Abstract: Abstract Societies are considered in which a non-transitive dominance relation exists between every pair of members, such as the peck-right in a flock of hens. A one-dimensional measure of the structure of such a society,h, is defined, withh=0 for equality andh=1 for the hierarchy. It is assumed that each member of the society is characterized by an ability vector whose components depend on individual characteristics such as size, concentration of sex hormone, etc., but not on social factors such as social rank. The distribution of abilities among members of the society is assumed to be given by a distribution function which is the same for all members, and the probability that one member dominates another is given by a function of the ability vectors of the two. On these assumptions formulas for the expected (mean) value and variance ofh are determined in terms of the distribution and dominance probability functions. Some special cases are calculated, especially that for normany distributed abilities and dominance probability given by the normal probability integral. Several conclusions are derived. If all members are of equal ability, so that dominance probability is 1/2, then any sizable society is much more likely to be near the equality than the hierarchy; and, as the size of the society increases, the probability that it will be near the hierarchy becomes vanishingly small. If the dominance probability is a weighted sum of several independent components, which make up the ability vector, then the society is less likely to be close to the hierarchy as the number of these components increases. The hierarchy is the prevalent structure only if unreasonably small differences in ability are decisive for dominance. From this it appears that the social factors, or psychological factors such as the previous history of dominance, which are not included in the present treatment, may be of great importance in explaining the observed prevalence of structures very close to the hierarchy in flocks of domestic hens.