  The role of skill (fitness) and luck (randomness) as driving forces on the
dynamics of resource accumulation in a myriad of systems have long puzzled
scientists. Fueled by undisputed inequalities that emerge from actual
competitions, there is a pressing need for better understanding the effects of
skill and luck in resource accumulation. When such competitions are driven by
externalities such as cumulative advantage (CA), the rich-get-richer effect,
little is known with respect to fundamental properties such as their duration
and intensity. In this work we provide a mathematical understanding of how CA
exacerbates the role of luck in detriment of skill in simple and well-studied
competition models. We show, for instance, that if two agents are competing for
resources that arrive sequentially at each time unit, an early stroke of luck
can place the less skilled in the lead for an extremely long period of time, a
phenomenon we call "struggle of the fittest". In the absence of CA, the more
skilled quickly prevails despite any early stroke of luck that the less skilled
may have. We prove that duration of a simple skill and luck competition model
exhibit power law tails when CA is present, regardless of skill difference,
which is in sharp contrast to exponential tails when CA is absent. Our findings
have important implications to competitions not only in complex social systems
but also in contexts that leverage such models.
