Convexity and Complementarity in Network Formation: Implications for the Structure of Pairwise Stable Networks

This paper studies the properties of convexity (concavity) and strategic complements (substitutes) in network formation and the implications for the structure of pairwise stable networks. First, different definitions of convexity (concavity) in own links from the literature are put into the context of diminishing marginal utility of own links. Second, it is shown that there always exists a pairwise stable network as long as the utility function of each player satisfies convexity in own links and strategic complements. For network societies with a profile of utility functions satisfying concavity in own links and strategic complements, a local uniqueness property of pairwise stable networks is derived. The results do neither require any specification on the utility function nor any other additional assumptions such as homogeneity.Networks, Network Formation, Game Theory, Supermodularity, Increasing Differences, Stability, Existence, Uniqueness

Under-connected and Over-connected Networks

Since the seminal contribution of Jackson & Wolinsky 1996 [A Strategic Model of Social and Economic Networks, JET 71, 44-74] it has been widely acknowledged that the formation of social networks exhibits a general conflict between individual strategic behavior and collective outcome. What has not been studied systematically are the sources of inefficiency. We approach this omission by analyzing the role of positive and negative externalities of link formation. This yields general results that relate situations of positive externalities with stable networks that cannot be “too dense” in a well-defined sense, while situations with negative externalities tend to induce “too dense” networks.Networks, Network Formation, Connections, Game Theory, Externalities, Spillovers, Stability, Efficiency

Under-connected and over-connected networks

Since the seminal contribution of Jackson & Wolinsky 1996 [A Strategic Model of Social and Economic Networks, JET 71, 44-74] it has been widely acknowledged that the formation of social networks exhibits a general conflict between individual strategic behavior and collective outcome. What has not been studied systematically are the sources of inefficiency. We approach this gap by analyzing the role of positive and negative externalities of link formation. We find general results that relate situations of positive externalities with stable networks that cannot be "too dense" in a well-defined sense, while situations with negative externalities, tend to induce "too dense" networks.networks, network formation, connections, game theory, externalities, spillovers, stability, efficiency

Under-connected and over-connected networks: the role of externalities in strategic network formation

Since the seminal contribution of Jackson and Wolinsky (J Econ Theory 71(1):44–74, 1996) it has been widely acknowledged that the formation of social networks exhibits a general conflict between individual strategic behavior and collective outcome. What has not been studied systematically are the sources of inefficiency. We approach this omission by analyzing the role of positive and negative externalities of link formation. This yields general results that relate situations of positive externalities with stable networks that cannot be “too dense” in a well-defined sense, while situations with negative externalities tend to induce “too dense” networks. Those results are neither restricted to specific assumptions on the agents’ preferences (e.g. homogeneity), nor to a specific notion of stability or efficiency

Stable Networks in Homogeneous Societies

Hellmann T, Landwehr J. Stable Networks in Homogeneous Societies. Center for Mathematical Economics Working Papers. Vol 517. Bielefeld: Center for Mathematical Economics, Bielefeld University; 2014.We study the structure of pairwise stable networks from a very general point. Rather than assuming a particular functional form of utility, we simply assume that the society is homogeneous, i.e. that agents' utilities differ only with respect to their network position while their names do not matter. Existence of certain stable network structures is then implied by fairly general assumptions on externalities between links. Depending on the form of link externalities, either the empty or complete network are always pairwise stable, stable symmetric networks exist, or stable networks with a connected subgroup exist. If the society becomes more homogeneous, then it is possible to characterize the set of all pairwise stable networks: they are nested split graphs (NSG). We illustrate these results with many examples from the literature, including utility profiles that depend on centrality measures such as Bonacich centrality. In particular, for low discount factors every pairwise stable network is an NSG if utility is given by Bonacich centrality

