An evolutionary model with best response and imitative rules

We formulate an evolutionary oligopoly model where quantity setting players produce following either the static expectation best response or a performance-proportional imitation rule. The choice on how to behave is driven by an evolutionary selection mechanism according to which the rule that brought the highest performance attracts more followers. The model has a stationary state that represents a heterogeneous population where rational and imitative rules coexist and where players produce at the Cournot–Nash level. We find that the intensity of choice, a parameter representing the evolutionary propensity to switch to the most profitable rule, the cost of the best response implementation as well as the number of players have ambiguous roles in determining the stability property of the Cournot–Nash equilibrium. This marks important differences with most of the results from evolutionary models and oligopoly competitions. Such differences should be referred to the particular imitative behavior we consider in the present modeling setup. Moreover, the global analysis of the model reveals that the above-mentioned parameters introduce further elements of complexity, conditioning the convergence toward an inner attractor. In particular, even when the Cournot–Nash equilibrium loses its stability, outputs of players little differ from the Cournot–Nash level and most of the dynamics is due to wide variations of imitators’ relative fraction. This describes dynamic scenarios where shares of players produce more or less at the same level alternating their decision mechanisms.     

Adaptive learning in an asymmetric auction: genetic algorithm approach

Agent-based simulations are performed to study adaptive learning in the context of asymmetric first-price auctions. Non-linearity of the Nash equilibrium strategies is used to investigate the effect of task complexity on adaptive learning by varying the degree of approximation the agents can handle. In addition, learning in different information environments is explored. Social learning allows agents to imitate each other’s bidding strategies based on their relative success. Under individual learning agents are limited to their own experience. We observe convergence to steady states near the predicted equilibrium in all cases. The ability to learn non-linear functions helps the agents with a non-linear equilibrium strategy but hurts the agents with an almost linear one. Better information about the opponent population has a relatively modest impact. A larger number of strategies to experiment with and an ability to systematically compare strategies by holding a number of factors constant have a comparatively stronger beneficial effect.     

Imitation and price competition in a differentiated market

We study the stable market outcome that evolves in a spatially differentiated market when price-competing firms choose actions by imitation of the most profitable firm. We compare and contrast the stable outcomes under two imitation procedures: one, where each firm immediately imitates the most profitable firm, and the other when a firm imitates another firm only if it is more profitable while being “sufficiently similar” (in context of the market segment it operates in) or “sufficiently close”. In either case, the symmetric pure strategy Nash equilibrium is always a stable outcome. However, when imitation of the most profitable firm is immediate and market differentiation is ‘moderate’, states with prices lower than the Nash equilibrium are also stable. In contrast, when imitation of the most profitable firm is more gradual and market differentiation is below a threshold, states with prices above the Nash equilibrium are also stable. Thus, while competitive evolutionary pressure in this imitation based model does result in the Nash equilibrium always being stable, other outcomes may be stable as well. Interestingly, the states that are stable under gradual imitation give the firms a higher profit than the stable states under immediate imitation.     

Public support to innovation and imitation in a non-scale growth model

This paper studies technology policy within a version of Jones's [1995. R&D-based models of economic growth. Journal of Political Economy 103, 759–784] non-scale R&D-based growth framework that incorporates imitation of foreign techniques. In the model, imitation is the most important source of productivity growth at the beginning of the convergence process, whereas innovation dominates later on. In addition, the transitional dynamics of the model can account for well-known empirical regularities regarding the relationship between the level of economic development and public support to technology innovation and imitation. The paper shows as well that, even though policy in Jones-type non-scale models has no long-run growth effects, level effects can be substantial.     

企业创新战略的博弈选择

文章通过对古诺模型与斯塔克博格模型的纳什均衡解的比较，得出了大企业领导创新可赢得比原来更加多的市场份额,从博弈论的角度证明了大企业应该主动创新的结论。同时,在第二部分通过对智猪模型改进后的博弈矩阵的分析，明确了一般情况下小企业应选择模仿创新为自己的创新战略。     

On the dimensionality of bounds generated by the Shapley–Folkman theorem

The Shapley–Folkman theorem places a scalar upper bound on the distance between a sum of non-convex sets and its convex hull. We observe that some information is lost when a vector is converted to a scalar to generate this bound and propose a simple normalization of the underlying space which mitigates this loss of information. As an example, we apply this result to the Anderson (1978) core convergence theorem, and demonstrate how our normalization leads to an intuitive, unitless upper bound on the discrepancy between an arbitrary core allocation and the corresponding competitive equilibrium allocation.     

