  Cumulative advantage (CA) refers to the notion that accumulated resources
foster the accumulation of further resources in competitions, a phenomenon that
has been empirically observed in various contexts. The oldest and arguably
simplest mathematical model that embodies this general principle is the P\'olya
urn process, which finds applications in a myriad of problems. The original
model captures the dynamics of competitions between two equally fit agents
under linear CA effects, which can be readily generalized to incorporate
different fitnesses and nonlinear CA effects. We study two statistics of
competitions under the generalized model, namely duration (i.e., time of the
last tie) and intensity (i.e., number of ties). We give rigorous mathematical
characterizations of the tail distributions of both duration and intensity
under the various regimes for fitness and nonlinearity, which reveal very
interesting behaviors. For example, fitness superiority induces much shorter
competitions in the sublinear regime while much longer competitions in the
superlinear regime. Our findings can shed light on the application of P\'olya
urn processes in more general contexts where fitness and nonlinearity may be
present.
