  Herbert Simon's classic rich-get-richer model is one of the simplest
empirically supported mechanisms capable of generating heavy-tail size
distributions for complex systems. Simon argued analytically that a population
of flavored elements growing by either adding a novel element or randomly
replicating an existing one would afford a distribution of group sizes with a
power-law tail. Here, we show that, in fact, Simon's model does not produce a
simple power law size distribution as the initial element has a dominant
first-mover advantage, and will be overrepresented by a factor proportional to
the inverse of the innovation probability. The first group's size discrepancy
cannot be explained away as a transient of the model, and may therefore be many
orders of magnitude greater than expected. We demonstrate how Simon's analysis
was correct but incomplete, and expand our alternate analysis to quantify the
variability of long term rankings for all groups. We find that the expected
time for a first replication is infinite, and show how an incipient group must
break the mechanism to improve their odds of success. We present an example of
citation counts for a specific field that demonstrates a first-mover advantage
consistent with our revised view of the rich-get-richer mechanism. Our findings
call for a reexamination of preceding work invoking Simon's model and provide
an expanded understanding going forward.
