Pair-wise Envy Free and Stable Matchings for Two-sided Systems with Techniques

Introduction: The two-sided matching model of Gale and Shapley (1962) can be interpreted as one where a non-empty finite set of firms need to employ a non-empty finite set of workers. Further, each firm can employ at most one worker and each worker can be employed by at most one firm. Each worker has preferences over the set of firms and each firm has preferences over the set of workers. An assignment of workers to firms is said to be stable if there does not exist a firm and a worker who prefer each other to the ones they are associated with in the assignment. Gale and Shapley (1962) proved that every two-sided matching problem admits at least one stable matching. In this paper we extend the above model by including a non-empty finite set of techniques. An assignment now comprises disjoint triplets, each triplet consisting of a firm, a worker and a technique. A technique can be likened to a machine that the firm and worker together use for production. Each firm has preferences over the set of ordered pairs of workers and techniques and each worker has preferences over the set of ordered pairs of firms and techniques. We call such models two-sided systems with techniques. There are two kinds of issues we address in the context of this model, now that concerns naturally extend beyond those of pair-wise stability as defined in Gale and Shapley (1962). The first issue is about the possibility of a pair of agents being better off than in their current assignment by perhaps using a different technique. The existence of such a possibility allows for a pair of agents to 'envy' the technique that may have been assigned to a different pair. It is natural to seek an assignment that excludes 'envy' and which may therefore be called 'pair-wise envy free'. The second issue that we address in this paper, pertains to a situation where each firm is initially endowed with a technique. In such a situation we are interested in proving the existence of an assignment such that no coalition can re-allocate the techniques that they have been endowed with, and consequently be better off. A matching which satisfies this property is called stable. Through out the paper, we assume as in Danilov (2003) (: though in a slightly different context) that the preferences of the workers are lexicographic, with firms …