On the Duration and Intensity of Competitions in Nonlinear P\'olya Urn Processes with Fitness

TL;DR

This paper analyzes how fitness and nonlinear cumulative advantage affect the duration and intensity of competitions modeled by Pólya urn processes, revealing diverse behaviors depending on the regime.

Contribution

It provides rigorous mathematical characterizations of the tail distributions of competition duration and intensity in generalized Pólya urn models with fitness and nonlinearity.

Findings

Fitness superiority shortens competitions in sublinear regimes.

Fitness superiority lengthens competitions in superlinear regimes.

The tail distributions exhibit regime-dependent behaviors.

Abstract

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.