The formation and stability of social hierarchies is a question of general
relevance. Here, we propose a simple generalized theoretical model for
establishing social hierarchy via pair-wise interactions between individuals
and investigate its stability. In each interaction or fight, the probability of
"winning" depends solely on the relative societal status of the participants,
and the winner has a gain of status whereas there is an equal loss to the
loser. The interactions are characterized by two parameters. The first
parameter represents how much can be lost, and the second parameter represents
the degree to which even a small difference of status can guarantee a win for
the higher-status individual. Depending on the parameters, the resulting status
distributions reach either a continuous unimodal form or lead to a totalitarian
end state with one high-status individual and all other individuals having
status approaching zero. However, we find that in the latter case long-lived
intermediary distributions often exist, which can give the illusion of a stable
society. As we show, our model allows us to make predictions consistent with
animal interaction data and their evolution over a number of years. Moreover,
by implementing a simple, but realistic rule that restricts interactions to
sufficiently similar-status individuals, the stable or long-lived distributions
acquire high-status structure corresponding to a distinct high-status class.
Using household income as a proxy for societal status in human societies, we
find agreement over their entire range from the low-to-middle-status parts to
the characteristic high-status "tail". We discuss how the model provides a
conceptual framework for understanding the origin of social hierarchy and the
factors which lead to the preservation or deterioration of the societal
structure.Comment: Added sections 4.1 and S2.A about agonistic interactions in animals,
added sections 4.2.1 and S2.B regarding potential proxies for societal status
in non-human animals, added references to sections 1 and 2. Main text: 34
pages, 11 figures. Supplementary appendices: 36 pages, 24 figure