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.drift, Nash equilibrium, similarity relations, replicator dynam, learning

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.similarity relations, drift, Nash equilibrium, selection dynamics, learning

Doubts and equilibria

In real life strategic interactions decision-makers are likely to entertain doubts about the degree of optimality of their play. To capture this feature of real choice-making, we present here a model based on the doubts felt by an agent about how well is playing a game. The doubts are coupled with (and mutually reinforced by) imperfect discrimination capacity, which we model here by means of similarity relations. We assume that each agent builds procedural preferences defined on the space of expected payoffsstrategy frequencies attached to his current strategy. These preferences, together with an adaptive learning process lead to doubt-based selection dynamic systems. We introduce the concepts of Mixed Strategy Doubt Equilibria, Mixed Strategy Doubt-Full Equilibria and Mixed Strategy Doubtless Equilibria and show the theoretical and the empirical relevance of these concept

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.Financial support, from the Basque Government, grant number PI -1998-68 and from CICYT, grant number PB96-1469-CO5-04

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.Financial support from the Basque Government, grant number PI - 1998-68, CICYT, grant number PB96-1469-CO5-04 and The University of the Basque Country, UPV 00043.321-15836/2004 is gratefully acknowledged

A Game-Theoreteic Analysis of Minority Language Use in Multilingual Societies

This chapter studies multilingual democratic societies with highly developed economies. These societies are assumed to have two languages with official status: language A, spoken by every individual, and language B, spoken by the bilingual minority. We emphasize that language rights are important, but the survival of the minority language B depends mainly on the actual use bilinguals make of B. The purpose of the present chapter is to study some of the factors affecting the bilingual speakers language choice behaviour. Our view is that languages with their speech communities compete for speakers just as fi rms compete for market share. Thus, the con ict among the minority languages in these societies does not take the rough expressions such as those studied in Desmet et al. (2012). Here the con flict is more subtle. We model highly plausible language choice situations by means of choice procedures and non-cooperative games, each with different types of information. We then study the determinants of the bilinguals ' strategic behaviour with regard to language. We observe that the bilinguals' use of B is shaped, essentially, by linguistic conventions and social norms that are developed in situations of language contact.Financial support provided by the Basque Government and the Ministerio de Economía y Competitividad (ECO2012-31626) are gratefully acknowledged

The Language Game: A Game-Theoretic Approach to Language Contact

We study a society inside which two official languages, the majority language A and the minority language B, are in contact and compete for the same social functions. We propose a non-cooperative game to capture some features of this competitive situation. In the game, there are two types of players: the bilingual one who speaks both A and B and the monolingual one who speaks only A. The information about which type is each player is private. A real life situation captured by the game is that in many interactions bilingual players must decide under incomplete information about which language to use. One implication of this information structure is that while A satisfies the main properties of a public good, B does not. Another implication is that it may have dangerous consequences on the language diversity of the society. We show that in many equilibria bilingual players fail to coordinate in their preferred language and end up using the majority language A.

Minority Language and the Stability of Bilingual Equilibria

We investigate a society with two official languages: A, shared by all individuals and B, spoken by a bilingual mirority. Thus, it is only B that needs t increase its population share, and therefore, only the language dynamics that derive from the intearctions that occur inside the bilingual population are both empirically and theoretically relevant. To this end, a model is developed in which the bilingual agents must make strategic decisions about the language to be used in a conversation. Decisions are taken under imperfect information about the linguistic type of the participants in the interaction. We first study all the posible equilibria the model might produce and the language used in each of them. Then, in a dynamic setting, we study the building of a language convention by the bilingual speakers. The main result is that there is a mixed strategy Nash equilibrium in which bilingual agents use both the A and B languages. This equilibrium is evolutionary stable, and dynamically, it is asymptotically stable for the one-population replicator dynamics. In this equilibrium, the use of B between bilingual individuals could be very low.imperfect information, majority/minority language, language competition

Doubts and Equilibria

In real life strategic interactions, decision-makers are likely to entertain doubts about the degree of optimality of their play. To capture this feature of real choice-making, we present here a model based on the doubts felt by an agent about how well is playing a game. The doubts are coupled with (and mutually reinforced by) imperfect discrimination capacity, which we model here by means of similarity relations. We assume that each agent builds procedural preferences de ned on the space of expected payoffs-strategy frequencies attached to his current strategy. These preferences, together with an adaptive learning process lead to doubt-based selection dynamic systems. We introduce the concepts of Mixed Strategy Doubt Equilibria, Mixed Strategy Doubt-Full Equilibria and Mixed Strategy Doubtless Equilibria and show the theoretical and the empirical relevance of these concepts.

Doubts and Equilibria

In real life strategic interactions, decision-makers are likely to entertain doubts about the degree of optimality of their play. To capture this feature of real choice-making, we present here a model based on the doubts felt by an agent about how well is playing a game. The doubts are coupled with (and mutually reinforced by) imperfect discrimination capacity, which we model here by means of similarity relations. We assume that each agent builds procedural preferences de ned on the space of expected payoffs-strategy frequencies attached to his current strategy. These preferences, together with an adaptive learning process lead to doubt-based selection dynamic systems. We introduce the concepts of Mixed Strategy Doubt Equilibria, Mixed Strategy Doubt-Full Equilibria and Mixed Strategy Doubtless Equilibria and show the theoretical and the empirical relevance of these concepts.Cabrales gratefully acknowledges the financial support of the Spanish Ministry of Science and Technology under grants CONSOLIDER 2010 (CSD 2006-0016) and SEJ2006-11665-C02-00. Uriarte gratefully acknowledges the financial support from the Spanish Ministry of Science and Technology and FEDER, grant number SEJ 2006-05455, and the University of the Basque Country, UPV 00043.321-15836/2004