Selection of a correlated equilibrium in Markov stopping games

This paper deals with an extension of the concept of correlated strategies to Markov stopping games. The Nash equilibrium approach to solving nonzero-sum stopping games may give multiple solutions. An arbitrator can suggest to each player the decision to be applied at each stage based on a joint distribution over the players’ decisions according to some optimality criterion. This is a form of equilibrium selection. Examples of correlated equilibria in nonzero-sum games related to the best choice problem are given. Several concepts of criteria for selecting a correlated equilibrium are used.     

Evolutionary dynamics of the spatial Prisoner’s Dilemma with self-inhibition

In this paper we study the influence of interventions on self-interactions in a spatial Prisoner’s Dilemma on a two-dimensional grid with periodic boundary conditions and synchronous updating of the dynamics. We investigate two different types of self-interaction modifications. The first type (FSIP) is deterministic, effecting each self-interaction of a player by a constant factor, whereas the second type (PSIP) performs a probabilistic interventions. Both types of interventions lead to a reduction of the payoff of the players and, hence, represent inhibiting effects. We find that a constant but moderate reduction of self-interactions has a very beneficial effect on the evolution of cooperators in the population, whereas probabilistic interventions on self-interactions are in general counter productive for the coexistence of the two different strategies.     

Evolutionary modelling in environmental economics

In this paper, we analyze some evolutionary models for the exploitation of renewable natural resources recently proposed in the specialized literature. The common feature shared by all these models is to consider the evolution of the resource, which is exploited by a population of heterogeneous interacting agents. We review several sources of agents’ heterogeneity: different attitudes towards exploitation (dynamics of cooperative vs. non-cooperative behaviours); harvesting in different patches where different rules are imposed (spatial heterogeneity); harvesting with different technologies (standard and ‘environmentally-friendly’ technologies); heterogeneity with respect to the targeted species.     

Lagrangean variables in infinite dimensional spaces for a dynamic economic equilibrium problem

This paper is focused on the study of a dynamic competitive equilibrium by using Lagrangean multipliers. This mathematical formulation allows us the improve the Walrasian model by considering the common possibility of an uncharged delayed payment in a given time (for example, by using a credit card). Firstly the economic equilibrium problem is reformulated as an evolutionary variational problem; then the Lagrangean theory in infinite dimensional spaces is applied. Thanks to the application of this theory we obtain the existence of Lagrangean multipliers, which allows us to give a computational procedure for the equilibrium solutions.     

Fast reconstruction of aerodynamic shapes using evolutionary algorithms and virtual nash strategies in a CFD design environment

This paper compares the performances of two different optimisation techniques for solving inverse problems; the first one deals with the Hierarchical Asynchronous Parallel Evolutionary Algorithms software (HAPEA) and the second is implemented with a game strategy named Nash-EA. The HAPEA software is based on a hierarchical topology and asynchronous parallel computation. The Nash-EA methodology is introduced as a distributed virtual game and consists of splitting the wing design variables-aerofoil sections-supervised by players optimising their own strategy. The HAPEA and Nash-EA software methodologies are applied to a single objective aerodynamic ONERA M6 wing reconstruction. Numerical results from the two approaches are compared in terms of the quality of model and computational expense and demonstrate the superiority of the distributed Nash-EA methodology in a parallel environment for a similar design quality.     

