Strategic Basins of Attraction, the Path Dominance Core, and Network Formation Games

Given the preferences of players and the rules governing network formation, what networks are likely to emerge and persist? And how do individuals and coalitions evaluate possible consequences of their actions in forming networks? To address these questions we introduce a model of network formation whose primitives consist of a feasible set of networks, player preferences, the rules of network formation, and a dominance relation on feasible networks. The rules of network formation may range from non-cooperative, where players may only act unilaterally, to cooperative, where coalitions of players may act in concert. The dominance relation over feasible networks incorporates not only player preferences and the rules of network formation but also assumptions concerning the degree of farsightedness of players. A specification of the primitives induces an abstract game consisting of (i) a feasible set of networks, and (ii) a path dominance relation defined on the feasible set of networks. Using this induced game we characterize sets of network outcomes that are likely to emerge and persist. Finally, we apply our approach and results to characterize the equilibrium of well known models and their rules of network formation, such as those of Jackson and Wolinsky (1996) and Jackson and van den Nouweland (2005).

[1]  M. D. Davis.  La theorie des jeux , 1973 .

[2]  Gabrielle Demange,et al.  On Group Stability in Hierarchies and Networks , 2004, Journal of Political Economy.

[3]  P. Jean-Jacques Herings,et al.  Farsightedly Stable Networks , 2006, Games Econ. Behav..

[4]  David Schmeidler,et al.  Collective Choice Correspondences as Admissible Outcomes of Social Bargaining Processes , 1976 .

[5]  M. Richardson,et al.  Solutions of Irreflexive Relations , 1953 .

[6]  The Partnered Core of a Game without Side Payments , 1996 .

[7]  C. Qin.  A conjecture of Shapley and Shubik on competitive outcomes in the cores of NTU market games , 1993 .

[8]  H. Moulin,et al.  Cores of effectivity functions and implementation theory , 1982 .

[9]  Jeroen Kuipers,et al.  Absorbing and generalized stable sets , 2005, Soc. Choice Welf..

[10]  M. Chwe.  Farsighted Coalitional Stability , 1994 .

[11]  M. Slikker,et al.  Network Formation, Costs, and Potential Games , 2002 .

[12]  M. Maschler,et al.  The Structure of the Kernel of a Cooperative Game , 1967 .

[13]  R. Rosenthal.  A class of games possessing pure-strategy Nash equilibria , 1973 .

[14]  M. Jackson.  STABLE NETWORKS , 2000 .

[15]  Myrna Holtz Wooders,et al.  Networks and farsighted stability , 2005, J. Econ. Theory.

[16]  R. Aumann.  Markets with a continuum of traders , 1964 .

[17]  A. Nouweland.  Group Formation in Economics: Models of Network Formation in Cooperative Games , 2005 .

[18]  Stef Tijs,et al.  Potential maximizers and network formation , 2000, Math. Soc. Sci..

[19]  M. Jackson,et al.  A Strategic Model of Social and Economic Networks , 1996 .

[20]  M. Breton,et al.  Equilibria in a Model with Partial Rivalry , 1997 .

[21]  Robert W. Rosenthal,et al.  Cooperative games in effectiveness form , 1972 .

[22]  Fred B. Schneider,et al.  A Theory of Graphs , 1993 .

[23]  Prakash P. Shenoy,et al.  A dynamic solution concept for abstract games , 1980 .

[24]  Oriol Carbonell-Nicolau.  Games and Economic Behavior , 2011 .

[25]  A. Mauleon,et al.  Farsightedness and Cautiousness in Coalition Formation , 2003 .

[26]  The partnered core of an economy and the partnered competitive equilibrium , 1996 .

[27]  O. Rozenfeld.  Strong Equilibrium in Congestion Games , 2007 .

[28]  On the existence of a pure strategy Nash equilibrium in group formation games , 2000 .

[29]  Dynamic Club Formation With Coordination , 2002 .

[30]  Antoni Calvó-Armengol,et al.  Pairwise-stability and Nash equilibria in network formation , 2005, Int. J. Game Theory.

[31]  Matthew O. Jackson,et al.  The Evolution of Social and Economic Networks , 2002, J. Econ. Theory.

[32]  Hans Peters,et al.  Chapters in Game Theory , 2004 .

[33]  M. Wooders.  The Tiebout Hypothesis: Near Optimality in Local Public Good Economies , 1980 .

[34]  L. Shapley,et al.  The kernel and bargaining set for convex games , 1971 .

[35]  Philip Wolfe,et al.  Contributions to the theory of games , 1953 .

[36]  G. Demange,et al.  Group Formation in Economics , 2005 .

[37]  Shlomo Weber,et al.  Pure Strategy Nash Equilibrium in a Group Formation Game with Positive Externalities , 1997 .

[38]  L. Xue,et al.  Coalitional stability under perfect foresight , 1998 .

[39]  E. Kalai,et al.  An Admissible Set Occurring in Various Bargaining Situations , 1977 .

[40]  Sanjeev Goyal,et al.  A Noncooperative Model of Network Formation , 2000 .

[41]  Shlomo Weber,et al.  Equilibrium in a Finite Local Public Goods Economy , 1998 .

[42]  Club Formation Games with Farsighted Agents , 2005 .

[43]  Roger B. Myerson,et al.  Graphs and Cooperation in Games , 1977, Math. Oper. Res..

[44]  L. Shapley,et al.  REGULAR ARTICLEPotential Games , 1996 .

[45]  Anne van den Nouweland,et al.  Strongly Stable Networks , 2002, Games Econ. Behav..

[46]  H. Scarf.  The Core of an N Person Game , 1967 .

[47]  E. Rowland.  Theory of Games and Economic Behavior , 1946, Nature.

[48]  L. Shapley,et al.  Potential Games , 1994 .

[49]  Donald B. Gillies,et al.  3. Solutions to General Non-Zero-Sum Games , 1959 .

[50]  Ad M. A. van Deemen,et al.  A note on generalized stable sets , 1991 .

[51]  John P. Conley,et al.  Tiebout Economies with Differential Genetic Types and Endogenously Chosen Crowding Characteristics , 2001, J. Econ. Theory.

[52]  Frank H. Page,et al.  Group Formation in Economics: Farsighted Stability in Network Formation , 2005 .

[53]  Networks and clubs , 2007 .

[54]  Bernard Guerrien.  La théorie des jeux , 1995 .

[55]  I. Milchtaich,et al.  Congestion Games with Player-Specific Payoff Functions , 1996 .

[56]  Myrna Holtz Wooders,et al.  Club Networks with Multiple Memberships and Noncooperative Stability , 2009, Games Econ. Behav..

[57]  Licun Xue,et al.  Farsighted stability in hedonic games , 2000, Soc. Choice Welf..

[58]  M. Jackson.  A Survey of Models of Network Formation: Stability and Efficiency , 2003 .

[59]  J. Harsanyi.  An Equilibrium-Point Interpretation of Stable Sets and a Proposed Alternative Definition , 1974 .

[60]  W. F. Lucas,et al.  A Game with No Solution , 1968 .