  Several tiers of social organization with varying economic and social
disparities have been observed. However, a quantitative characterization of the
types and the causal mechanisms for the transitions have hardly been explained.
While anthropologists have emphasized that gift exchange, rather than market
exchange, prevails in traditional societies and shapes social relations, few
mathematical studies have explored its consequences for social organizations.
In this study, we present a simple model of competitive gift-giving that
describes how gifts bring goods to the recipient and honor to the donor, and
simulate social change. Numerical simulations and an analysis of the
corresponding mean-field theory demonstrate the transitions between the
following four phases with different distribution shapes of wealth and social
reputation: the band, without economic or social disparities; the tribe, with
economic but without social disparities; the chiefdom, with both; and the
kingdom, with economic disparity and weak social disparity except for an
outlier, namely, the ``monarch''. The emergence of strong disparities is
characterized by power law distributions and is attributed to the ``rich get
richer'' process. In contrast, the absence of such a process leads to
exponential distributions due to random fluctuations. The phases depend on the
parameters characterizing the frequency and scale of gift interactions. Our
findings provide quantitative criteria for classifying social organizations
based on economic and social disparities, consistent with anthropological
theory and empirical observations. Thus, we propose empirically measurable
explanatory variables and characteristics for the evolution of social
organizations. The constructive model, guided by social scientific theory, can
provide the basic mechanistic explanation of social evolution and integrate
theories of the social sciences.
