Essays on Segregation, Minorities, and Imitation

The thesis is composed by three chapters. The common theme of the three essays is the identification of long run equilibria of games played on regular networks where interactions are at local level. In the first and second chapter I contribute to distinct literatures applying technics and results from the literature on evolutionary game theory with local interactions. The study of economic segregation within towns, and the tendency of ethnic minorities to live in ethnic enclaves are at the center of the first and the second chapter respectively. In the third chapter a refinement of previous results in the theoretical literature on evolutionary game theory with local interactions is given. In the thesis I develop models with different specifications of revision opportunities, error models, and behavior of agents, which require different techniques for the selection of the long run equilibria. The spatial structure is common to the three chapters, in fact agents are deployed on lattices with periodic boundary conditions. In the first chapter the spatial structure is given by a two dimensional lattice, i.e a torus, in the third chapter a one dimensional lattice, i.e. a ring, is used, while in the second chapter I use both the specifications. The network is always fixed and exogenously given, and the concept of continuous neighborhood is used. Neighborhoods are said to be continuous when are based on the individual perception of agents. Each agent is in the middle of his own neighborhood, and then the neighborhoods of different agents may partially overlap but never coincide. This in contrast with the concept of bounded neighborhoods where agents belonging to the same neighborhood share all the neighbors. In this last case neighborhoods form a partition of the set of agents. Following the literature on continuous neighborhood the two main specifications are considered, in fact are used both the Moore and the Von Neumann neighborhoods in the two dimensional lattice. In the one dimensional lattice models different dimensions of the neighborhood are considered. In the first chapter agents move within the network exchanging position each other. In every period two agents are selected and switch position if both will be better off in the new neighborhood. Differently in the second and third chapter agents do not move within the network while instead they revise their strategy based on the strategies of neighbors. The timing of revision opportunities is another fundamental ingredient of the three models. In the first and the second chapter asynchronous revision opportunities are modeled, in fact in the first chapter only one couple per time has the possibility to switch position, while in the second chapter only one agent per turn revises the strategy. In the third chapter the revision opportunities are simultaneous, and all the agents revise their strategy in each period. In the three chapter, as usual in evolutionary game theory, agents have bounded rationality. In fact agents have a myopic behavior, they are unable to make any prevision about the future states, and then revise their strategy only considering the actual state. In the third chapter a further level of irrationality is given by the fact that agents, instead of best replying to the actual situation, as in the first and second chapter, imitate the action of the best performing neighbor. A stochastically stable state is a state that is observed with a positive probability in the long run in presence of a small perturbation. The perturbation is at the individual level, in fact each agent, in every moment, has a small but positive chance to make a decision differently from that prescribed by his behavioral rule. The perturbation in biology is used to model mutations, while in economics the noise is represented by mistakes and experimentations. Through the introduction of a small amount of noise is possible to define a perturbed transition matrix for which is possible to move from any state of the world to any other in a finite number of steps. In the first chapter the behavior of agents is described by a logit choice function, for which costly mistakes are less likely. Each agent has a positive probability to accept or not an exchange, depending on the variation of utility obtained with the exchange. The game can be described by a potential function defined on the set of strategies of agents, for which every change in the utility of agents is reflected in a variation of the potential function. Assuming the logit choice rule and in the presence of asynchronous revision opportunities, in a potential game the stochastically stable states coincide with states having maximum potential. In this setting is sufficient to study the potential function to select the set of long run equilibria. In the second and third chapter a uniform error model is implemented. Each error can occur with the same probability, independently from how costly it is. In the second chapter three versions of the model are developed. In the first version there is not a spatial structure and interactions are at the global level. The identification of stochastically stable states is obtained using the technique developed by Young (1993), based on results by Freidlin and Wentzell (1984). At the basis of this technique there is the construction of rooted trees, made by the least resistance paths from each absorbing set to each other, where an absorbing set is a minimal set from which the unperturbed process can not escape. Between the paths connecting an absorbing set to another one the least resistance path is the one that is more likely to be observed. The summation of the resistance of all the least resistance paths ending in an absorbing set, E, and starting from all the other absorbing sets is the stochastic potential of the absorbing set E. The absorbing sets with minimal stochastic potential are the stochastically stable sets. The application is relatively simple having only two absorbing sets connected by only one path. In the second version of the model agents are deployed on a two dimensional lattice and interact only with a set of neighbors and the result is obtained via simulations. There are many advantages in the use of simulations, but also some limitations. In fact when the probability of a transition out from an absorbing set is very low it is unlikely to observe it during a simulation, and may be difficult to asses which transition is more likely. When the error rate is higher transitions are more likely but the perturbation may affect too much the dynamics. In the third version agents are deployed on one dimensional lattice, as in the third chapter. The radius coradius technique proposed by Ellison (2000) is used to identify the stochastically stable sets. The basic idea is to compute the radius and the coradius of all the absorbing sets and compare them. The radius represents how is difficult to leave the basin of attraction of an absorbing set, while the difficulty to enter into the basin of attraction of an absorbing set is measured by the coradius. The stochastically stable set of the model is contained in the subset of absorbing sets having radius greater then coradius.