The Limits of the Core

Players 1, 2 and 3 form a cooperative game with transferable utility. A single player alone obtains 0. The pairs obtain v(1,2) = 60, v(1,3) = 80 and v(2,3) = 40, and the grand coalition obtains v(1,2,3) = 100. An allocation (x₁, x₂, x₃) with x₁ + x₂ + x₃ = 100 is in the *core* if no coalition S receives less than v(S) in total (no coalition can profitably break away). What is the largest payoff that player 2 can receive in a core allocation?

• game theory

Source: Core of a transferable-utility cooperative game (Gillies, 1953); the three-player characteristic function is invented for AxiomIQ. Statement written for AxiomIQ.