A Mathematical Model of Status Hierarchy

The emergence of status hierarchy, defined as a social order that ranks individuals in society from top to bottom, is one of the main topics in sociology. Although many empirical studies on this topic have been conducted, the complete theoretical understanding of it remains lacking. In recent years, Gould proposed a groundbreaking theory of status hierarchy. Using game theory and social network theory, he showed that status hierarchy can be considered as a type of equilibrium when players assign attachments to all other players on the basis of their qualities. The major difficulty in Gould’s model, however, is the unwarranted assumption of limitless resources such as time and emotional costs that players must pay in executing their attachment strategies. Here we extend Gould’s model to be theoretically more coherent and empirically more valid by incorporating multidimensional choices for resource constraints. Unlike Gould’s model, in our model, a player choose only one allocation strategy as a multidimensional choice; that is, a player must determine the attachment levels for all the other players at the same time. Our main result is to show what is called “the relationality of social status”. Specifically, we show that an individual’s status is entirely determined by the individual’s relative quality in the social system. This observation contrasts with that of Gould’s model in which an individual’s status is dyadically determined. Our model can be considered as a type of network formation model having broad applicability.