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In this note we develop a generalisation of the $\lambda$-Core solution for non-cooperative games in normal form. We show that this generalised $\lambda$-Core is non-empty for the class of separable games that admit a socially optimal Nash equilibrium. Examples are provided that indicate that non-emptiness of the generalised $\lambda$-Core cannot be expected for large classes of normal form games.

We introduce a new network centrality measure founded on the Gately value for cooperative games with transferable utilities. A directed network is interpreted as representing control or authority relations between players--constituting a hierarchical network. The power distribution of a hierarchical network can be represented through a TU-game. We investigate the properties of this TU-representation and investigate the Gately value of the TU-representation resulting in the Gately power measure. We establish when the Gately measure is a Core power gauge, investigate the relationship of the Gately with the $\beta$-measure, and construct an axiomatisation of the Gately measure.

We investigate Gately's solution concept for cooperative games with transferable utilities. Gately's conception introduced a bargaining solution that minimises the maximal quantified ``propensity to disrupt'' the negotiation process of the players over the allocation of the generated collective payoffs. Gately's solution concept is well-defined for a broad class of games. We also consider a generalisation based on a parameter-based quantification of the propensity to disrupt. Furthermore, we investigate the relationship of these generalised Gately values with the Core and the Nucleolus and show that Gately's solution is in the Core for all regular 3-player games. We identify exact conditions under which generally these Gately values are Core imputations for arbitrary regular cooperative games. Finally, we investigate the relationship of the Gately value with the Shapley value.

We study a class of non-cooperative aggregative games -- denoted as \emph{social purpose games} -- in which the payoffs depend separately on a player's own strategy (individual benefits) and on a function of the strategy profile which is common to all players (social benefits) weighted by an individual benefit parameter. This structure allows for an asymmetric assessment of the social benefit across players. We show that these games have a potential and we investigate its properties. We investigate the payoff structure and the uniqueness of Nash equilibria and social optima. Furthermore, following the literature on partial cooperation, we investigate the leadership of a single coalition of cooperators while the rest of players act as non-cooperative followers. In particular, we show that social purpose games admit the emergence of a stable coalition of cooperators for the subclass of \emph{strict} social purpose games. Due to the nature of the partial cooperative leadership equilibrium, stable coalitions of cooperators reflect a limited form of farsightedness in their formation. As a particular application, we study the tragedy of the commons game. We show that there emerges a single stable coalition of cooperators to curb the over-exploitation of the resource., Comment: 27 pages

A network game assigns a level of collectively generated wealth to every network that can form on a given set of players. A variable network game combines a network game with a network formation probability distribution, describing certain restrictions on network formation. Expected levels of collectively generated wealth and expected individual payoffs can be formulated in this setting. We investigate properties of the resulting expected wealth levels as well as the expected variants of well-established network game values as allocation rules that assign to every variable network game a payoff to the players in a variable network game. We establish two axiomatizations of the Expected Myerson Value, originally formulated and proven on the class of communication situations, based on the well-established component balance, equal bargaining power and balanced contributions properties. Furthermore, we extend an established axiomatization of the Position Value based on the balanced link contribution property to the Expected Position Value.

This survey explores the literature on game-theoretic models of network formation under the hypothesis of mutual consent in link formation. The introduction of consent in link formation imposes a coordination problem in the network formation process. This survey explores the conclusions from this theory and the various methodologies to avoid the main pitfalls. The main insight originates from Myerson's work on mutual consent in link formation and his main conclusion that the empty network (the network without any links) always emerges as a strong Nash equilibrium in any game-theoretic model of network formation under mutual consent and positive link formation costs. Jackson and Wolinsky introduced a cooperative framework to avoid this main pitfall. They devised the notion of a pairwise stable network to arrive at equilibrium networks that are mainly non-trivial. Unfortunately, this notion of pairwise stability requires coordinated action by pairs of decision makers in link formation. I survey the possible solutions in a purely non-cooperative framework of network formation under mutual consent by exploring potential refinements of the standard Nash equilibrium concept to explain the emergence of non-trivial networks. This includes the notions of unilateral and monadic stability. The first one is founded on advanced rational reasoning of individuals about how others would respond to one's efforts to modify the network. The latter incorporates trusting, boundedly rational behaviour into the network formation process. The survey is concluded with an initial exploration of external correlation devices as an alternative framework to address mutual consent in network formation.

This paper considers the core of a competitive market economy with an endogenous social division of labour. The theory is founded on the notion of a "consumer-producer", who consumes as well as produces commodities. First, we show that the Core of such an economy with an endogenous social division of labour can be founded on deviations of coalitions of arbitrary size, extending the seminal insights of Vind and Schmeidler for pure exchange economies. Furthermore, we establish the equivalence between the Core and the set of competitive equilibria for continuum economies with an endogenous social division of labour. Our analysis also concludes that self-organisation in a social division of labour can be incorporated into the Edgeworthian barter process directly. This is formulated as a Core equivalence result stated for a Structured Core concept based on renegotiations among fully specialised economic agents, i.e., coalitions that use only fully developed internal divisions of labour. Our approach bridges the gap between standard economies with social production and coalition production economies. Therefore, a more straightforward and natural interpretation of coalitional improvement and the Core can be developed than for coalition production economies., Comment: 20 pages

This paper studies critical nodes or "middlemen" that intermediate flows in a directed network. The contestability of a node is introduced as a network topological concept of competitiveness meaning that an intermediary's role in the brokering of flows in the network can be substituted by a group of other nodes. We establish the equivalence of uncontested intermediaries and middlemen. The notion of node contestability gives rise to a measure that quantifies the control exercised by a middleman in a network. We present a comparison of this middleman centrality measures with relevant, established network centrality measures. Furthermore, we provide concepts and measures expressing the robustness of a middleman as the number of links or nodes that have to be added to or removed from the network to nullify the middleman's power. We use these concepts to identify and measure middleman power and robustness in two empirical networks: Krackhardt's advice network of managers in a medium sized corporation and the well-known Florentine marriage network as a proxy of power brokerage between houses in Renaissance Florence., Comment: 30 pages, 8 figures, 5 tables. arXiv admin note: substantial text overlap with arXiv:1401.0655

Critical nodes or "middlemen" have an essential place in both social and economic networks when considering the flow of information and trade. This paper extends the concept of critical nodes to directed networks. We identify strong and weak middlemen. Node contestability is introduced as a form of competition in networks; a duality between uncontested intermediaries and middlemen is established. The brokerage power of middlemen is formally expressed and a general algorithm is constructed to measure the brokerage power of each node from the networks adjacency matrix. Augmentations of the brokerage power measure are discussed to encapsulate relevant centrality measures. We use these concepts to identify and measure middlemen in two empirical socio-economic networks, the elite marriage network of Renaissance Florence and Krackhardt's advice network., Comment: 28 pages, 6 figures, 2 tables

Recent research in industrial organisation has investigated the essential place that middlemen have in the networks that make up our global economy. In this paper we attempt to understand how such middlemen compete with each other through a game theoretic analysis using novel techniques from decision-making under ambiguity. We model a purposely abstract and reduced model of one middleman who pro- vides a two-sided platform, mediating surplus-creating interactions between two users. The middleman evaluates uncertain outcomes under positional ambiguity, taking into account the possibility of the emergence of an alternative middleman offering intermediary services to the two users. Surprisingly, we find many situations in which the middleman will purposely extract maximal gains from her position. Only if there is relatively low probability of devastating loss of business under competition, the middleman will adopt a more competitive attitude and extract less from her position., Comment: 23 pages, 3 figures