Development of structure in a society with a dominance relation when new members are added successively

A society with a dominance relation is considered to be built up by starting with a small society and adding new members in succession. As each member is added he engages in contests with each of the older members to determine the dominance relation between them. The probability that the older member dominates is considered to depend on the size of the society and linearly on the older members score. A recurrence relation for the hierarchy index is derived. The approach of the society to a hierarchical structure is considered for various special cases of this probability. Reasonable assumptions concerning this dominance probability are shown to lead to structures close to the hierarchy. If the new member dominates all the older ones below a certain rank, and is dominated by all those above this rank, then the hierarchy will persist if it is the initial structure, or the structure will tend to hierarchy as the size increases, if it is not the initial structure.