Stable Roommate Problem with Diversity Preferences

In the multidimensional stable roommate problem, agents have to be allocated to rooms and have preferences over sets of potential roommates. We study the complexity of finding good allocations of agents to rooms under the assumption that agents have diversity preferences [Bredereck et al., 2019]: each agent belongs to one of the two types (e.g., juniors and seniors, artists and engineers), and agents' preferences over rooms depend solely on the fraction of agents of their own type among their potential roommates. We consider various solution concepts for this setting, such as core and exchange stability, Pareto optimality and envy-freeness. On the negative side, we prove that envy-free, core stable or (strongly) exchange stable outcomes may fail to exist and that the associated decision problems are NP-complete. On the positive side, we show that these problems are in FPT with respect to the room size, which is not the case for the general stable roommate problem. Moreover, for the classic setting with rooms of size two, we present a linear-time algorithm that computes an outcome that is core and exchange stable as well as Pareto optimal. Many of our results for the stable roommate problem extend to the stable marriage problem.Comment: accepted to IJCAI'2

Information Technology and the Dynamics of Joint Innovation

We develop a model of an innovative industry to examine how information technology, by both enhancing matching efficiency and knowledge sharing, can have an ambiguous effect on the total amount of innovation. We consider a population of firms holding different knowledge expertise, and forming partnerships to conduct joint R&D. We assume that bringing together different expertise has positive value for innovating but also that joint innovation implies a partial convergence of the partners'' expertise. We study how the distribution of firms changes and thus how the innovative potential of the economy evolves. We show that as heterogeneity is used as an input by the innovative process, the industry must eventually collapse to a unique expertise, but how fast this takes place depends on the quality of IT. As a result of falling dispersion, a tension arises between static and dynamic efficiency. JEL Classification Numbers: C78,O33,O38.mathematical economics and econometrics ;

Stable Matchings for the Room-mates Problem

We show that, given two matchings for a room-mates problem of which say the second is stable, and given a non-empty subset of agents S if (a) no agent in S prefers the first matching to the second, and (b) no agent in S and his room-mate in S under the second matching prefer each other to their respective room-mates in the first matching, then no room-mate of an agent in S prefers the second matching to the first. This result is a strengthening of a result originally due to Knuth (1976). In a paper by Sasaki and Toda (1992) it is shown that if a marriage problem has more than one stable matchings, then given any one stable matching, it is possible to add agents and thereby obtain exactly one stable matching, whose restriction over the original set of agents, coincides with the given stable matching. We are able to extend this result here to the domain of room-mates problems. We also extend a result due to Roth and Sotomayor (1990) originally established for two-sided matching problems in the following manner: If in a room-mates problem, the number of agents increases, then given any stable matching for the old problem and any stable matching for the new one, there is at least one agent who is acceptable to this new agent who prefers the new matching to the old one and his room-mate under the new matching prefers the old matching to the new one. Sasaki and Toda (1992) shows that the solution correspondence which selects the set of all stable matchings, satisfies Pareto Optimality, Anonymity, Consistency and Converse Consistency on the domain of marriage problems. We show here that if a solution correspondence satisfying Consistency and Converse Consistency agrees with the solution correspondence comprising stable matchings for all room-mates problems involving four or fewer agents, then it must agree with the solution correspondence comprising stable matchings for all room-mates problems.Stable matchings, Room-mate problem

Credible Group Stability in Multi-Partner Matching Problems

It is known that in two-sided many-to-many matching markets, pair-wise stability is not logically related with the (weak) core, unlike in many-to-one matching markets (Blair, 1988). In this paper, we seek a theoretical foundation for pairwise stability when group deviations are allowed. Group deviations are defined in graphs on the set of agents. We introduce executable group deviations in order to discuss the credibility of group deviations and to defined credibly group stable matchings. We show, under responsive preferences, that credible group stability is equivalent to pairwise stability in the multi-partner matching problem that includes two-sided matching problems as special cases. Under the same preference restriction, we also show the equivalence between the set of pairwise stable matchings and the set of matchings generated by coalition-proof Nash equilibria of an appropriately defined strategic form game. However, under a weaker preference restriction, substitutability, these equivalences no longer hold, since pairwise stable matchings may be strictly Pareto-ordered, unlike under responsiveness.Multi-partner matching problem, Pairwise stable matching network, Credible group deviation

How (not) to Choose Peers in Studying Groups

This paper analyzes social group formation when agents are subject to peer effects within groups increasing human capital and instantaneous utility. When agents are heterogeneous on two dimensions, ability and social skills, and monetary payments are not feasible the model predicts segregation at the top and at the bottom of the attribute space and bunching for heterogeneous intermediate types. Groups may be heterogeneous in taste types and more heterogeneous types are more likely to participate. The equilibrium allocation does not induce cost-efficient human capital accumulation. Introducing ability tracking may produce beneficial results despite decreasing differences in human capital production.Education, Peer-effects, Matching, Group Formation