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Strategic complementarities, network games and endogenous network formation

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Abstract

This paper investigates the role of strategic complementarities in the context of network games and network formation models. In the general model of static games on networks, we characterize conditions on the utility function that ensure the existence and uniqueness of a pure-strategy Nash equilibrium, regardless of the network structure. By applying the game to empirically-relevant networks that feature nestedness—Nested Split Graphs—we show that equilibrium strategies are non-decreasing in the degree. We extend the framework into a dynamic setting, comprising a game stage and a formation stage, and provide general conditions for the network process to converge to a Nested Split Graph with probability one, and for this class of networks to be an absorbing state. The general framework presented in the paper can be applied to models of games on networks, models of network formation, and combinations of the two.

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Notes

  1. For instance, Watts and Strogatz (1998) show that social networks are characterized by high clustering and short average path lengths and Barabási and Albert (1999) argue that the degree distribution follows a power-law. Examples of networks that are nested, i.e. in which the neighborhood of nodes are contained within the neighborhoods of nodes with higher degrees, are identified by König et al. (2014). See also, e.g., Foster and Rosenzweig (1995), Durlauf (2004), for network effects.

  2. In Jackson and Rogers (2007), an endogenous network formation model that generates networks consistent with the empirically observed properties of many networks, except nestedness, is proposed. The model laid out by König et al. (2014) is able to generate also nestedness.

  3. Under mild technical conditions the concept is equivalent to supermodularity, see for instance Milgrom and Roberts (1990).

  4. There is ample empirical evidence of complementarities in networks. Case and Katz (1991) finds that the probability of a youth being involved in criminal activities increases when the family moves to a neighborhood with more crime. Strategic complementarities in actions are also present in labor markets, in R & D activity and within families. See Durlauf (2004) for an extensive survey of neighborhood effects in economics.

  5. Our methodology applies the work by Karamardian (1969a, b) and Rosen (1965) to a network setting, where payoffs depend only directly on own and neighbors’ actions.

  6. Nested Split Graphs have been studied in the fields of physics and mathematics. They also go under the name Threshold Graphs, for a review see Mahadev and Peled (1995).

  7. This part is also related to dynamic models of network formation, such as Jackson and Wolinsky (1996), Bala and Goyal (2000), Watts (2001), Dutta et al. (2005), where the shape of the network is dynamically changing. Although the resulting networks are the outcome of endogenous link choices, these models do not feature the combined element of strategic interactions and network formation present in this paper.

  8. We will refer to this mapping as a utility function for simplicity, but as the framework is general and applicable to different economic and social settings, it may well represent other types of payoffs as well.

  9. This representation implies both that the utility of own effort is constant across players and that the utility from neighbors’ actions is the same.

  10. We focus on differentiable utility functions allowing us to state supermodularity in partial derivatives.

  11. In case the process prescribes adding a link to an agent who is connected to everyone, or removing a link from an agent who has no connections, payoffs are realized and the period ends.

  12. If remaining idle when getting the opportunity to form a link were part of the agents’ choice sets, our results would not be affected, but endowing players with the possibility of not removing a link when removal is imposed, would. The theorems would still hold under these assumptions, but the resulting network formation process would be uninteresting since no agent would ever want to remove a link if given the opportunity to remain idle. Due to strict positive externalities, the network would converge to the complete network if this choice were endogenous, as a player’s utility is increasing in degree.

  13. An important contribution to the network-formation literature is Barabási and Albert (1999), who provide a framework in which links are created randomly with the probability of creating a link proportional to the degree. Preferential attachment models have the feature that as nodes obtain more links, their probability of obtaining a new link increases, thereby generating a positive feedback loop. The virtue of these networks is that they exhibit scale-free degree distributions—a feature common in empirically observed networks. See Jackson (2008) and Goyal (2007) for a review of preferential-attachment models.

  14. A continuous random variable \(X\) satisfies the power law if its density is of the form \(f(x)=\kappa x^{-\gamma }\) for positive scalars \(\kappa \) and \(\gamma \).

  15. Empirical properties of networks—the ones above included—are recapitulated in Jackson and Rogers (2007), König et al. (2014).

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Acknowledgments

We would like to thank Gabriel Carroll, Matthew O. Jackson, Michael D. König, Marco van der Leij, Ismael Y. Mourifié, Anna Larsson Seim, Yves Zenou and an anonymous referee for helpful comments and discussions as well as seminar participants at the MIT/Harvard Network Economics Workshop, the EEA meetings in 2010 and the SAET conference in 2012.

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Authors and Affiliations

  1. Department of Mathematics, Stockholm University, Stockholm, Sweden

    Andreas Lagerås

  2. Department of Economics, University of Toronto, Toronto, Canada

    David Seim

  3. The Research Institute for Industrial Economics, Stockholm, Sweden

    David Seim

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  1. Andreas Lagerås
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Correspondence to David Seim.

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Lagerås, A., Seim, D. Strategic complementarities, network games and endogenous network formation. Int J Game Theory 45, 497–509 (2016). https://doi.org/10.1007/s00182-015-0466-x

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