The symmetric phase of the Minority Game

GABRIEL ACOSTA; INÉS CARIDI; SEBASTIÁN GUALA; JAVIER MARENCO

Ciudad Autónoma de Buenos Aires

Workshop; Latin American School and Workshop on Data Analysis and Mathematical Modeling of Social Science; 2016

Centro latinoamericno de formación interdisciplinaria (CELFI)

The Minority Game (MG) was introduced in 1997 by Challet and Zhang in an attempt to catch essential characteristics of a population competing for limited resources. As in the case of a traffic problem in which people have to decide between two routes, in the MG an individual achieves the best result when she manages to be in the minority group. In the model, there are N agents, that at each step of the game must choose one of two sides, 0 or 1. The only information available for the agents is the system state, which stored the best side choices for the last m steps and that is updated after each step of the game. The parameter m defines the information-processing capacity of the agents. Agents take decisions based on strategies. The Full Strategy Minority Game (FSMG) is an instance of the Minority Game (MG) which includes a single copy of every potential agent (a combination of strategies). In this work we present some results in which the FSMG helps to understand some aspects of the MG in the symmetric phase. We explicitly solve the FSMG thanks to certain symmetries of this game. Then, by considering the MG as a statistical sample of the FSMG, we computed approximated values of the key variable observed in the MG, which measures the waste of population resources, in accordance with computational results. Another property is the quasi-periodicity of the sequence of minority sides, which turn out to be  periodic in the case of the FSMG. Moreover, we characterize these sequences as the eulerian paths on a De Bruijn graphs connecting states of the system . On the other hand, although there are no explicit interaction among MG agents, it is known that they interact through the global magnitudes of the model and through their strategies. We have formalized the implicit interactions among MG agents as if they were links on a complex network. We have defined the link between two agents by quantifying the similarity between them, in terms of their strategies. We have analyzed the structure of the resulting network for different MG parameters, such as the number of agents (N) and the agent?s capacity to process information (m). In the region of crowd-effects of the model, the resulting networks structure is a small world network, whereas in the region where the behavior of the MG is the same as in a game of random decisions, MG networks become a random network of Erdos-Renyi. We have explicitly calculated the degree distribution of the FSMG network and, on the basis of this analytical result, we have estimated the degree distribution of the MG network, which is in accordance with computational results.