ON THE ECONOMIC INTERPRETATION OF THE BRÓDY CONJECTURE

The aim of the paper is to study the economic aspects of the Bródy conjecture: an increase in the size of a (random) input matrix causes a decline in the ratio of its subdominant and dominant eigenvalues and implies faster convergence to equilibrium [Bródy, A. (1997) The Second Eigenvalue of the Leontief Matrix. Economic Systems Research, 9, 253–258]. Simulation results provide evidence that this ratio depends inversely on the level of data aggregation and can therefore not be a good indicator of the speed of convergence of an economy to its equilibrium path. We show that this is consistent with findings based on actual input–output tables of EU member states. These results imply that theorems about the speed of convergence of random matrices are not useful in describing the cyclical dynamics of real economies.     

Biased Bayesian learning with an application to the risk-free rate puzzle

Based on the axiomatic framework of Choquet decision theory, we develop a closed-form model of Bayesian learning with ambiguous beliefs about the mean of a normal distribution. In contrast to rational models of Bayesian learning the resulting Choquet Bayesian estimator results in a long-run bias that reflects the agent's ambiguity attitudes. By calibrating the standard equilibrium conditions of the consumption based asset pricing model we illustrate that our approach contributes towards a resolution of the risk-free rate puzzle. For a plausible parameterization we obtain a risk-free rate in the range of 3.5–5%. This is 1–2.5% closer to the empirical risk-free rate than according calibrations of the rational expectations model.     

Coefficients of ergodicity for Markov chains with uncertain parameters

One of the central considerations in the theory of Markov chains is their convergence to an equilibrium. Coefficients of ergodicity provide an efficient method for such an analysis. Besides giving sufficient and sometimes necessary conditions for convergence, they additionally measure its rate. In this paper we explore coefficients of ergodicity for the case of imprecise Markov chains. The latter provide a convenient way of modelling dynamical systems where parameters are not determined precisely. In such cases a tool for measuring the rate of convergence is even more important than in the case of precisely determined Markov chains, since most of the existing methods of estimating the limit distributions are iterative. We define a new coefficient of ergodicity that provides necessary and sufficient conditions for convergence of the most commonly used class of imprecise Markov chains. This so-called weak coefficient of ergodicity is defined through an endowment of the structure of a metric space to the class of imprecise probabilities. Therefore we first make a detailed analysis of the metric properties of imprecise probabilities.     

On the uniqueness of local equilibria

The purpose of this paper is to define a new notion of local equilibrium in an exchange economy, where the consumers face lower bounds on net trades. Then, we show that the local equilibrium is unique if the lower bounds are closed enough to 0. By the way, we also provide a convergence result of local equilibrium price toward Walras equilibrium price of a suitable tangent linear economy.     

Piecewise linear exchange equilibrium

It is known that if a pure exchange model in which all consumers have linear utility functions has an equilibrium it has one which is rational in the initial data. Examples show on the other hand that this is no longer true in general if utility functions are merely piecewise linear. In this note we observe that for the case of two traders the rationality result carries over and we sketch a finite procedure for calculating the equilibrium.     

Universally converging adjustment processes—a unifying approach

Both in game theory and in general equilibrium theory, there exists a number of universally converging adjustment processes. In game theory, these processes typically serve the role of selecting a Nash equilibrium. Examples are, the tracing procedure of Harsanyi and Selten or the equilibrium selection procedure proposed by McKelvey and Palfrey. In general equilibrium, the processes are adjustment rules by which an auctioneer can clear all markets. Examples are the processes studied by Smale, Kamiya, van der Laan and Talman, and Herings. The underlying reasons for convergence have remained rather mysterious in the literature, and convergence of different processes has seemed unrelated. This paper shows that convergence of all these processes relies on Browder’s fixed point theorem.     

