We study a generalization of Shapley-Scarf's (1974) economy in which multiple types of indivisible goods are traded. We show that many of the distinctive results from the Shapley-Scarf economy do not carry over to this model, even if agents' preferences are strict and can be represented by additively separable utility functions. The core may be empty. The strict core, if nonempty, may be multi-valued, and might not coincide with the set of competitive allocations. Furthermore, there is no Pareto efficient, individually rational, and strategy-proof social choice rule. We also show that the core may be empty in the class of economies with a single type of indivisible good and agents consuming multiple units, even if no complementarity exists among the goods.

We consider the n-player houseswapping game of Shapley-Scarf (1974), with indifferences in preferences allowed. It is well-known that the strict core of such a game may be empty, single-valued, or multivalued. We define a condition on such games called "segmentability", which means that the set of players can be partitioned into a "top trading segmentation". It generalizes Gale's well-known idea of the partition of players into "top trading cycles" (which is used to find the unique strict core allocation in the model with no indifference). We prove that a game has a nonempty strict core if and only if it is segmentable. We then use this result to devise an O(n 3) algorithm which takes as input any houseswapping game, and returns either a strict core allocation or a report that the strict core is empty. Finally, we are also able to construct a linear inequality system whose feasible region's extreme points precisely correspond to the allocations of the strict core. This last result parallels the results of Vande Vate (1989) and Rothblum (1991) for the marriage game of Gale and Shapley (1962).

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This paper considers von Neumann-Morgenstern (vNM) stable sets in marriage games. Ehlers (Journal of Economic Theory 134: 537-547, 2007) showed that if a vNM stable set exists in a marriage game, the set is a maximal distributive lattice of matchings that includes all core matchings. To determine what matchings form a vNM stable set, we propose a polynomial-time algorithm that finds a man-optimal matching and a woman-optimal matching in a vNM stable set of a given marriage game. This algorithm also generates a modified preference profile such that a vNM stable set is obtained as the core of a marriage game with the modified preference profile. It is well known that cores of marriage games are nonempty. However, the nonemptiness of cores does not imply the existence of a vNM stable set. It is proved using our algorithm that there exists a unique vNM stable set for any marriage game.

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