Competing Urn Model Dynamics

• Competing urn models are a class of stochastic processes where interacting urns or ball types exhibit competitive dynamics through mechanisms like occupancy constraints and feedback loops.
• They utilize diverse methodologies such as occupancy-based negative association, graph-induced annihilation, and crossed removal rules to analyze phase transitions and survival properties.
• These models offer practical insights into network feedback, synchronization, and criticality, helping researchers predict extinction probabilities and long-term system behavior.

In the literature, the expression competing urn model does not denote a single standardized stochastic process. It is used for several non-equivalent urn constructions in which competition is expressed through different mechanisms: occupancy constraints in independent ball-to-urn allocation, reinforcement-plus-annihilation on graphs, crossed removal rules, internal negative feedback between reinforcing and balancing tendencies, and cross-urn feedback in interacting systems. A precise reading therefore depends on the model class under discussion. At the same time, these models share a common structural theme: the state evolution of one colour, type, or urn is not autonomous, but is constrained or redirected by the presence of others (Kahn et al., 2010, Ahlberg et al., 2016, Yadav, 2017, Maulik et al., 2022). A useful boundary case is the Diaconis group-multiplication urn, which the 2022 revisit explicitly contrasts with standard Pólya and generalized Friedman urns and describes as a single-urn multiplicative random-label process rather than a classical competing urn model (Yang et al., 2022).

1. Occupancy-based competing urns and negative dependence

In one classical usage, competing urns refers to the experiment in which $m$ balls are dropped, randomly and independently, into urns $1,\dots,n$. Writing $\sigma:[m]\to[n]$ for the random assignment map and $B_j=|\sigma^{-1}(j)|$ for the occupancy of urn $j$, one studies indicators such as the ordinary occupancy variable $x_j=\mathbf 1_{\{B_j\ge 1\}}$, the threshold urn variable $x_j=\mathbf 1_{\{B_j\ge t_j\}}$, and the more general interval urn encoding

$$x_j(\sigma)=t \quad\Longleftrightarrow\quad a_j(t)\le B_j<a_j(t+1).$$

The 2010 analysis proves that threshold urn measures are conditionally negatively associated and, more generally, that interval urn measures are conditionally negatively associated in the i.i.d. case. Its proof reduces conditional negative association to conditional negative correlation statements of the form $X\downarrow Y\mid Q$, then uses a ratio inequality for the conditional law $\mu_k(l)=\Pr(Y=l\mid X=k)$, strong log-concavity of an aggregated occupancy count $1,\dots,n$0, convexity of support, and an orientation-counting lemma on multigraphs (Kahn et al., 2010).

The same paper formulates the broader generalized urn measure by allowing non-identically distributed balls through a weight array $1,\dots,n$1 with

$1,\dots,n$2

Here the ball assignments remain independent, but their marginal laws may differ from ball to ball. The i.i.d. competing-urn theorems are obtained as a specialization of this generalized setting, but the conditional negative association problem for the full non-i.i.d. model remained open in that work (Kahn et al., 2010).

That open problem was resolved in 2025. For the generalized independent urn model, where $1,\dots,n$3 are independent but not necessarily identically distributed, the law of both the occupation indicators and the full occupancy vector was shown to satisfy conditional negative association. The key intermediate notion is the normalized matching property

$1,\dots,n$4

for increasing events, applied to the conditional law of the set of balls landing in a specified urn. The proof rewrites the problem in terms of weighted bipartite graphs $1,\dots,n$5, weights $1,\dots,n$6, admissible orientations, and then passes from normalized matching to the conditional Feder–Mihail property and finally to conditional negative association. This settles the Kahn–Neiman question affirmatively for non-identical ball distributions (Chan, 8 Sep 2025).

