Abstract
Consider a setting in which agents can take one of two ordered actions and in which the incentive to take the high action increases in the number of other agents taking it. Furthermore, assume that we do not know anything else about the game being played. What can we say about the details of the interaction between actions and incentives when we observe a set or a subset of all possible equilibria? In this paper, we study this question by exploring three nested classes of games: (a) binary games of strategic complements; (b) games in (a) that admit a network representation; and (c) games in (b) in which the network is complete. Our main results are the following: 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 agents can be seen as a lattice on the set of all subsets of N under the partial order defined by the set inclusion relation (C). If the game happens to be strict in the sense that agents are never indifferent among outcomes (games in (a)), then the resulting lattice of equilibria satisfies a straightforward sparseness condition. (1) We show that, in fact, for each such lattice, L, there is a game in (a), such that its set of equilibria is L (we say that such a game expresses L); (2) We show that there exists a game in (b), whose set of equilibria contains a given collection, C, of subsets of N, if and only C satisfies the sparseness condition, and the smallest game in (a) expressing C is trade robust; (3) We show that there exists a game on the complete graph (games in (c)), whose set of equilibria coincides with some collection, C, if and only if C is a chain satisfying the sparseness condition.
Highlights
Games of strategic complements represent situations in which actions admit a natural ordering, and in which the incentives of any one agent to take a higher action are increasing in the actions taken by other agents
The set of equilibria of these games have several very nice properties stemming from the fact that they are complete lattices (See Zhou (1994) [8] for a proof of the fact that the set of equilibria of any supermodular game is a complete lattice, and Topkis (1998) [9] for an extensive survey of supermodularity, including games of strategic complements), and these properties often imply that they can be profitably analyzed using monotone comparative statics techniques
Any observed set of behaviors implies some restrictions on the best response correspondence of the game that may have given rise to them as equilibria, and in the case of games of strategic complements, any such best response correspondence can be seen as stemming from a simple model of peer influence
Summary
Games of strategic complements represent situations in which actions admit a natural ordering, and in which the incentives of any one agent to take a higher action (given that ordering) are increasing in the actions taken by other agents. Since their approach builds up directly on the description of influence (either the model of influence or the command game) and not on first predicting possible behaviors based on a notion of equilibrium, their theorems are not directly applicable to our work The richness of these two cooperative frameworks and the simple technical bridge with the non-cooperative literature noted in this paper, allowing us to derive our results regarding weighted graphical games, suggests that there may be very useful further connections between the cooperative and non-cooperative approaches to the question of influence.
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