The transmission of continuous cultural traits in endogenous social networks

Hellmann T, Panebianco F. The transmission of continuous cultural traits in endogenous social networks. Center for Mathematical Economics Working Papers. Vol 579. Bielefeld: Center for Mathematical Economics; 2018.We study a model of transmission of continuous cultural traits across generations where children learn their cultural trait from their parents and their social environment modeled by a network. Parents can engage in the socialization process of their children by biasing links in the network in order for their children to adopt a cultural trait similar to their own. In this endogenous network, we study the emergence of positive and negative role models, the existence of a steady state cultural trait, its characterization in terms long-run influence of each dynasty, and the speed of convergence

Local and Global Pollution and International Environmental Agreements in a Network Approach

Günther M, Hellmann T. Local and Global Pollution and International Environmental Agreements in a Network Approach. Center for Mathematical Economics Working Papers. Vol 545. Bielefeld: Center for Mathematical Economics; 2015.Increasing concerns about climate change have given rise to the formation of International Environmental Agreements (IEAs) as a possible solution to limit global pollution effects. In this paper, we study the stability of IEAs in a repeated game framework where we restrict to strategies which are simple and invariant to renegotiation. Our main contribution to the literature on IEAs is that we allow for heterogeneous patterns of pollution such that additional to a global effect of pollution there are local pollution effects represented by a network structure. We show that stable IEAs exist if the network structure is balanced. Too large asymmetries in the degree of local spillovers may however lead to non-existence of stable structures. The generality of our approach allows for several applications to general problems in the provision of public goods

Evolution of Social networks

Hellmann T, Staudigl M. Evolution of Social networks. Working Papers. Institute of Mathematical Economics. Vol 470. Bielefeld: Universität Bielefeld; 2012.Modeling the evolution of networks is central to our understanding of modern large communication systems, such as theWorld-Wide-Web, as well as economic and social networks. The research on social and economic networks is truly interdisciplinary and the number of modeling strategies and concepts is enormous. In this survey we present some modeling approaches, covering classical random graph models and game-theoretic models, which may be used to provide a unified framework to model and analyze the evolution of networks

R&D Investments Under Endogenous Cluster Formation

Dawid H, Hellmann T. R&D Investments Under Endogenous Cluster Formation. Working Papers in Economics and Management. Vol 04-2016. Bielefeld: Bielefeld University, Department of Business Administration and Economics; 2016.We study investments in R&D and the formation of R&D clusters of firms which are competitors in the market. In a three stage game, firms first decide on the budget allocated to their R&D department, then form research clusters and finally compete in quantities. The second stage cluster formation is modeled by the unanimity game introduced in Bloch(1995). We show that for any distribution of R&D investments, an equilibrium of the second stage cluster formation exists and is generically unique up to a permutation of firms which chose the same investment. Restricting to two investment levels in the first stage, we provide a complete characterization of the equilibria of the three stage game. We show that for some range of investment costs, equilibria with no-investment co-exist with equilibria where a large fraction or even all firms invest in R&D. Furthermore, in the high-investment equilibrium firms over{invest compared to a scenario where research clusters are ex-ante fixed and also compared to the welfare optimum

Pairwise stable networks in homogeneous societies

Hellmann T, Landwehr J. Pairwise stable networks in homogeneous societies. Center for Mathematical Economics Working Papers. Vol 517 Version Januar 2018. Bielefeld: Center for Mathematical Economics; 2018.We study general properties of pairwise stable networks in homogeneous societies, i.e. when agents' utilities differ only with respect to their network position while their names do not matter. Rather than assuming a particular functional form of utility, we impose general link externality conditions on utility such as ordinal convexity and ordinal strategic complements. Depending on these rather weak notions of link externalities, we show that pairwise stable networks of various structure exist. For stronger versions of the convexity and strategic complements conditions, we are even able to characterize all pairwise stable networks: they are nested split graphs (NSG). We illustrate these results with many examples from the literature, including utility funtions that arise from games with strategic complements played on the network and utility funtions that depend on centrality measures such as Bonacich centrality