On the duration and intensity of cumulative advantage competitions

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.