Aumann-Maschler Contested-Garment Bankruptcy Formula

Understand why the Talmudic nucleolus splits only the contested estate!

In Mishnah Bava Metzia 2a, two claimants hold verified claims of c_1 = 50 (half a garment) and c_2 = 100 (the whole garment) against an estate worth E = 100 — and receive 25 and 75 rather than pari-passu 33.33 and 66.67! Robert Aumann and Michael Maschler (1985) proved that this ancient rule coincides identically with the game-theoretic Nucleolus of the bankruptcy coalition game! Its logic is universal: Creditor 2 cannot claim more than c_2, so MAX(0, E - c_2) is uncontestedly conceded to Creditor 1; Creditor 1 cannot claim more than c_1, so MAX(0, E - c_1) is conceded to Creditor 2; and the Contested Remainder E - MAX(0, E - c_2) - MAX(0, E - c_1) is split 50 / 50, simplifying to x_1 = 0.5 * (E + MAX(0, E - c_2) - MAX(0, E - c_1))!

Verify on Bankruptcy_Estate E = 120 ($k, B2), Small_Creditor_Claim c_1 = 100 ($k, B3), and Large_Creditor_Claim c_2 = 300 ($k, B4): Conceded to Creditor 1 is MAX(0, 120 - 300) = 0; Conceded to Creditor 2 is MAX(0, 120 - 100) = 20; Contested Remainder is 120 - 20 = 100 (split 50 + 50), awarding Small Creditor x_1 = 50.00 (50.0% recovery) and Large Creditor x_2 = 70.00 (23.3% recovery), compared to Pari-Passu Proportional Award 120 * (100 / 400) = 30.00 (+20.00 small-creditor protection)! Test Mishnah inputs E = 100, c_1 = 50, c_2 = 100 -> x_1 = 25.00, x_2 = 75.00; and test high estate E = 350 -> equal-loss regime x_1 = 75.00, x_2 = 275.00 (each loses 25.00)!

Universal Contested-Garment Nucleolus Award Formula

Validate 0 <= E <= c_1 + c_2 and c_1 + c_2 > 0: if Estate E exceeds total claims c_1 + c_2, the debtor is solvent rather than bankrupt!

=LET(es_vl, B2, cl_1, B3, cl_2, B4, IF(OR(es_vl < 0, cl_1 < 0, cl_2 < 0, cl_1 + cl_2 <= 0, es_vl > cl_1 + cl_2), "Invalid Insolvent Estate", ROUND(0.5 * (es_vl + MAX(0, es_vl - cl_2) - MAX(0, es_vl - cl_1)), 2)))

Pari-Passu Comparison & Small-Claimant Protection Wedge Formula

Notice that x_1 + x_2 = E holds identically across all three Aumann-Maschler regimes (Equal Gains E <= c_1, Half-Small-Claim c_1 < E <= c_2, and Equal Losses E > c_2)!

=IF(ISTEXT(E2), E2, ROUND(E2 - B2 * (B3 / (B3 + B4)), 2))

Further reading

• Arte Sheets Lesson — Shapley Value Three-Player Coalition Marginal Contribution Formula
• Arte Sheets Lesson — Kalai-Smorodinsky Monotonic Bargaining vs Asymmetric Nash Split