“…. Observe that if graph is collectively profitable then is well defined by (5), as logarithm is defined for positive real numbers and utilities are positive for all = 1, 2, … , (assuming that log 0 = −∞). Let ∈ ℝ ×1 be the vector of signed limit prices, defined by: (6) = { − ∈ , ∈ ℬ, 0 ∈ ℐ.…”