When Does Learning in Games Generate Convergence to Nash Equilibria? The Role of Supermodularity in an Experimental Setting

This study clarifies the conditions under which learning in games produces convergence to Nash equilibria in practice. Previous work has identified theoretical conditions under which various stylized learning processes achieve convergence. One technical condition is supermodularity, which is closely related to the more familiar concept of strategic complementarities. We experimentally investigate the role of supermodularity in achieving convergence through learning. Using a game from the literature on solutions to externalities, we systematically vary a free parameter below, close to, at and beyond the threshold of supermodularity to assess its effects on convergence. We find that supermodular and ¡°near-supermodular¡± games converge significantly better than those far below the threshold. From a little below the threshold to the threshold, the improvement is statistically insignificant. Within the class of supermodular games, increasing the parameter far beyond the threshold does not significantly improve convergence. Simulation shows that while most experimental results persist in the long run, some become more pronounced.learning, supermodular games

Some Notes on Learning in Games with Strategic Complementarities

Fictitious play is the classical myopic learning process, and games with strategic complementarities are an important class of games including many economic applications. Knowledge about convergence properties of fictitious play in this class of games is scarce, however. Beyond dominance solvable games, global convergence has only been established for games with strategic complementarities and diminishing marginal returns (Krishna, 1992, HBSWorking Paper 92-073). This result is known to depend critically on the assumption of a tie-breaking rule. We show that restricting the analysis to nondegenerate games allows us to drop this assumption. More importantly, an ordinal version of strategic complementarities turns out to suffice. As a byproduct, we also obtain global convergence in generalized ordinal potential games with diminishing marginal returns.Fictitious Play, Learning Process, Strategic Complementarities, Supermodular Games

Two More Classes of Games with the Fictitious Play Property

Fictitious play is the oldest and most studied learning process for games. Since the already classical result for zero-sum games, convergence of beliefs to the set of Nash equilibria has been established for some important classes of games, including weighted potential games, supermodular games with diminishing returns, and 3x3 supermodular games. Extending these results, we establish convergence for ordinal potential games and quasi-supermodular games with diminishing returns. As a by-product we obtain convergence for 3xm and 4x4 quasi-supermodular games.Fictitious Play, Learning Process, Ordinal Potential Games, Quasi-Supermodular Games

Mean Field Equilibrium in Dynamic Games with Complementarities

We study a class of stochastic dynamic games that exhibit strategic complementarities between players; formally, in the games we consider, the payoff of a player has increasing differences between her own state and the empirical distribution of the states of other players. Such games can be used to model a diverse set of applications, including network security models, recommender systems, and dynamic search in markets. Stochastic games are generally difficult to analyze, and these difficulties are only exacerbated when the number of players is large (as might be the case in the preceding examples). We consider an approximation methodology called mean field equilibrium to study these games. In such an equilibrium, each player reacts to only the long run average state of other players. We find necessary conditions for the existence of a mean field equilibrium in such games. Furthermore, as a simple consequence of this existence theorem, we obtain several natural monotonicity properties. We show that there exist a "largest" and a "smallest" equilibrium among all those where the equilibrium strategy used by a player is nondecreasing, and we also show that players converge to each of these equilibria via natural myopic learning dynamics; as we argue, these dynamics are more reasonable than the standard best response dynamics. We also provide sensitivity results, where we quantify how the equilibria of such games move in response to changes in parameters of the game (e.g., the introduction of incentives to players).Comment: 56 pages, 5 figure

Comparative Statics by Adaptative Dynamics and the Correspondence Principle

This paper formalizes the relation between comparative statics and the out-of-equilibrium explanation for how a system evolves after a change in parameters. The paper has two main results. First, an increase in an exogenous parameter sets o learning dynamics that involve larger values of the endogenous variables. Second, equilibrium selections that are not monotone increasing in the exogenous variables must be predicting unstable equilibria. Moreover, under some conditions monotone comparative statics and stability are equivalent.

On continuous ordinal potential games

If the preferences of the players in a strategic game satisfy certain continuity conditions, then the acyclicity of individual improvements implies the existence of a Nash equilibrium. Moreover, starting from any strategy profile, an arbitrary neighborhood of the set of Nash equilibria can be reached after a finite number of individual improvements.potential game; compact-continuous game; finite improvement property

Learning in games with strategic complementarities revisited

Fictitious play is a classical learning process for games, and games with strategic complementarities are an important class including many economic applications. Knowledge about convergence properties of fictitious play in this class of games is scarce, however. Beyond games with a unique equilibrium, global convergence has only been claimed for games with diminishing returns [V. Krishna, Learning in games with strategic complementarities, HBS Working Paper 92-073, Harvard University, 1992]. This result remained unpublished, and it relies on a specific tie-breaking rule. Here we prove an extension of it by showing that the ordinal version of strategic complementarities suffices. The proof does not rely on tie-breaking rules and provides some intuition for the result

The Evolutionary Processes for the Populations of Firms and Workers

This paper analyzes the cultural evolution of firms and workers. Following an imitation rule, each firm and worker decides whether to be innovative (or not) and skilled (or unskilled). We apply evolutionary game theory to find the system of replicator dynamics, and characterize the low-level and high-level equilibria as Evolutionarily Stable Strategies (ESS) “against the field.” Hence, we study how a persistent state of underdevelopment can arise in strategic environments in which players are imitative rather than rational maximizers. We show that when the current state of the economy is in the basin of attraction of the poverty trap, players should play against the field if they want to change their status quo. The threshold level to overcome the poverty trap can be lowered if there is an appropriate policy using income taxes, education costs and skill premia. Hence, we study the replicator dynamics with a subsidy and payoff taxation to overcome the poverty trap.Imitative behavior, conformism, poverty traps, skill premium, strategic complementarities