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Publisher : Elsevier - Science Direct (الزویر - ساینس دایرکت)
Journal : Journal of Mathematical Economics, Volume 48, Issue 1, January 2012, Pages 14–20
We study a labor market described by a many-to-one matching market with externalities among firms in which each firm’s preferences depend not only on workers whom it hires, but also on workers whom its rival firms hire. We define a new stability concept called weak stability and investigate its existence problem. We show that when the preferences of firms satisfy an extension of substitutability and two new conditions called increasing choice and no external effect by an unchosen worker, then a weakly stable matching exists. We also show that a weakly stable matching may fail to exist without these restrictions.
Since the seminal paper of Gale and Shapley (1962), matching markets have been extensively analyzed by many researchers. We refer to Roth and Sotomayor (1990) for a detailed study of the literature. A typical application of matching markets is a labor market. Such a market is often formulated by a many-to-one matching market in which each firm can hire multiple workers, and each worker can work for at the most one firm (cf. Kelso and Crawford, 1982, Echenique and Oviedo, 2004 and Hatfield and Milgrom, 2005). Their studies focus on stable matchings in which firm and workers cannot deviate profitably. In standard models, it is assumed that each firm’s preferences depend only on workers it hires. In a labor market, however, it is natural to consider instances in which each firm’s preferences depend on workers whom its rival firms hire, since they compete among themselves in a market. This paper takes the externalities among firms into account.1 We define a new stability concept called weak stability, and provide restrictions on the preferences of firms which guarantee the existence of a weakly stable matching. There are several studies on one-to-one matching markets with externalities. Sasaki and Toda (1986) first introduced one-to-one matching markets with externalities. When externalities are present, the stability of a matching depends on how deviating agents predict the reaction of the other agents. They considered three stability concepts, and investigated their existence in general one-to-one matching markets.2Sasaki and Toda (1996) generalized the results of Sasaki and Toda (1986) by using what they define as an estimation function, which describes the set of possible matchings that the deviating pair predicts will result from their deviation. Given such estimation functions, a pair of agents deviates from a matching, if they are made better off under all matchings predicted by their estimation functions. They showed that a stable matching exists in all markets if and only if each agent has a universal estimation function which considers all matchings possible. This result is due to the assumption that estimation functions are given exogenously. Hafalir (2008) introduced an endogenous estimation function which depends on preferences of the other agents. He provided a sufficient condition for estimation functions to be compatible with the existence of a stable matching, and showed that a particular notion of endogenous beliefs called sophisticated expectations guarantees the existence of a stable matching. Mumcu and Saglam (2010), on the other hand, considered PP-stability of Sasaki and Toda (1986), which is defined on the assumption that a pair of agents deviates from a matching, keeping the other agents’ matchings fixed. They provided some restrictions on preferences for the existence of a PP-stable matching. Our study builds on Mumcu and Saglam (2010) by considering a many-to-one market. We first analyze a small market by using a strong stability which is the many-to-one analogue of the solution concept used in Mumcu and Saglam (2010). A strongly stable matching may not always exist even in a small market due to incredible deviations. A deviation can be regarded as incredible when the deviation of a firm and workers yields a further deviation within the deviating members and workers who are hired by the deviating firm before the deviation. We introduce the notion of “strongly blocking” to rule out incredible deviations, and define a weakly stable matching that cannot be strongly blocked by any pair.3 The set of weakly stable matchings is nonempty under a certain condition on the external effect. We show that a weakly stable matching exists if the preferences of firms satisfy an extension of substitutability and two new conditions called increasing choice and no external effect by an unchosen worker. Substitutability is originally introduced by Kelso and Crawford (1982) for the matching model without externalities, and is a sufficient condition for the existence of a stable matching. An extension of substitutability for our model, however, does not guarantee the existence of a weakly stable matching in a market with externalities. Increasing choice requires that (i) the choice set of a firm depends only on the set of workers hired by its rival firms, and (ii) the choice set of a firm expands when the set of workers hired by the rival firms expands. No external effect by an unchosen worker means that if firm ff does not choose worker ww from a subset of workers, then firm ff’s choice from another subset of workers in which worker ww is excluded is not affected by a rival firm additionally hiring worker ww. In other words, an external effect in firm ff’s choice is caused only by an important worker for firm ff. We show, by example, that a weakly stable matching may not exist without increasing choice or no external effect by an unchosen worker. By imposing an additional assumption, we also give a sufficient condition for the existence of a strongly stable matching. The sufficient condition is substantially different from that of Mumcu and Saglam (2010), because applying their condition to our model restricts not only on the preferences of firms, but also those of workers. The rest of this paper is organized as follows. Section 2introduces the model of a many-to-one matching market with externalities among firms and defines two stability concepts: strong stability and weak stability. In this section, restrictions on preferences for firms are also discussed. In Section 3, a relationship between the strong and weak stability is discussed and the existence of a weakly stable matching is proved. Section 4 concludes.
نتیجه گیری انگلیسی
In this paper, we define a new stability concept called the weak stability in a many-to-one matching market with externalities among firms. The weak stability is defined on the choice set and rules out incredible deviations which is caused by negative externalities. Whether a weakly stable matching exists depends on how the choice set varies according to how the other firms match with workers. We provide a sufficient condition for the existence of a weakly stable matching: substitutability, increasing choice and no external effect by an unchosen worker.