Table 3 Characteristic function V(S) and the maximal global utility M(S) for each coalition S ⊆ N (Example 1)

{2}

{u ∈ R^{2}|u ^r₂  ≤ 30}

u ^r₂ (9.58) = 30

{3}

{u ∈ R^{3}|u ^r₃  ≤ 30}

u ^r₃ (10.00) = 30

{4}

{u ∈ R^{4}|u ^r₄  ≤ 15}

u ^r₄ (10.45) = 15

{2,3}

{u ∈ R^{2,3}|u ^r₂  + u ^r₃  ≤ 60}

u ^r₂ (9.58) + u ^r₃ (10.00) = 30 + 30

{2,4}

{u ∈ R^{2,4}|u ^r₃  + u ^r₄  ≤ 45}

u ^r₂ (9.58) + u ^r₄ (10.45) = 30 + 15

{3,4}

{u ∈ R^{3,4}|u ^r₃  + u ^r₄  ≤ 57}

u ^r₃ (10.02) + u ^r₃ (10.00) = 28 + 29

{2,3,4}

{u ∈ R^{2,3,4}|u ^r₂  + u ^r₃  + u ^r₄  ≤ 89}

$\begin{array}{l}{u}_2^r\left(9.43\right)+u_3^r\left(9.45\right)+u_4^r\left(10.30\right)=\hfill \\ \begin{array}{lllll}\hfill & \hfill & \hfill & \hfill & =29.50+29.50+30\hfill \end{array}\hfill \end{array}$