Embedding Games with Strategic Complements into Games with Strategic Substitutes

Games with strategic substitutes (GSS) are generally less tractable than games with strategic complements (GSC). This paper revisits the GSC versus GSS comparison by establishing a novel connection between them. We show through a network perspective that, when the strategy set of each player is the product of some linearly ordered sets that are order isomorphic to subsets of the real space, every GSC can be embedded into a GSS, such that the set of pure strategy Nash equilibria of the former is a projection of that of the latter. In comparison, no GSS with multiple pure strategy Nash equilibria can be embedded into any GSC. In this sense, the class of GSS is broader than the class of GSC.

[1]  Éva Tardos,et al.  Introduction to computer science and economic theory , 2015, J. Econ. Theory.

[2]  F. Harary,et al.  STRUCTURAL BALANCE: A GENERALIZATION OF HEIDER'S THEORY1 , 1977 .

[3]  R. Amir.  Supermodularity and Complementarity in Economics: An Elementary Survey , 2003 .

[4]  David M. Kreps.  Notes On The Theory Of Choice , 1988 .

[5]  Lin Zhou,et al.  The Set of Nash Equilibria of a Supermodular Game Is a Complete Lattice , 1994 .

[6]  Yann Bramoullé,et al.  Games Played on Networks , 2015 .

[7]  M. Jackson,et al.  Games on Networks , 2014 .

[8]  Federico Echenique,et al.  Mixed equilibria in games of strategic complementarities , 2003 .

[9]  L. Shapley,et al.  The assignment game I: The core , 1971 .

[10]  X. Vives.  Oligopoly Pricing: Old Ideas and New Tools , 1999 .

[11]  J. Geanakoplos,et al.  Multimarket Oligopoly: Strategic Substitutes and Complements , 1985, Journal of Political Economy.

[12]  A. Tarski.  A LATTICE-THEORETICAL FIXPOINT THEOREM AND ITS APPLICATIONS , 1955 .

[13]  Joseph P. S. Kung,et al.  Bimatroids and invariants , 1978 .

[14]  Sunanda Roy,et al.  On the (non-)lattice structure of the equilibrium set in games with strategic substitutes , 2007 .

[15]  D. M. Topkis.  Supermodularity and Complementarity , 1998 .

[16]  R. Kranton,et al.  Strategic Interaction and Networks , 2010 .

[17]  Federico Echenique,et al.  Finding all equilibria in games of strategic complements , 2007, J. Econ. Theory.

[18]  Bruno Strulovici,et al.  Comparative Statics, Informativeness, and the Interval Dominance Order , 2009 .

[19]  L. Shapley,et al.  College Admissions and the Stability of Marriage , 1962 .

[20]  Mohamed Belhaj,et al.  Network games under strategic complementarities , 2014, Games Econ. Behav..

[21]  Federico Echenique,et al.  Mixed equilibria are unstable in games of strategic complements , 2004, J. Econ. Theory.

[22]  Asuman E. Ozdaglar,et al.  Optimal Pricing in Networks with Externalities , 2011, Oper. Res..

[23]  F. Heider.  Attitudes and cognitive organization. , 1946, The Journal of psychology.

[24]  Lloyd S. Shapley,et al.  MARKETS AS COOPERATIVE GAMES , 1955 .

[25]  X. Vives.  Nash equilibrium with strategic complementarities , 1990 .

[26]  Paul R. Milgrom,et al.  Monotone Comparative Statics , 1994 .

[27]  Richard W. Cuthbertson,et al.  The Logic of Logistics: Theory, Algorithms and Applications for Logistics Management , 1998, J. Oper. Res. Soc..

[28]  Francis Bloch,et al.  Pricing in social networks , 2013, Games Econ. Behav..

[29]  Andrew J. Monaco,et al.  Games with strategic complements and substitutes , 2015 .

[30]  D. M. Topkis.  Equilibrium Points in Nonzero-Sum n-Person Submodular Games , 1979 .

[31]  X. Vives.  Complementarities and Games: New Developments , 2004 .

[32]  Zhigang Cao,et al.  Shapley's Conjecture on the Cores of Abstract Market Games , 2017, Games Econ. Behav..

[33]  Wei Chen,et al.  Optimal Pricing in Social Networks with Incomplete Information , 2010, WINE.

[34]  Alexander Schrijver,et al.  Matroids and linking systems , 1979, J. Comb. Theory B.

[35]  A Characterization of Strategic Complementarities , 2003 .

[36]  Paul R. Milgrom,et al.  Rationalizability, Learning, and Equilibrium in Games with Strategic Complementarities , 1990 .