Emergence of social hierarchies in a society of two competitive classes

In recent years, the statistical physics approach is giving us a unique position for understanding a large variety of socioeconomic systems and, in particular, cities. The need for better interpretable models for cities is critical: understanding the micro-motives behind human behaviour is a necessary step to explain their macroscopic social behaviour and to be useful in the decision-making process. One of the topics which is worth tackling is the observation, study and understanding of the formation of different kinds of hierarchies which could appear inside cities. The hierarchy of social organization is an omnipresent property of animal and human aggregations, related to many features of the system such as collective decisions, intrinsic properties of the individuals, spatial characteristics, and so on. We here follow the footsteps of the Bonabeau model introduced by E Bonabeau et al. in 1995 and add a second class of agents into the society. The variation of an agent fitness can just change by competition, while the total fitness in the society always remains constant. The main particularity of the model is that only a pairwise interaction between agents of opposite classes are allowed. Furthermore, to proceed with a simple interaction the fitness of each involved agent is standardized under the minimum value of its class, that is to say, it could be understood as the prestige/reputation which an agent has in its own community. Finally, when an interaction occurs, the exchange of fitness is fixed with a given proportion x of the opponent. To solve the model numerically, a stochastic random walk under a square lattice LxL for the movement of the agents, a residence time algorithm to reproduce the time that a jump happens and a simple Monte Carlo method are applied. The main result is that for a broad range of values of the probability to win a fight η, the fitness of the agents of each class show a clear decays in time except for one or very few agents which capture almost all the fitness of the system. Consequently, the results show a behavioural change in several observables of the system. The studied observables are: the maximum agent fitness of each class normalized by the sum of all agent fitness of the same class, the maximum agent fitness now normalized by the total fitness in the society, and finally the total fitness of each class under the total fitness in the society. This behavioural change could be understood as a phase transition from egalitarian to hierarchical society for each class as a function of the control parameter η, which plays the role of the temperature of the system. We prove that the results are invariant in the system size, and they just depend on the number of agents of each class. As a matter of fact, for a particular value of η, the results do not depend on the number of agents in the society. In addition, a universal scaling function is a good candidate for the data collapsing on the same curve independent of the agents’ quantity. Further variations of the model, such as other winners’ probabilities, different kind of interactions between agents and even additional more classes could also be implemented and studied in detail