Dynamic equilibria of group vaccination strategies in a heterogeneous population

In this paper we present an evolutionary variational inequality model of vaccination strategies games in a population with a known vaccine coverage profile over a certain time interval. The population is considered to be heterogeneous, namely its individuals are divided into a finite number of distinct population groups, where each group has different perceptions of vaccine and disease risks. Previous game theoretical analyses of vaccinating behaviour have studied the strategic interaction between individuals attempting to maximize their health states, in situations where an individual’s health state depends upon the vaccination decisions of others due to the presence of herd immunity. Here we extend such analyses by applying the theory of evolutionary variational inequalities (EVI) to a (one parameter) family of generalized vaccination games. An EVI is used to provide conditions for existence of solutions (generalized Nash equilibria) for the family of vaccination games, while a projected dynamical system is used to compute approximate solutions of the EVI problem. In particular we study a population model with two groups, where the size of one group is strictly larger than the size of the other group (a majority/minority population). The smaller group is considered much less vaccination inclined than the larger group. Under these hypotheses, considering that the vaccine coverage of the entire population is measured during a vaccine scare period, we find that our model reproduces a feature of real populations: the vaccine averse minority will react immediately to a vaccine scare by dropping their strategy to a nonvaccinator one; the vaccine inclined majority does not follow a nonvaccinator strategy during the scare, although vaccination in this group decreases as well. Moreover we find that there is a delay in the majority’s reaction to the scare. This is the first time EVI problems are used in the context of mathematical epidemiology. The results presented emphasize the important role played by social heterogeneity in vaccination behaviour, while also highlighting the valuable role that can be played by EVI in this area of research.      

Complex dynamics in equilibrium asset pricing models with boundedly rational,heterogeneous agents

We study a simple model based upon the Lucas framework where heterogeneous agents behave rationally in a fully intertemporal setting but do not know other investors' personal preferences, wealth or investment portfolios. As a consequence, agents initially do not know the equilibrium asset pricing function and must make guesses, which they update via adaptive learning with constant gain. We demonstrate that even in this simple environment the economy can, depending on parameters, exhibit either stable convergence to equilibrium, or chaotic dynamical behavior of asset prices and trading volume without converging to the rational expectations equilibrium of the Lucas model. This contradicts the assertion that the Lucas model is stable in the face of modest deviations from the strong assumptions required to compute the equilibrium. © 2013 Wiley Periodicals, Inc. Complexity 19: 38–55, 2014     

Improvement dynamics in games with strategic complementarities

In a finite game with strategic complementarities, every strategy profile is connected to a Nash equilibrium with a finite individual improvement path. If, additionally, the strategies are scalar, then every strategy profile is connected to a Nash equilibrium with a finite individual best response improvement path.The first author acknowledges financial support from the Russian Foundation for Basic Research (grant 02-01-00854) and from a presidential grant for the state support of the leading scientific schools (NSh-1843.2003.01); he also thanks Universidad Carlos III de Madrid, Departamento de Economía, and personally Francisco Marhuenda for their hospitality. The other authors would like to thank Michihiro Kandori, Akihiko Matsui, Jim Friedman, Claudio Mezzetti, and Federico Echenique for helpful comments. All the three thank an anonymous referee and the Editor of this journal for bringing us together and suggesting the merger of two independent works.     

On equilibrium in pure strategies in games with many players

We demonstrate that, if there are sufficiently many players, any Bayesian equilibrium of an incomplete information game can be “ε-purified” . That is, close to any Bayesian equilibrium there is an approximate Bayesian equilibrium in pure strategies. Our main contribution is obtaining this result for games with a countable set of pure strategies. In order to do so we derive a mathematical result, in the spirit of the Shapley–Folkman Theorem, permitting countable strategy sets. Our main assumption is a “large game property,” dictating that the actions of relatively small subsets of players cannot have large affects on the payoffs of other players. E. Cartwright and M. Wooders are indebted to Phillip Reny, Frank Page and two anonymous referees for helpful comments.     

Optimal and equilibrium balking strategies in the single server Markovian queue with catastrophes

We consider a Markovian queue subject to Poisson generated catastrophes. Whenever a catastrophe occurs, all customers are forced to abandon the system, the server is rendered inoperative and an exponential repair time is set on. We assume that the arriving customers decide whether to join the system or balk, based on a natural reward-cost structure. We study the balking behavior of the customers and derive the corresponding Nash equilibrium and social optimal strategies.     