2. Graph-based competition, annihilation, and single-survivor theorems

A very different meaning of competing urn model appears in graph-based systems where urns live on the vertices of a finite connected graph $1,\dots,n$7, each urn is monochromatic, and colours compete through local reproduction and annihilation. In the two-type model of Griffiths, Janson, Morris, and the first author, the state is encoded by a signed vector $1,\dots,n$8, where $1,\dots,n$9 is the number of red balls at $\sigma:[m]\to[n]$0 minus the number of blue balls at $\sigma:[m]\to[n]$1. At each step, a ball is chosen uniformly from all balls in the system, a ball of the same colour is added to each neighbouring urn, and opposite colours annihilate one-for-one when they meet. The core theorem states that for every finite connected graph $\sigma:[m]\to[n]$2 and any finite initial configuration, almost surely only one colour survives. More sharply, if $\sigma:[m]\to[n]$3 is the Perron–Frobenius eigenvalue of the adjacency matrix and $\sigma:[m]\to[n]$4 the corresponding positive eigenvector, then

$\sigma:[m]\to[n]$5

almost surely and in $\sigma:[m]\to[n]$6, with $\sigma:[m]\to[n]$7. Since $\sigma:[m]\to[n]$8 has strictly positive coordinates, the limit is either entirely positive or entirely negative, forcing eventual monochromatic survival (Ahlberg et al., 2016).

That two-type theorem is used in the same paper as the engine for extinction results in growth models on $\sigma:[m]\to[n]$9. The reduction is geometric: in the two-type growth model, boundary segment lengths evolve exactly like balls in an urn process on a cycle, so the urn theorem implies that one of the colours infects only finitely many sites almost surely. The paper also notes that the strong single-survivor property is special to the two-type case on finite connected graphs; for $B_j=|\sigma^{-1}(j)|$0 types, coexistence can occur on some finite graphs (Ahlberg et al., 2016).

For $B_j=|\sigma^{-1}(j)|$1 types, the cycle graph $B_j=|\sigma^{-1}(j)|$2 was later shown to retain the single-survivor property. In that model each ball has an independent Poisson clock of rate $B_j=|\sigma^{-1}(j)|$3; when its clock rings, it sends a copy of itself to each neighbouring vertex, and different types annihilate upon contact. The main theorem states that for every $B_j=|\sigma^{-1}(j)|$4 and any nonzero initial configuration, the $B_j=|\sigma^{-1}(j)|$5-type competing urn scheme on the cycle of length $B_j=|\sigma^{-1}(j)|$6 has almost surely a single surviving type. The proof exploits the cycle-specific decomposition into tribes and fronts, together with an auxiliary signed process on a cycle with a sign-reversing edge and modified reinforcement matrix $B_j=|\sigma^{-1}(j)|$7. For that auxiliary process, the leading asymptotic is harmonic: $B_j=|\sigma^{-1}(j)|$8 with $B_j=|\sigma^{-1}(j)|$9. The reduction from tribes/fronts to this signed process shows that the number of tribes eventually drops to one (Ahlberg et al., 2022).

3. Crossed removal models: OK Corral urns and urns with removals

Another competitive mechanism is not annihilation on a graph but crossed removal. In Kuba’s general-weight formulation, two colours start with $j$0 white and $j$1 black balls, and the process stops when one colour disappears. Urn model I generalizes sampling without replacement: $j$2 whereas urn model II is the OK Corral urn model with general weights: $j$3 The paper studies the random variable $j$4, the number of white balls remaining when all black balls have been drawn, derives explicit probability mass functions for both models, and shows that model I with weights $j$5 is dual to model II with reciprocal weights $j$6, where $j$7 and $j$8. In this sense, competition is encoded by the fact that in the OK Corral rule the removal probability of one colour is proportional to the opposing colour class weight (Kuba, 2010).

A related discrete-time competition mechanism appears in generalized Pólya urns with removals. If $j$9 is the vector of colour counts, then at each step one chooses a ball of type $x_j=\mathbf 1_{\{B_j\ge 1\}}$0 with probability proportional to $x_j=\mathbf 1_{\{B_j\ge 1\}}$1, returns it together with $x_j=\mathbf 1_{\{B_j\ge 1\}}$2 additional balls of the same type, and removes

$x_j=\mathbf 1_{\{B_j\ge 1\}}$3

balls of every competing type $x_j=\mathbf 1_{\{B_j\ge 1\}}$4. The corresponding interaction graph is defined by $x_j=\mathbf 1_{\{B_j\ge 1\}}$5 if $x_j=\mathbf 1_{\{B_j\ge 1\}}$6. The main theorem states that, with probability one, the system eventually retains only a random subset of mutually non-interacting colours, while all other colours become extinct. The surviving colours then evolve as independent Yule processes with parameter $x_j=\mathbf 1_{\{B_j\ge 1\}}$7. Here the long-run state is neither coexistence of all colours nor a deterministic monopoly of a single one, but a random antichain of colours in the interaction graph (Popov et al., 2020).

