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Since stable matchings may not exist, we adopt a weaker notion of stability for solving the roommate problem: the bargaining set. Klijn and Massó (2003) show that the bargaining set coincides with the set of weakly stable and weakly efficient matchings in the marriage problem. First, we show that a weakly stable matching always exists in the roommate problem. However, weak stability is not sufficient for a matching to be in the bargaining set. Second, we prove that the bargaining set is always non-empty. Finally, as Klijn and Massó (2003) get for the marriage problem, we show that the bargaining set coincides with the set of weakly stable and weakly efficient matchings in the roommate problem.

We study the stability of overlapping group structures where each group possesses a constitution that contains the rules governing both the composition of the group and the conditions needed to leave the group and/or to become a new member of the group. We provide a necessary and sufficient condition on preferences that guarantees the existence and the emergence of constitutionally stable group structures. We show that although more blocking power for the individuals might enlarge the set of constitutionally stable group structures, it could happen that the society will never reach a stable group structure.

We propose a framework of network formation where players can form two types of links: public links are observed by everyone and shadow links are only observed by neighbors. We introduce a novel solution concept called rationalizable peer-confirming pairwise stability, which generalizes Jackson and Wolinsky (1996)'s pairwise stability notion to accommodate shadow links. We then study the case when public links and shadow links are perfect substitutes and relate our concept to pairwise stability. Finally, we consider two specific models and show how false beliefs about others' behavior may lead to segregation in friendship networks with homophily, reducing social welfare.

We introduce the concept of a horizon-K farsighted set to study the influence of the degree of farsightedness on network stability. The concept generalizes existing concepts where all players are either fully myopic or fully farsighted. We show that a horizon-K farsighted set always exists and that the horizon-1 farsighted set G₁ is always unique. For generic allocation rules, the set G₁ always contains a horizon-K farsighted set for any K. We provide easy to verify conditions for a set of networks to be a horizon-K farsighted set and we consider the efficiency of networks in horizon-K farsighted sets. We discuss the effects of players with different horizons in an example of criminal networks.

We study the formation of R&D networks when each firrm benefits from the research done by other firms it is connected to. Firms can be either myopic or farsighted when deciding about the links they want to form. We propose the notion of myopic-farsighted stable set to determine the R&D networks that emerge in the long run. When the majority of firms is myopic, stability leads to R&D networks consisting of either two asymmetric components with the largest component comprises three-quarters of firms or two symmetric components of nearly equal size with the largest component having only myopic firms. But, once the majority of firms becomes farsighted, only R&D networks with two asymmetric components remain stable. Firms in the largest component obtain greater profits, with farsighted firms having in average more collaborations than myopic firms that are either loose-ends or central for spreading the innovation within the component. Besides myopic and farsighted firms, we introduce yes-firms that always accept the formation of any link and never delete a link subject to the constraint of non- negative profits. We show that yes-firms can stabilize R&D networks consisting of a single component that maximize the social welfare. Finally, we look at the evolution of R&D networks and we find that R&D networks with two symmetric components will be rapidly dismantled, single component R&D networks will persist many periods, while R&D networks consisting of two asymmetric components will persist forever.

Mauleon, Roehl and Vannetelbosch (GEB, 2018) develop a general theoretical framework to study the stability of overlapping coalition settings. Each group possesses a constitution that contains the rules governing both the composition of the group and the conditions needed to leave the group and/or to become a new member of the group. They propose the concept of constitutional stability to study the group structures that are going to emerge at equilibrium in overlapping coalition settings. They combine requirements on constitutions and preferences for guaranteeing both the existence and the emergence of constitutionally stable group structures. In this paper, we show that an alternative way to exclude the occurrence of closed cycles is to look for constitutions that allow for a common ranking.

