A Behavioral Foundation for Models of Evolutionary Drift

Binmore and Samuelson (1999) have shown that perturbations (drift) are crucial to study the stability properties of Nash equilibria. We contribute to this literature by providing a behavioural foundation for models of evolutionary drift. In particular, this article introduces a microeconomic model of drift based on the similarity theory developed by Tversky (1977), Kahneman and Tversky (1979) and Rubinstein (1988),(1998). An innovation with respect to those works is that we deal with similarity relations that are derived from the perception that each agent has about how well he is playing the game. In addition, the similarity relations are adapted to a dynamic setting. We obtain different models of drift depending on how we model the agent´s assessment of his behaviour in the game. The examples of the ultimatum game and the chain-store game are used to show the conditions for each model to stabilize elements in the component of Nash equilibria that are not subgame- perfect. It is also shown how some models approximate the laboratory data about those games while others match the data.

The Supercore for Normal Form Games

We study the supercore of a system derived from a normal form game. For the case of a finite game with pure strategies, we define a sequence of games and show that the supercore of that system coincides with the set of Nash equilibrium strategy profiles of the last game in the sequence. This result is illustrated with the characterization of the supercore for the n-person prisoners’ dilemma. With regard to the mixed extension of a normal form game, we show that the set of Nash equilibrium profiles coincides with the supercore for games with a finite number of Nash equilibria. For games with an infinite number of Nash equilibria this need not be no longer the case. Yet, it is not difficult to find a binary relation which guarantees the coincidence of these two sets.

A Model of Evolutionay Drift

Drift appears to be crucial to study the stability properties of Nash equilibria in a component specifying different out-of-equilibrium behaviour. We propose a new microeconomic model of drift to be added to the learning process by which agents find their way to equilibrium. A key feature of the model is the sensitivity of the noisy agent to the proportion of agents in his player population playing the same strategy as his current one. We show that, 1. Perturbed Payoff-Positive and PayoffMonotone selection dynamics are capable of stabilizing pure non strict Nash equilibria in either singleton or nonsingleton component of equilibria; 2. The model is relevant to understand the role of drift in the behaviour observed in the laboratory for the Ultimatum Game and for predicting outcomes that can be experimentally tested. Hence, the selection dynamics model perturbed with the proposed drift may be seen as well as a new learning tool to understand observed behaviour.

The von Neumann-Morgenstern stable sets for 2x2 games

We analyze the von Neumann and Morgenstern stable sets for the mixed extension of 2 2 games when only single profitable deviations are allowed. We show that the games without a strict Nash equilibrium have a unique vN&M stable set and otherwise they have infinite sets.

Stability of Switched Feedback Time-Varying Dynamic Systems Based on the Properties of the Gap Metric for Operators

The stabilization of dynamic switched control systems is focused on and based on an operator-based formulation. It is assumed that the controlled object and the controller are described by sequences of closed operator pairs (L, C) on a Hilbert space H of the input and output spaces and it is related to the existence of the inverse of the resulting input-output operator being admissible and bounded. The technical mechanism addressed to get the results is the appropriate use of the fact that closed operators being sufficiently close to bounded operators, in terms of the gap metric, are also bounded. That philosophy is followed for the operators describing the input-output relations in switched feedback control systems so as to guarantee the closed-loop stabilization.

Effective and Robust Generalized Predictive Speed Control of Induction Motor

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The Economics of "Why is it so hard to save a threatened Language?"

We study the language choice behavior of bilingual speakers in modern societies, such as the Basque Country, Ireland andWales. These countries have two o cial languages:A, spoken by all, and B, spoken by a minority. We think of the bilinguals in those societies as a population playing repeatedly a Bayesian game in which, they must choose strategically the language, A or B, that might be used in the interaction. The choice has to be made under imperfect information about the linguistic type of the interlocutors. We take the Nash equilibrium of the language use game as a model for real life language choice behavior. It is shown that the predictions made with this model t very well the data about the actual use, contained in the censuses, of Basque, Irish and Welsh languages. Then the question posed by Fishman (2001),which appears in the title, is answered as follows: it is hard, mainly, because bilingual speakers have reached an equilibrium which is evolutionary stable. This means that to solve fast and in a re ex manner their frequent language coordination problem, bilinguals have developed linguistic conventions based chie y on the strategy 'Use the same language as your interlocutor', which weakens the actual use of B.1

Gain Scheduling Control of an Islanded Microgrid Voltage

The aim of this research study has been to design a gain scheduling (GS) digital controller in order to control the voltage of an islanded microgrid in the presence of fast varying loads (FVLs), and to compare it to a robust controller. The inverter which feeds the microgrid is connected to it through an inductance-capacitor-inductance (LCL) filter. The oscillatory and nonlinear behaviour of the plant is analyzed in the whole operating zone. Afterwards, the design of the controllers which contain two loops in cascade are described. The first loop concerns the current control, while the second is linked to the voltage regulation. Two controllers, one defined as Robust and another one as GS controller, are designed for the two loops, emphasizing in their robustness and their ability to damp the oscillatory plant behaviour. To finish, some simulations are carried out to study and compare the two kinds of controllers in different operating points. The results show that both controllers damp the oscillatory behaviour of the plant in closed loop (CL), and that the GS controller ensures a better rejection of current disturbances from FVLs.

Robust Sliding Control of SEIR Epidemic Models

This paper is aimed at designing a robust vaccination strategy capable of eradicating an infectious disease from a population regardless of the potential uncertainty in the parameters defining the disease. For this purpose, a control theoretic approach based on a sliding-mode control law is used. Initially, the controller is designed assuming certain knowledge of an upper-bound of the uncertainty signal. Afterwards, this condition is removed while an adaptive sliding control system is designed. The closed-loop properties are proved mathematically in the nonadaptive and adaptive cases. Furthermore, the usual sign function appearing in the sliding-mode control is substituted by the saturation function in order to prevent chattering. In addition, the properties achieved by the closed-loop system under this variation are also stated and proved analytically. The closed-loop system is able to attain the control objective regardless of the parametric uncertainties of the model and the lack of a priori knowledge on the system.