Existence of equilibrium stationary strategies in discounted noncooperative stochastic games with uncountable state space

This paper considers discounted noncooperative stochastic games with uncountable state space and compact metric action spaces. We assume that the transition law is absolutely continuous with respect to some probability measure defined on the state space. We prove, under certain additional continuity and integrability conditions, that such games have -equilibrium stationary strategies for each >0. To prove this fact, we provide a method for approximating the original game by a sequence of finite or countable state games. The main result of this paper answers partially a question raised by Parthasarathy in Ref. 1.     

Scaling limit of fluctuations for the equilibrium Glauber dynamics in continuum

The Glauber dynamics investigated in this paper are spatial birth and death processes in a continuous system having a grand canonical Gibbs measure of Ruelle type as an invariant measure. We prove that such processes, when appropriately scaled, have as scaling limit a generalized Ornstein-Uhlenbeck process. First we prove convergence of the corresponding Dirichlet forms. This convergence requires only very weak assumptions. The interaction potential ? only has to be stable (S), integrable (I), and we have to assume the low activity high temperature regime. Under a slightly stronger integrability condition (I_∞) and a conjecture on the Percus-Yevick equation we even can prove strong convergence of the corresponding generators. Finally, we prove that the scaled processes converge in law. Here the hardest part is to show tightness of the scaled processes (note that the processes only have càdlàg sample path). For the proof we have to assume that the interaction potential is positive (P). The limiting process then is identified via the associated martingale problem. For this the above mentioned strong convergence of generators is essential.     

Existence of stationary equilibrium strategies in non-zero sum discounted stochastic games with uncountable state space and state-independent transitions

Non-zero sum discounted stochastic games with uncountable state space and state in-dependent transitions have stationary equilibrium strategies.     

Evolutionary dynamics in a Lotka–Volterra competition model with impulsive periodic disturbance

In this paper, we develop a theoretical framework to investigate the influence of impulsive periodic disturbance on the evolutionary dynamics of a continuous trait, such as body size, in a general Lotka–Volterra‐type competition model. The model is formulated as a system of impulsive differential equations. First, we derive analytically the fitness function of a mutant invading the resident populations when rare in both monomorphic and dimorphic populations. Second, we apply the fitness function to a specific system of asymmetric competition under size‐selective harvesting and investigate the conditions for evolutionarily stable strategy and evolutionary branching by means of critical function analysis. Finally, we perform long‐term simulation of evolutionary dynamics to demonstrate the emergence of high‐level polymorphism. Our analytical results show that large harvesting effort or small impulsive harvesting period inhibits branching, while large impulsive harvesting period promotes branching. Copyright © 2015 John Wiley & Sons, Ltd.     

Reciprocity between the cerebellum and the cerebral cortex: Nonlinear dynamics in microscopic modules for generating voluntary motor commands

The cerebellum and basal ganglia are reciprocally connected with the cerebral cortex, forming many loops that function as distributed processing modules. Here we present a detailed model of one microscopic loop between the motor cortex and the cerebellum, and we show how small arrays of these microscopic loops (CB modules) can be used to generate biologically plausible motor commands for controlling movement. A fundamental feature of CB modules is the presence of positive feedback loops between the cerebellar nucleus and the motor cortex. We use nonlinear dynamics to model one microscopic loop and to investigate its bistable properties. Simulations demonstrate an ability to program a motor command well in advance of command generation and an ability to vary command duration. However, control of command intensity is minimal, which could interfere with the control of movement velocity. To assess these hypotheses, we use a minimal nonlinear model of the neuromuscular (NM) system that translates motor commands into actual movements. Simulations of the combined CB‐NM modular model indicate that movement duration is readily controlled, whereas velocity is poorly controlled. We then explore how an array of eight CB‐NM modules can be used to control the direction and endpoint of a planar movement. In actuality, thousands of such microscopic loops function together as an array of adjustable pattern generators for programming and regulating the composite motor commands that control limb movements. We discuss the biological plausibility and limitations of the model. We also discuss ways in which an agent‐based representation can take advantage of the modularity in order to model this complex system. © 2008 Wiley Periodicals, Inc. Complexity, 2008