The role of genetic risk markers for Alzheimer's disease (AD) in mediating the neurocognitive endophenotypes (NEs) of subjects with mild cognitive impairment (MCI) has rarely been studied. The aim of the present study was to investigate the relationship between well-known AD-associated single-nucleotide polymorphisms (SNPs) and individual NEs routinely evaluated during diagnosis of MCI, AD, and other dementias. The Fundacio ACE (ACE) dataset, comprising information from 1245 patients with MCI, was analyzed, including the total sample, amnestic MCI (aMCI) (n = 811), and non-amnestic MCI (naMCI) (n = 434). As probable-MCI (Pr-MCI) patients with memory impairment have a higher risk of AD, which could influence the statistical power to detect genetic associations, the MCI phenotype was also stratified into four related conditions: Pr-aMCI (n = 262), Pr-naMCI (n = 76), possible (Pss)-aMCI (n = 549), and Pss-naMCI (n = 358). Validation analyses were performed using data from the German study on Aging, Cognition, and Dementia in primary care patients (AgeCoDe), and the German Dementia Competence Network (DCN). SNP associations with NEs were calculated in PLINK using multivariate linear regression analysis adjusted for age, gender, and education. In the total MCI sample, APOE-epsilon 4 was significantly associated with the memory function NEs delayed recall (DR) (beta = -0.76, p = 4.1 x 10(-10)), learning (beta = -1.35, p = 2.91 x 10(-6)), and recognition memory (beta = -0.58, p = 9.67 x 10(-5)); and with DR in the aMCI group (beta = -0.36, p = 2.96 x 10(-5)). These results were confirmed by validation in the AgeCoDe (n = 503) and DCN (n = 583) datasets. APOE-epsilon 4 was also significantly associated with the NE learning in individuals classified as having Pss-aMCI (beta = -1.37, p = 5.82 x 10(-5)). Moreover, there was a near study-wide significant association between the HS3ST1 locus (rs6448799) and the backward digits working memory NE (beta = -0.52, p = 7.57 x 10(-5))

We introduce the concept of minimal strong curb sets which is a set-theoretic coarsening of the notion of strong Nash equilibrium. Strong curb sets are product sets of pure strategies such that each player’s set of recommended strategies contains all actions she may rationally select in every coalition she might belong to, for any belief each coalition member may have that is consistent with the recommendations to the other players. Minimal strong curb sets are shown to exist and are compared with other well-known solution concepts. We provide a dynamic learning process leading the players to play strategies from a minimal strong curb set only.

The paper analyzes the role of the structure of communication—i.e. who is talking with whom—in a coordination game. We run an experiment in a three-player game with Pareto ranked equilibria, where a pair of players has a profitable joint deviation from the Pareto- superior equilibrium. We show that specific communication structures lead to different 'coalition-proof' equilibria in this game. Results match the theoretical predictions. Subjects communicate and play the Pareto-superior equilibrium when communication is public. When pairs of players exchange messages privately, subjects play the Pareto-inferior equilibrium. Even in these latter cases, however, players’ beliefs and choices tend to react to messages, despite the fact that these are not credible.

We study stable sets for marriage problems under the assumption that players can be both myopic and farsighted. We introduce the new notion of the myopic-farsighted stable set, which is based on the notion of a myopic-farsighted improving path. A myopic-farsighted stable set is the set of matchings such that there is no myopic-farsighted improving path from any matching in the set to another matching in the set (internal stability) and there is a myopic-farsighted improving path from any matching outside the set to some matching in the set (external stability). For the special cases where all players are myopic and where all players are farsighted, our concept predicts the set of matchings in the core. When all men are myopic and the top choice of each man is a farsighted woman, we show that the singleton consisting of the woman-optimal stable matching is a myopic-farsighted stable set. The same result holds when all women are farsighted. We present examples where this is the unique myopic-farsighted stable set as well as examples where this is the unique myopic-farsighted stable set as well as examples of myopic-farsighted stable sets consisting of a core element different from the woman-optimal matching or even of a non-core element.

We study von Neumann Morgenstern stable sets for one-to-one matching problems under the assumption of coalitional sovereignty (C), meaning that a deviating coalition of players does not have the power to arrange the matches of agents outside the coalition. We study both the case of pairwise and coalitional deviations. We argue further that dominance has to be replaced by path dominance (P) along the lines of van Deemen (1991) and Page and Wooders (2009). This results in the pairwise CP vNM set in the case of pairwise deviations and the CP vNM set in the case of coalitional deviations. We obtain a unique prediction for both types of stable sets: the set of matchings that belong to the core.

We analyze a model of information centralization in teams where players can only exchange information through an endogenous network. Over several periods each player can either pass or not pass her information to her neighbors. Once one player has centralized all the information, all players receive some payoff. The winner who collects all the information gets an additional reward. Since each player discounts payoffs over time, she faces the dilemma of either letting another player centralizing all the information fast, or trying to collect all the information by herself and to overtake the leadership. We find that there is always a single winner who centralizes the information at equilibrium and that only minimally connected networks can be pairwise stable. We also characterize the winner and the duration for any network and for any discount factor. We show that the star network is always pairwise stable. More surprisingly, we find that even networks in which the centralization takes a long time can be pairwise stable.

