Solution
The Limits of the Core
Show the problem again
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?
Worked solution
Group-rationality of the pair {1, 3} says x₁ + x₃ ≥ 80. Since the three payoffs sum to 100, this means x₂ = 100 - (x₁ + x₃) ≤ 20. The value 20 is attainable: take (x₁, x₂, x₃) = (50, 20, 30). Check the pairs: x₁ + x₂ = 70 ≥ 60, x₁ + x₃ = 80 ≥ 80, x₂ + x₃ = 50 ≥ 40, and each single payoff is nonnegative. So the largest core payoff for player 2 is **20**. (Enumerating all integer allocations shows the core is x₁ in [40, 60], x₂ in [0, 20], x₃ in [20, 40], restricted by the pairwise constraints.)
Source: Core of a transferable-utility cooperative game (Gillies, 1953); the three-player characteristic function is invented for AxiomIQ. Statement written for AxiomIQ.