We discuss the traditional interpretation of Edgeworth’s conception of competition neo-Walrasian authors developed following the Debreu-Scarf ’s theorem legacy. This interpretation presents Edgeworth as the forerunner of the cooperative-games approach based on the notion of the core as a solution concept. We appraise those notions using Edgeworth’s texts. This allows us to plead in favour of Edgeworth’s originality and of the need to rescue his theory of perfect competition from a misleading way. Textual evidence allows us to assert that the language of the cooperative game theory, and the subsequent modern theory of the core, do not correspond to Edgeworth’s theory. The modern notion of coalition seems to be an interpretation of Edgeworth’s concept of “combination” and of “cooperative associations.” Both notions deeply differ from the traditional concept of coalition when one examines them more carefully. Edgeworth clearly establishes a difference between a group of agents entering in a series of bilateral contract relations and a “combination.” The logic of the former is purely individual in the sense of an individual improvement accepting bilateral contracts within a new group rather than the bilateral contracts she have formerly agreed within another group. In fact, within the new improving-group all contracts continue to be bilaterally established and, more important, these kind of association does not necessary imply giving up all previous exchanges outside this group. From this alternative interpretation about Edgeworth’s notion of “combination” we extract a completely different conclusion than that implied by the coalitional formation presented in the cooperative game theoretical framework. In this framework, the core of an economy is simply reduced by the presence of an increasing number of agents forming this kind of coalitions. From the Egeworthian point of view, the number of agents does not per se guaranteed the increasing of competition when this kind.