We develop a general theoretical framework that allows us to study the group structures that are going to emerge at equilibrium in overlapping coalition settings. We introduce the notion of constitution in order to model for each group the rules governing both the composition of the group and the conditions needed to leave the group and/or to become a new member of the group. We propose the concept of constitutional stability to study the group structures that are going to emerge at equilibrium. This concept generalizes previous stability concepts in the literature in which the constitutional rules were exogenously given or not explicitly considered. We combine requirements on constitutions and preferences for guaranteeing both the existence and the emergence of constitutionally stable group structures. Finally, we show how these results are useful to identify constitutionally stable group structures in many-to-many matchings.

We study stable sets for marriage problems under the assumption that players can be both myopic and farsighted. We introduce the new notion of the myopic-farsighted stable set. For the special cases where all players are myopic and where all players are farsighted, our concept predicts the set of matchings in the core. When all men are myopic and the top choice of each man is a farsighted woman, we show that the singleton consisting of the woman-optimal stable matching is a myopic-farsighted stable set. The same result holds when all women are farsighted.

We develop a general theoretical framework that allows us to study the group structures that are going to emerge at equilibrium when individuals are allowed to engage in several groups at the same time. We introduce the notion of constitution in order to model for each group the rules governing both the composition of the group and the conditions needed to leave the group and/or to become a new member of the group. We then propose the concept of con- stitutional stability to study the group structures that are going to emerge at equilibrium in overlapping coalition settings. This concept generalizes previ- ous stability concepts in the literature in which the constitutional rules were exogenously given or not explicitly considered. We combine requirements on constitutions and preferences for guaranteeing both the existence and the emergence of constitutionally stable group structures. Furthermore, by em- bedding many-to-many matchings into our setting, we show how these results could be useful to identify the constitutionally stable group structures in this particular environment.

We study von Neumann Morgenstern stable sets for one-to-one matching problems under the assumption of coalitional sovereignty (C), meaning that a deviating coalition of players does not have the power to arrange the matches of agents outside the coalition. We study both the case of pairwise and coalitional deviations. We argue further that dominance has to be replaced by path dominance (P) along the lines of van Deemen (1991) and Page and Wooders (2009). This results in the pairwise CP vNM set in the case of pairwise deviations and the CP vNM set in the case of coalitional deviations. We obtain a unique prediction for both types of stable sets: the set of matchings that belong to the core. (C) 2017 Elsevier B.V. All rights reserved.

The paper analyzes the role of the structure of communication—i.e. who is talking with whom—in a coordination game. We run an experiment in a three-player game with Pareto ranked equilibria, where a pair of players has a profitable joint deviation from the Pareto-superior equilibrium. We show that specific communication structures lead to different ‘coalition-proof’ equilibria in this game. Results match the theoretical predictions. Subjects communicate and play the Pareto-superior equilibrium when communication is public. When pairs of players exchange messages privately, subjects play the Pareto-inferior equilibrium. Even in these latter cases, however, players’ beliefs and choices tend to react to messages, despite the fact that these are not credible.

We analyze how different rules for exiting an alliance affect the formation of strategic alliances. We adopt the concept of contractual stability to predict the alliances that are going to emerge in the long run. We find that any asymmetric alliance structure consisting of two alliances is contractually stable under the unanimity decision rule. In addition, the grand alliance which is the efficient structure is stable. If we allow for side payments to compensate former partners, then some less efficient structures that were stable without side payments are no more stable. Moreover, we show that the stability of alliances under the unanimity rule to exit is robust to the type of firms, myopic or farsighted. Finally, there is no contractually stable alliance structure under the simple majority decision rule.

In many economic networks, the network is not exogenous but instead agents decide what links they want to build. A central question is predicting the networks that agents will form. This chapter presents myopic and farsighted concepts for modeling network formation when the formation of a link requires the consent of both agents. The chapter illustrates the bites that these solution concepts have on economic applications and investigates whether the networks formed by farsighted agents are efficient and different from those formed by myopic agents. Three models of network formation are analyzed: networks of R&D collaborations, networks of free trade agreements, and criminal networks. It is shown that, depending on the application, myopia and farsightedness may lead to divergent predictions, and farsightedness can help to support the emergence of efficient networks.

