Cross-Efficiency: A Critique
In this article:
2. Formal Definition of Cross-Efficiency
3. A Critique of the Cross-Efficiency Approach
The starting point in cross-efficiency studies is the “ratio model” formulation in non-parametric DEA efficiency studies, using a weighted sum of outputs on a weighted sum of inputs as objective function when maximising the productivity of a unit with weights as endogenous variables. The constraint is that no unit can have a productivity greater than one (a normalisation) when using the solutions for the endogenous weights of the unit under investigation. Inserting the optimal solutions for the weights, the productivity expression is defined as the technical efficiency score for the unit. Having the solutions for the weights of all units, cross-efficiency scores for a specific unit are defined as the productivity ratio of the unit calculated by successively using the weights of all other units (peers’ appraisal), or using the specific unit’s own weight for all other units (appraisal of peers).
The stated main purpose of introducing cross-efficiency is to get a better discrimination between efficient units. Another aim is to eliminate unrealistic weight schemes. The purpose of this paper is to investigate the merits of the cross-efficiency approach viewed in the perspective of efficiency measurement for production activities with well-defined inputs and outputs. In the cross-efficiency literature there has been no strict formal analysis supporting the view that the cross-efficiency numbers play an un-ambiguous role in ranking of the efficiency of the units or eliminating unrealistic weighting schemes.