Characterizing Stability Properties in Games with Strategic Substitutes

In games with strategic substitutes (GSS), convergence of the best response dynamic starting from the inf (or sup) of the strategy space is equivalent to global stability (convergence of every adaptive dynamic to the same pure strategy Nash equilibrium). Consequently, in GSS, global stability can be analyzed using a single best response dynamic. Moreover, in GSS, global stability is equivalent to dominance solvability, showing that in this class of games, two different foundations for robustness of predicted outcomes are equivalent, and both can be checked using a single best response dynamic. These equivalences are useful to study stability of equilibria in a variety of applications. Furthermore, in parameterized GSS, under natural conditions, dynamically stable equilibrium selections can be viewed in terms of monotone selections of equilibria. Several examples are provided.Global stability, adaptive dynamics, strategic substitutes, dominance solvable, learning, monotone comparative statics, Correspondence principle

Endogenous Timing in General Rent-Seeking and Conflict Models

This paper examines simultaneous versus sequential choice of effort in a two player contest with a general contest success function. The timing of moves, determined in a pre-play stage prior to the contest-subgame, as well as the value of the prize is allowed to be endogenous. Contrary to endogenous timing models with an exogenously fixed prize the present paper finds the following. (1) Players may decide to choose their effort simultaneously in the subgame perfect equilibrium (SPE) of the extended game. (2) The SPE does not need to be unique, (3) in particular, there is no unique SPE with sequential moves if costs of effort are exclusively endogenously determined. (4) If the unique SPE is sequential play, the win probability in the NE is in no way crucial for the determination of an endogenous leadership. (5) Finally, symmetry among players does not rule out incentives for precommitment to effort locally away from the Nash-Cournot level.Contests, Endogenous timing, Endogenous prize

A non-robustness in the order structure of the equilibrium set in lattice games

The order and lattice structure of the equilibrium set in games with strategic complements do not survive a minimal introduction of strategic substitutes: in a lattice game in which all-but-one players exhibit strategic complements (with one player exhibiting strict strategic complements), and the remaining player exhibits strict strategic substitutes, no two equilibria are comparable. More generally, in a lattice game, if either (1) just one player has strict strategic complements and another player has strict strategic substitutes, or (2) just one player has strict strategic substitutes and has singleton-valued best-responses, then without any restrictions on the strategic interaction among the other players, no two equilibria are comparable. In such cases, the equilibrium set is a non-empty, complete lattice, if, and only if, there is a unique equilibrium. Moreover, in such cases, with linearly ordered strategy spaces, the game has at most one symmetric equilibrium. Several examples are presented.Lattice games, strategic complements, strategic substitutes, equilibrium set

Strategic Interaction and Networks

This paper brings a general network analysis to a wide class of economic games. A network, or interaction matrix, tells who directly interacts with whom. A major challenge is determining how network structure shapes overall outcomes. We have a striking result. Equilibrium conditions depend on a single number: the lowest eigenvalue of a network matrix. Combining tools from potential games, optimization, and spectral graph theory, we study games with linear best replies and characterize the Nash and stable equilibria for any graph and for any impact of players’ actions. When the graph is sufficiently absorptive (as measured by this eigenvalue), there is a unique equilibrium. When it is less absorptive, stable equilibria always involve extreme play where some agents take no actions at all. This paper is the first to show the importance of this measure to social and economic outcomes, and we relate it to different network link patterns.Networks, potential games, lowest eigenvalue, stable equilibria, asymmetric equilibria

Endogenous Timing in General Rent‐Seeking and Conflict Models

This paper examines simultaneous versus sequential choice of effort in a twoplayer contest with a general contest success function. The timing of moves, determined in a pre‐play stage prior to the contest‐subgame, as well as the value of the prize is allowed to be endogenous. Contrary to endogenous timing models with an exogenously fixed prize the present paper finds the following. (1) Players may decide to choose their effort simultaneously in the subgame perfect equilibrium (SPE) of the extended game, (2) the SPE does not need to be unique, (3) in particular, there is no unique SPE with sequential moves if costs of effort are exclusively endogenously determined, (4) if the unique SPE is sequential play, the win probability in the NE is in no way crucial for the determination of an endogenous leadership, (5) and symmetry among players does not rule out incentives for precommitment to effort locally away from the Nash‐Cournot levelContests, Endogenous timing, Endogenous prize

Coalition Formation in Games without Synergies

This paper establishes sufficient conditions for the existence of a stable coalition structure in the ”coalition unanimity” game of coalition formation, first defined by Hart and Kurz (1983) and more recently studied by Yi (1997, 2000). Our conditions are defined on the strategic form game used to derive the payoffs the game of coalition formation. We show that if no synergies are generated by the formation of coalitions, a stable coalition structure always exists provided that players are symmetric and either the game exhibits strategic complementarity or, if strategies are substitutes, the best reply functions are contractions. We illustrate the role of synergies in a Cournot oligopoly example with cost reducing R&D.Coalition formation, Synergies, Strong Nash equilibrium

Diffusion of Behavior and Equilibrium Properties in Network Games

Situations in which agents’ choices depend on choices of those in close proximity, be it social or geographic, are ubiquitous. Selecting a new computer platform, signing a political petition, or even catching the flu are examples in which social interactions have a significant role. While some behaviors or states propagate and explode within the population (e.g., Windows OS, the HIV virus) others do not (e.g., certain computer viruses). Our goal in this paper is twofold. First, we provide a general dynamic model in which agents’ choices depend on the underlying social network of connections. Second, we show the usefulness of the model in determining when a given behavior expands within a population or disappears as a function of the environment’s fundamentals. We study a framework in which agents face a choice between two actions, 0 and 1 (e.g., whether to pursue a certain level of education, switch to Linux OS, etc.). Agents are linked through a social network, and an agent’s payoffs from each action depend on the number of neighbors she has and her neighbors’ choices. The diffusion process is defined so that at each period, each agent best responds to the actions taken by her neighbors in the previous period, assuming that her neighbors follow the population distribution of actions (a mean-field approximation). Steady states correspond to equilibria of the static game. Under some simple conditions, equilibria take one of two forms. Some are stable, so that a slight perturbation to any such equilibrium would lead the diffusion process to converge back to that equilibrium point. Other equilibria are unstable, so that a slight change in the distribution of actions leads to a new distribution of actions and eventually to a stable steady state. We call such equilibria tipping points. We analyze how the environment’s fundamentals (cost distribution, payoffs, and network structure) affect the set of equilibria, and characterize the adoption patterns within the network. The paper relates to recent work on network games and network diffusion, including work by Stephen Morris (2000); Pastor-Satorras and Vespignani (2000); Mark E. J. Newman (2002); Dunia López-Pintado (2004); Jackson and Brian W. Rogers (2007); Jackson and Yariv (2005); and Andrea Galeotti et al. (2005, henceforth GGJVY). Its contribution is in characterizing diffusion of strategic behavior and analyzing the stability properties of equilibria, and employing methods that allow us to make comparisons across general network structures and settings. Given that social networks differ substantially and systematically in structure across settings (e.g., ethnic groups, professions, etc.), understanding the implications of social structure on diffusion is an important undertaking for a diverse set of applications