Cost sharing solutions defined by non-negative eigenvectors
We consider a cost sharing problem, where each individual is identi ed by a characteristic (a positive real number) ci: The two main positions on how to share a common cost M are the Egalitarian and the Proportional solutions. These solutions can be obtained as the Perron's eigenvectors (right and left, respectively) of a characteristics matrix A, with rk(A) = 1; de fined from the individuals' characteristics ci: Then, the right Perron's eigenvector associated to any Levinger's transformation L(α) = α A'+(1-α )A; α є [0;1] ; provides a dif ferent solution to the cost sharing problem (from the egalitarian one, for α = 0; to the proportional one, for α = 1). We are interested in analyzing the properties of the components of these Perron's eigenvectors as (non linear) functions of the parameter α that de fines the convex combination of matrices A and A': These components de fine the solution of the cost sharing problem, that could be understood as a compromise between the egalitarian and proportional approaches. We prove that when the associated positive eigenvector is normalized, its components have a monotone behaviour in the unit interval [0;1]: Additional properties and a way of selecting a particular compromise solution are provided.
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