Abstract
This paper examines a normal form game of network formation due to Myerson (Game theory: analysis of conflict, Harvard University Press, Cambridge, 1991). All players simultaneously announce the links they wish to form. A link is created if and only if there is mutual consent for its formation. The empty network is always a Nash equilibrium of this game. We define a refinement of Nash equilibria that we call trial perfect. We show that the set of networks which can be supported by a pure strategy trial perfect equilibrium coincides with the set of pairwise-Nash equilibrium networks, for games with link-responsive payoff functions.
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Notes
But, this is not demanding robustness to bilateral moves, as pairwise-Nash equilibrium does not allow pairs of players to coordinate fully in their strategies.
Any network, except the complete network and networks where all absent links are beneficial to both parties involved, can be supported by multiple pure strategy Nash equilibria.
We adopt the network and link notation from Bloch and Jackson (2006).
To quote Myerson: “ Now consider a link-formation process in which each player independently writes down a list of players with whom she wants to form a link (...) and the payoff allocation is (...) for the graph that contains a link for every pair of players who have named each other” (p. 448).
Although this is a very simple game, the number of pure strategies of a player, \(2^{n-1}\), increases exponentially with the number of players. Baron et al. (2008) shows that it is NP-hard to check whether there exists a Nash equilibrium that guarantees a minimum payoff to all players.
Jackson and Rogers (2004) deals with random graphs in strategic network formation, though in a different context.
When nobody announces any link.
See also, Calvó-Armengol (2004) for an application of this equilibrium notion.
See Calvó-Armengol and İlkılı (2009) for a characterization of proper equilibria of the Myerson network formation game.
Though the technique used in the proof is similar to that of Proposition 3 of Calvó-Armengol and İlkılı (2009), in fact, the result in this paper is stronger and implies that proposition.
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We thank the editor, the associate editor and two anonymous reviewers for their thoughtful and constructive comments. We would also like to thank Sjaak Hurkens, Matthew Jackson, Joan de Marti, Jordi Massó, Andrew McLennan, Ted Turocy and Sergio Vicente for helpful discussions and comments. We dedicate this paper to the memory of the late Antoni Calvó-Armengol. The usual disclaimer applies. Rahmi İlkılıç acknowledges the support from CONICYT (FONDECYT No. 1181955), the Institute for Research in Market Imperfections and Public Policy, MIPP, ICM IS130002, Ministerio de Economía, Fomento y Turismo and the Complex Engineering Systems Institute, ISCI (ICM-FIC: P05-004-F, CONICYT: FB0816). This article is partially based on the research Hüseyin Ikizler has conducted for his Ph.D. dissertation at Bilkent University, Department of Economics.
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İlkılıç, R., İkizler, H. Equilibrium refinements for the network formation game. Rev Econ Design 23, 13–25 (2019). https://doi.org/10.1007/s10058-019-00218-y
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DOI: https://doi.org/10.1007/s10058-019-00218-y