u_2 = 5 > 4.58 = u_i(e^i): individually rational, so {1}\{1\}, {2}\{2\} cannot block.

  2. MRS1(5,5)=MRS2(5,5)=1MRS_1(5,5) = MRS_2(5,5) = 1: on the contract curve and Pareto efficient, so {1,2}\{1, 2\} cannot block either. Hence (5,5)(5, 5) is in the core (indeed, it is the competitive equilibrium).

14. Foundations of Competitive Equilibrium