We study von Neumann Morgenstern stable sets for one-to-one matching problems under the assumption of coalitional sovereignty, meaning that a deviating coalition of players does not have the power to arrange the matches of agents outside the coalition. We study both the case of pairwise and coalitional deviations. We argue further that dominance has to be replaced by path dominance along the lines of van Deemen (1991) and Page and Wooders (2009). This results in the pairwise myopic vNM set and the myopic vNM set, respectively. We obtain a unique prediction for both types of stable sets: the set of matchings that belong to the core. We also show that the pairwise and coalitional analogues of the level-1 farsighted set yield the core as the unique prediction.

Pairwise stability (Jackson and Wolinsky, 1996) is the standard stability concept in network formation. It is a myopic notion in the sense that it only considers the immediate benefits of the players. A different perspective investigates perfect farsightedness, proposing related stability concepts. We design a simple network formation experiment to test these extreme theories, but find evidence against both of them: both myopically and farsightedly stable networks fail to emerge when they are not immune to limitedly farsighted deviations. The selection among multiple pairwise stable networks (and the performance of farsighted stability) crucially depends on the level of farsightedness needed to sustain them, and not on efficiency or cooperative considerations.

We study a model in which heterogeneous agents first form a trading network where linking costs are positive but infinitesimally small. Then, a seller and a buyer are randomly selected among the agents to bargain through a chain of intermediaries. We determine both the trading path and the allocation of the surplus among the seller, the buyer and the intermediaries at equilibrium. We show that, under the initiator bargaining protocol, a trading network is pairwise stable if it is a core–periphery network where the core consists of all impatient agents who are linked to each other and the periphery consists of all patient agents who have a single link towards an impatient agent. Once agents do not know the impatience of other agents, each bilateral bargaining session may involve delay. Then, core–periphery networks may not be pairwise stable because agents may prefer to add links for reducing the length of trading paths and so avoiding costly delays in reaching a global agreement.

The objective of the paper is to analyze the formation of groups where individuals are allowed to engage in several groups at the same time. Each group is supposed to have specific rules or constitutions governing which members may join or leave it. Given these constitutions, we consider a group structure to be stable if no group is modified any more. We provide requirements on constitutions and players' preferences under which stable group structures are induced for sure. Furthermore, by embedding many-to-many matchings into our setting, we apply our model to job markets with labor unions. We find a variation of Roth's "polarization of interests" (cf. Roth, 1984) between employers and employees.

We study von Neumann Morgenstern stable sets for one-to-one matching problems under the assumption of coalitional sovereignty, meaning that a deviating coalition of players does not have the power to arrange the matches of agents outside the coalition. We study both the case of pairwise and coalitional deviations. We argue further that dominance has to be replaced by path dominance along the lines of van Deemen (1991) and Page and Wooders (2009). This results in the pairwise myopic vNM set and the myopic vNM set, respectively. We obtain a unique prediction for both types of stable sets: the set of matchings that belong to the core. We also show that the pairwise and coalitional analogues of the level-1 farsighted set yield the core as the unique prediction.

We introduce the concept of a level-K farsighted set to study the influence of the degree of farsightedness on network stability. The concept generalizes existing concepts where all players are either fully myopic or fully farsighted. We show that a level-K farsighted set always exists and that the level-1 farsighted set G1 is always unique. For generic allocation rules, the set G1 always contains a level-K farsighted set for any K: We provide easy to verify conditions for a set of networks to be a level-K farsighted set and we consider the efficiency of networks in level-K farsighted sets. We show the tractability of the concept by applying it to a model of criminal networks.

We develop a general theoretical framework that allows us to study the group structures that are going to emerge at equilibrium when individuals are allowed to engage in several groups at the same time. We introduce the notion of constitution in order to model for each group the rules governing both the composition of the group and the conditions needed to leave the group and/or to become a new member of the group. We then propose the concept of con- stitutional stability to study the group structures that are going to emerge at equilibrium in overlapping coalition settings. This concept generalizes previ- ous stability concepts in the literature in which the constitutional rules were exogenously given or not explicitly considered. We combine requirements on constitutions and preferences for guaranteeing both the existence and the emergence of constitutionally stable group structures. Furthermore, by em- bedding many-to-many matchings into our setting, we show how these results could be useful to identify the constitutionally stable group structures in this particular environment.