The Nucleolus

Game Theory · Axiom Academy

Unit 6 - Cooperative Game Theory The excess of a coalition S under allocation x measures how much the coalition "complains" about the allocation. It represents the difference between what the coalition could achieve on its own and what it actually receives. A positive excess means the coalition is unhappy (could do better on its own). A negative excess means the coalition is satisfied (doing better than going alone). The nucleolus seeks to minimize the maximum excess across all coalitions. 2. Minimizing Maximum Unhappiness The nucleolus uses a lexicographic minimization approach: it first minimizes the largest excess, then (holding that constant) minimizes the second-largest excess, and so on. This means: compare the largest excess first. If tied, compare the second-largest, and so on. This ensures that no coalition is disproportionately unhappy compared to others. 3. Comparing Different Allocations Let's see how different allocations compare by their excess vectors. Consider a simple game with three players where v( 1,2,3 ) = 100, v( 1,2 ) = 80, v( 1,3 ) = 60, v( 2,3 ) = 70. Allocation A: (50, 30, 20) - gives more to player 1 Allocation B: (40, 35, 25) - more balanced Nucleolus: minimizes maximum excess The animation shows how different allocations produce different excess patterns. The nucleolus balances these excesses to achieve fairness. 4. Properties and Relationship to the Core The nucleolus has several important properties that make it a compelling solution concept:

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