Andreas Hefti

On the relationship between uniqueness and stability in sum-aggregative, symmetric and general differentiable games

This article explores the relationship between uniqueness and stability in differentiable regular games, with a major focus on the important classes of sum-aggregative, two-player and symmetric games. We consider three types of popular dynamics, continuous-time gradient dynamics as well as continuous- and discrete-time best-reply dynamics, and include aggregate-taking behavior as a non-strategic behavioral variant. We show that while in general games stability conditions are only sufficient for uniqueness, they are likely to be necessary as well in models with sum-aggregative or symmetric payoff functions. In particular, a unique equilibrium always verifies the stability conditions of all dynamics if strategies are equilibrium complements, and this also holds for both continuous-time dynamics if strategies are equilibrium substitutes with bounded slopes. These findings extend to the case of aggregate-taking equilibria. We further analyze the stability relations between the various dynamics, and demonstrate that the restrictive nature of the discrete dynamics originates from simultaneity of adjustments. Asynchronous decisions or heterogeneous forward thinking may stabilize the adjustment process.

On uniqueness and stability of symmetric equilibria in differentiable symmetric games

Higher-dimensional symmetric games become of more and more importance for applied micro- and macroeconomic research. Standard approaches to uniqueness of equilibria have the drawback that they are restrictive or not easy to evaluate analytically. In this paper I provide some general but comparably simple tools to verify whether a symmetric game has a unique symmetric equilibrium or not. I distinguish between the possibility of multiple symmetric equilibria and asymmetric equilibria which may be economically interesting and is useful to gain further insights into the causes of asymmetric equilibria in symmetric games with higher-dimensional strategy spaces. Moreover, symmetric games may be used to derive some properties of the equilibrium set of certain asymmetric versions of the symmetric game. I further use my approach to discuss the relationship between stability and (in)existence of multiple symmetric equilibria. While there is an equivalence between stability, inexistence of multiple symmetric equilibria and the unimportance of strategic effects for the comparative statics, this relationship breaks down in higher dimensions. Stability under symmetric adjustments is a minimum requirement of a symmetric equilibrium for reasonable comparative statics of symmetric changes. Finally, I present an alternative condition for a symmetric equilibrium to be a local contraction which is more general than the conventional approach of diagonal dominance and yet simpler to evaluate than the eigenvalue condition of continuous adjustment processes.

Equilibria in symmetric games: theory and applications

This article presents a new approach to analyze the equilibrium set of symmetric, differentiable games by separating between multiple symmetric equilibria and asymmetric equilibria. This separation allows to investigate, for example, how various parameter constellations affect the scope for multiple symmetric or asymmetric equilibria, or how the equilibrium set depends on the nature of the strategies. The approach is particularly helpful in applications because (1) it allows to reduce the complexity of the uniqueness-problem to a two-player game, (2) boundary conditions are less critical compared to standard procedures, and (3) best-replies need not be everywhere differentiable. The usefulness of the separation approach is illustrated with several examples, including an application to asymmetric games and to a two-dimensional price-information game.

Targeted Information and Limited Attention

We examine the implications of limited consumer attention for the targeting decisions of competing firms. Limited attention alters the strategic role of information provision as firms may become incentivized to behave as mass advertisers, despite perfect targeting abilities. We analyze the consequences of limited attention for targeting, strategic pricing, market shares, attention competition between firms, and the value of marketing data to firms. Accounting for limited attention in an otherwise standard targeting framework can explain several recent key issues from the advertising industry, such as consumer-side information overload or the increased usage of ad blocking tools.

Mental capabilities, trading styles, and asset market bubbles: theory and experiment

We propose that heterogeneous asset trading behavior is the result of two distinct, non-convertible mental dimensions: analytical (“quantitative”) capability and mentalizing (“perspective-taking”) capability. We develop a framework of mental capabilities that yields testable predictions about individual trading behavior, revenue distribution and aggregate outcomes. The two-dimensional structure of mental capabilities predicts the existence of four mental types with distinguishable trading patterns and revenues. Individuals will trade most successfully if and only if they have both capabilities. On the other hand, subjects who can mentalize well but have poor analytical capability will suffer the largest losses. As a consequence, being able in just one dimension does not assure trading success. We test these implications in a laboratory environment, where we first independently elicit subjects’ capabilities in both dimensions and then conduct a standard asset market experiment. We find that individual trading gains and patterns are consistent with our theoretical predictions. Our results suggest that two mental dimensions are necessary to encompass the complex heterogeneous behaviors in asset markets; a one-dimensional measure of mental capability will lead to biased conclusions. The findings have potential implications for financial institutions, which can use the measures to select successful traders, or for policy-makers, helping them to prevent the formation of asset bubbles. Finally, our conceptual framework and the empirical screening method could be applied to explain heterogeneous behavior in other games.

Local Contraction-Stability and Uniqueness

This paper investigates the relationship between uniqueness of Nash equilibria and local stability with respect to the best-response dynamics in the cases of sum-aggregative and symmetric games. If strategies are equilibrium complements, local stability and uniqueness are the same formal properties of the game. With equilibrium substitutes, local stability is stronger than uniqueness. If players adjust sequentially rather than simultaneously, this tends towards making a symmetric equilibria of symmetric games more stable. Finally, the relationship between the stability of the Nash best-response dynamics is compared to the stability of the response-dynamics induced by aggregate-taking behavior.