We develop a theory of stability in many-to-many matching markets. We give conditions under which the setwise-stable set, a core-like concept, is nonempty and can be approached through an algorithm. The usual core may be empty. The setwise-stable set coincides with the pairwise-stable set and with the predictions of a non-cooperative bargaining model. The setwise-stable set possesses the conflict/coincidence of interest properties from many-to-one and one-to-one models. The theory parallels the standard theory of stability for many-to-one, and one-to-one, models. We provide results for a number of core-like solutions, besides the setwise-stable set.
The paper proposes an algorithm to compute the full set of many-to-many stable matchings when agents have substitutable preferences. The algorithm starts by calculating the two optimal stable matchings using the deferred-acceptance algorithm. Then, it computes each remaining stable matching as the firm-optimal stable matching corresponding to a new preference profile which is obtained after modifying the preferences of a previously identified sequence of firms.
Scarf (1960) proposed a market environment and a model of dynamic adjustment in .which the standard tatonnement price adjustment process orbits around, rather than converges to, the competitive equilibrium. Hirota ( 1981) characterized the price paths by the configuration of endowments. We explore the predictions of Scarf's model in a non tatonnement experimental double auction. We find that the average transaction prices in each period do follow the path predicted by the Scarf and Hirota models. vVhen the model predicts prices will converge to the competitive equilibrium, our data converge; when the model predicts prices will orbit. our data orbit the equilibrium, and in the direction predicted by the model. Moreover. we observe a weak tendency for prices within a period to follow the path predicted by the model. Global Instability in ExperimentalGeneral Equilibrium:. The Scarf ExampleChristopher M. Anderson� Sander Granat! Charles R. Plott+and Ken-Ichi Shimomura § Int roduct ion 1This paper is motivated by two issues. The first is the nature of the equilibra tion process in multiple, interdependent, continuous, double auction markets.Not only are such markets in widespread use, their study has been of special scientific value. It has been known for years that this particular form of in dustrial organization creates an effective price discovery mechanism. Prices typically result near the competitive equilibrium. However, exactly how this price discovery takes place, the process itself, is unknown. The second moti vating issue is the extent to which models based on tatonnement help with understanding this process. Almost all general equilibrium, if not almost.all economics, is based on the tatonnement tool. The tool itself was devel oped to help theorists abstract from the complexity caused by disequilibrium trading patterns. In practice, economic systems are then analyzed as if the mechanism of price discovery were tatonnement even when it is obviouslynot. To what extent are tatonnement models reliable when applied in such a manner?. Existing experiment.al results suggest that such models are more reliable than might be believed.A natural way to study the dynamics of price discovery is to focus on sta bility, or more specifically, on instability. It is within an unstable system that the forces at. work to move prices and quantities are most clearly identifiable.The classic paper by Scarf (1960) uses a tatonnement. model to produce some striking predictions about the types of behavior a multiple market system might exhibit. Using a three commodity example he constructs an exchange environment in which price convergence never takes place. Specifically, under certain distributions of endowments the prices exhibited by a tat.onnement process travel only in closed orbits. Hirota (1981) characterized the price movements in terms of endowments. The combined theoretical result is that.prices can move in a systematic fashion without equilibration. The direction of the orbit can be reversed by a change in the en...
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