Asymptotics of generalized Pólya urns with non-linear feedback

TL;DR

This paper analyzes generalized Polya urns with nonlinear feedback functions $F_i$ to model reinforcement-driven competition among agents. It delivers a comprehensive regime classification, distinguishing polynomial-type (P) and exponential-type (E) feedback, and proves that asymptotic attraction domains are polytopes that partition the market simplex; it also establishes a scaling limit for the full evolution of market shares as the initial size grows, together with a Law of Large Numbers and a Functional Central Limit Theorem describing fluctuations via time-inhomogeneous SDEs. The results unify and extend prior work, showing when strong or weak monopoly occurs, and how near-linear feedback leads to Dirichlet limits, almost-linear convergence, or phase transitions between regimes. Practically, the findings provide a parameter- and initial-condition-dependent map from feedback structure to long-run market shares and dynamic fluctuations, with robust predictions in large markets.

Abstract

Generalized Pólya urns with non-linear feedback are an established probabilistic model to describe the dynamics of growth processes with reinforcement, a generic example being competition of agents in evolving markets. It is well known which conditions on the feedback mechanism lead to monopoly where a single agent achieves full market share, and various further results for particular feedback mechanisms have been derived from different perspectives. In this paper we provide a comprehensive account of the possible asymptotic behaviour for a large general class of feedback, and describe in detail how monopolies emerge in a transition from sub-linear to super-linear feedback via hierarchical states close to linearity. We further distinguish super- and sub-exponential feedback, which show conceptually interesting differences to understand the monopoly case, and study robustness of the asymptotics with respect to initial conditions, heterogeneities and small changes of the feedback mechanisms. Finally, we derive a scaling limit for the full time evolution of market shares in the limit of diverging initial market size, including the description of typical fluctuations and extending previous results in the context of stochastic approximation.

Figures (7)

• Figure 1: Simulated evolution of the market shares for the first 100 steps of a generalized Pólya urn with different feedback functions. Here $A=3$ and $X(0)=(1, 1, 1)$.
• Figure 2: Asymptotic attraction domains in the case $A=3$ with various feedback functions.
• Figure 3: The vector field $G$ for different feedback functions and $A=3$. Here $\bullet$ marks the stable and $\circ$ the unstable fixed points of the dynamics \ref{['eq: ODE']}. In addition, Figure (a) shows the asymptotic attraction domains as derived in Example \ref{['example: sMonPolynom']}.
• Figure 4: The processes $\sqrt{N}\left(Z^{(N)}(t)-Z(t)\right)$ for $A=3$ and $N=10.000$.
• Figure 5: Realisations of the process $M$ for different feedback functions and $A=3$ generated by the Euler-Maruyama method for \ref{['eq:sdes']} with bandwidth $\frac{1}{100}$.

Theorems & Definitions (95)

• Definition 2.1
• Theorem 2.2
• Proposition 2.3
• proof
• Theorem 2.4
• Theorem 2.5
• Theorem 2.6
• Theorem 2.7
• Definition 3.1
• Proposition 3.2