Cognitive capacity and cognitive hierarchy: a study based on beauty contest experiments

Recent developments in behavioral experiments, in particular game experiments, have placed human cognition in a pivotal place. Two related ideas are proposed and are popularly used in the literature, namely, cognitive hierarchy and cognitive capacity. While these two often meet in the same set of experiments and observations, few studies have formally addressed their relationship. In this study, based on six series of 15- to 20-person beauty contest experiments and the associated working memory tests, we examine the effect of cognitive capacity on the observed cognitive hierarchy. It is found that cognitive capacity has a positive effect on the observed cognitive hierarchy. This effect is strong in the initial rounds, and may become weaker, but without disappearing, in subsequent rounds, which suggests the possibility that cognitive capacity may further impact learning. We examine this possibility using the Markov transition dynamics of cognitive hierarchy. There is evidence to show that subjects with different cognitive capacities may learn differently, which may cause strong convergence to be difficult to observe.     

The random-time binomial model

In this paper we study a binomial model with random time steps and explain how to calculate values for European and American call and put options. We prove both weak convergence of the discrete processes to the Black–Scholes setup and convergence of the values for European and American put options. Computational experiments exhibit a smooth convergence structure and suggest that we can obtain a quadratic order of convergence via an extrapolation procedure. Approximations to jump-diffusions are straightforward.     

Good luck or good policy? An expectational theory of macro volatility switches

In an otherwise unique-equilibrium model, agents are segmented into a few informational islands according to the signal they receive about others' expectations. Even if agents perfectly observe fundamentals, rational-exuberance equilibria (REX) can arise as they put weight on expectational signals to refine their forecasts. Constant-gain adaptive learning can trigger jumps between the equilibrium where only fundamentals are weighted and a REX. This determines regime switching in macro volatility despite unchanged monetary policy and time-invariant distribution of exogenous shocks. In this context, a tight inflation-targeting policy can lower expectational complementarity preventing rational exuberance, although its effect is non-monotone.     

Further remarks on the Global Newton method

In this work, we are concerned with the behavior of Smale's ‘Global Newton’ process, starting from points not necessarily on the boundary. We find that the process is locally system stable in the sense that, sufficiently near an equilibrium, one is almost always led to some equilibrium of the system. However, the ‘Global Newton’ process is found typically not to be globally stable; instead, families of cycles appear, preventing convergence to equilibrium.     

A limit theorem for systems of social interactions

In this paper, we establish a convergence result for equilibria in systems of social interactions with many locally and globally interacting players. Assuming spacial homogeneity and that interactions between different agents are not too strong, we show that equilibria of systems with finitely many players converge to the unique equilibrium of a benchmark system with infinitely many agents. We prove convergence of individual actions and of average behavior. Our results also apply to a class of interaction games.     

A stochastic dynamic model of trade and growth: Convergence and diversification

There is a growing literature that studies the properties of models that combine international trade and neoclassical growth theory, but mostly in a deterministic setting. In this paper we introduce uncertainty in a dynamic Heckscher-Ohlin model and characterize the equilibrium of a small open economy in such an environment. We show that, when trade is balanced period-by-period, the per capita output and consumption of a small open economy converge to an invariant distribution that is independent of the initial wealth. Further, at the invariant distribution, there are periods in which the small economy diversifies. Numerical simulations show that the speed of convergence increases with the size of the shocks. In the limit, when there is no uncertainty, there is no convergence and countries may specialize permanently. The paper highlights the role of market incompleteness, as a result of the period-by-period trade balance, in this setup. Through an analytical example we also illustrate the importance of country specific risk in delivering our results.     

Italian economic dualism and convergence clubs at regional level

This paper gives a contribution to the debate on regional convergence by comparing the long run prediction of convergence clubs introduced by Quah (1996 and 1997) with the actual observed dynamics of the Italian regions during the period 1970–2004. To this end we analyze the evolution of per capita income levels for the Italian regions using a non-traditional (non-parametric) statistical model. In addition we segment the set of economies by the mean of hierarchical clustering methodologies and compare the trajectories of the regions by introducing different notions of distance. The general conclusion is that the average distance identifies a clear division between a high performance club consisting of regions from the Center North, and a low performance club composed by regions from the South and islands. The presence of a cluster composed by Center North regions is substantially confirmed by the distance correlation analysis, suggesting an homogeneous response to external shocks. By contrast Southern regions display the same dynamical evolution but difference in co-movements. Our analysis provides hints about the fundamentals that link the regions in their process of divergence. In fact the performance clubs pattern we discovered reflects the distribution of economic activities as well as the structural attributes of the regional economies.