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An estate cannot cover everyone’s claims. The Talmud, two thousand years before game theory, prescribed divisions that looked ad hoc for centuries. They turn out to be exactly the nucleolus: the split that makes the most aggrieved party as content as possible.

The nucleolus

When the core is empty, no allocation is unblockable, so stability alone cannot choose. The nucleolus picks the allocation that minimizes the largest complaint, then, among those, the next-largest, and so on. It always exists, is unique, and reconstructs a famous ancient bankruptcy rule.

From earlier lessons

Two quick ones before we start.

A coalition's excess at an allocation is what it could get minus...

The Shapley value divides by averaging each player's...

Predict, then play

A three-player majority game: any two players can seize the whole prize, so the core is empty. Adjust the split and watch the coalition complaints. Try to shrink the worst one.

Why did that happen?

Take a guess, then reveal the reason

With an empty core, someone always has a grievance: every pair could have grabbed the whole prize, so any pair that is underpaid feels cheated. The nucleolus does not try to eliminate complaints (impossible here) but to be as fair as possible about them, shrinking the loudest one first. In a symmetric game that means treating everyone alike, so the equal split leaves the biggest complaint smallest. Push any share up and some pair’s complaint grows.

The math

Definition, the nucleolus

For an allocation $x$ , each coalition $S$ has an excess $e(S, x) = v(S) - \sum_{i \in S} x_i$ , its complaint. Sort all excesses from largest to smallest. The nucleolus is the allocation that lexicographically minimizes this sorted vector: first minimize the largest excess, then among ties the next, and so on. It always exists, is unique, and lies in the core whenever the core is nonempty.

In the majority game, $v(S) = 1$ for any coalition of two or more. For a pair $\{i,j\}$ the excess is $1 - (x_i + x_j) = x_k$ , the share of the excluded player. To minimize the largest such excess is to minimize $\max_k x_k$ , which by symmetry and efficiency ( $x_1 + x_2 + x_3 = 1$ ) is achieved at $(\tfrac13, \tfrac13, \tfrac13)$ .

The Talmud connection: for claims $(100, 200, 300)$ against an estate of $200$ , the Talmud prescribes $(50, 75, 75)$ , which is exactly the nucleolus of the associated bankruptcy game. Aumann and Maschler showed the ancient rule and the nucleolus agree across all the Talmud’s cases.

Work it out

Minimize the worst complaint

In the majority game a pair’s complaint equals the excluded player’s share: the pair could have taken the whole $1$ but got $1 - x_k$ , so its excess is $x_k$ . The three pairwise complaints are therefore $x_1$ , $x_2$ , and $x_3$ . The largest is $\max(x_1, x_2, x_3)$ .

Minimizing that maximum, subject to $x_1 + x_2 + x_3 = 1$ , forces the shares equal: $(\tfrac13, \tfrac13, \tfrac13)$ , where the largest complaint is $\tfrac13$ . Any deviation raises some share above $\tfrac13$ , increasing the worst complaint. So the nucleolus is the equal split, which here matches the Shapley value because the game is symmetric. In asymmetric games, such as the Talmud estate, the nucleolus and Shapley value diverge.

Key idea

The nucleolus lexicographically minimizes coalition complaints, giving a unique division even when the core is empty. Symmetry forces it to agree with the Shapley value in this majority game, but symmetry alone does not make the two rules identical in every game.

Check yourself

Check yourselfapply

In the majority game, why is the equal split the nucleolus?

Check yourselfanalyse

What advantage does the nucleolus have over the core here?

Check yourselfbrings back: Shapley value

The nucleolus and the Shapley value both gave (1/3,1/3,1/3) here. Do they always agree?

Sources

Schmeidler, D. (1969). The nucleolus of a characteristic function game. SIAM Journal on Applied Mathematics, 17(6), 1163-1170.
Aumann, R. J., and Maschler, M. (1985). Game theoretic analysis of a bankruptcy problem from the Talmud. Journal of Economic Theory, 36(2), 195-213.