The Structure of the Lattices of Pure Strategy Nash Equilibria of Binary Games of Strategic Complements
25 Pages Posted: 17 Mar 2012 Last revised: 14 Feb 2013
Date Written: March 2012
Abstract
It has long been established in the literature that the set of pure strategy Nash equilibria of any binary game of strategic complements among a set N of players can be seen as a lattice on the set of all subsets of N under the partial order defined by the set inclusion relation (⊆). If the game happens to be strict in the sense that players are never indifferent among outcomes then the resulting lattice of equilibria satisfies a straightforward sparseness condition. In this paper we show that in fact this class of games expresses all such lattices. In particular we prove that any lattice under set inclusion on 2N satisfying this sparseness condition is the set of pure strategy Nash equilibria of some binary game of strategic complements with no indifference. This fact then suggests an interesting way of studying some subclasses of games of strategic complements: By attempting to characterize the subcollections of lattices that each of these classes is able to express. In the second part of the paper we study subclasses of binary games of strategic complements with no indifference, defined by restrictions that capture particular social influence structures: 1) simple games, 2) nested games, 3) hierarchical games 4) clan-like games and 5) graphical games of thresholds.
Keywords: Peer Effects, Social Networks, Implementation
JEL Classification: C70, C72, C00, D70
Suggested Citation: Suggested Citation
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