4. Internal competition: negative feedback, criticality, and competing strategies

In the critical Pólya urn, the competition is internal to a single two-colour urn. The state is $x_j=\mathbf 1_{\{B_j\ge 1\}}$8, with $x_j=\mathbf 1_{\{B_j\ge 1\}}$9 black balls and $x_j=\mathbf 1_{\{B_j\ge t_j\}}$0 white balls. At each elementary update, one ball is selected at random, returned, and one new ball is added according to

$x_j=\mathbf 1_{\{B_j\ge t_j\}}$1

The novelty is that $x_j=\mathbf 1_{\{B_j\ge t_j\}}$2 is not fixed: with probability $x_j=\mathbf 1_{\{B_j\ge t_j\}}$3 it is chosen from a memory-dependent strategy $x_j=\mathbf 1_{\{B_j\ge t_j\}}$4, and with probability $x_j=\mathbf 1_{\{B_j\ge t_j\}}$5 from a competing rule $x_j=\mathbf 1_{\{B_j\ge t_j\}}$6. In urn-I,

$x_j=\mathbf 1_{\{B_j\ge t_j\}}$7

The paper interprets this as a competition between suppression of growth of the dominant colour and enhancement of dormant character of the less represented colour. In the difference variable $x_j=\mathbf 1_{\{B_j\ge t_j\}}$8, the induced one-dimensional walk has step probabilities

$x_j=\mathbf 1_{\{B_j\ge t_j\}}$9

and drift

$$x_j(\sigma)=t \quad\Longleftrightarrow\quad a_j(t)\le B_j<a_j(t+1).$$0

so the imbalance is opposed by state-dependent negative feedback (Yadav, 2017).

The first-passage observable is the first tie $$x_j(\sigma)=t \quad\Longleftrightarrow\quad a_j(t)\le B_j<a_j(t+1).$$1, for which

$$x_j(\sigma)=t \quad\Longleftrightarrow\quad a_j(t)\le B_j<a_j(t+1).$$2

For urn-I the leading result is

$$x_j(\sigma)=t \quad\Longleftrightarrow\quad a_j(t)\le B_j<a_j(t+1).$$3

so the first-passage exponent varies continuously from $$x_j(\sigma)=t \quad\Longleftrightarrow\quad a_j(t)\le B_j<a_j(t+1).$$4 to $$x_j(\sigma)=t \quad\Longleftrightarrow\quad a_j(t)\le B_j<a_j(t+1).$$5. The same framework yields avalanche-type observables such as

$$x_j(\sigma)=t \quad\Longleftrightarrow\quad a_j(t)\le B_j<a_j(t+1).$$6

with power-law exponents related by scaling laws. For the size exponent the paper finds

$$x_j(\sigma)=t \quad\Longleftrightarrow\quad a_j(t)\le B_j<a_j(t+1).$$7

At $$x_j(\sigma)=t \quad\Longleftrightarrow\quad a_j(t)\le B_j<a_j(t+1).$$8, $$x_j(\sigma)=t \quad\Longleftrightarrow\quad a_j(t)\le B_j<a_j(t+1).$$9 and $X\downarrow Y\mid Q$0 define one universality class, while for the nonlinear size definition $X\downarrow Y\mid Q$1, the same choice gives $X\downarrow Y\mid Q$2, matching the mean-field branching-process universality class. The paper emphasizes that criticality is self-organized: except for the trivial $X\downarrow Y\mid Q$3 case, no external parameter tuning is required in the long-time regime (Yadav, 2017).

A second internal-competition construction appears in urn models with two types of strategies. There, a balanced two-colour urn alternates between a generalized Pólya/Bagchi–Pal-type strategy, chosen with probability $X\downarrow Y\mid Q$4, and an i.i.d. strategy, chosen with probability $X\downarrow Y\mid Q$5. The first strategy depends on the current urn composition; the second ignores it. The mean replacement matrix has eigenvalues

$X\downarrow Y\mid Q$6

and the asymptotics exhibit a phase transition at

$X\downarrow Y\mid Q$7

Below that threshold the centred process has $X\downarrow Y\mid Q$8-Gaussian fluctuations; at criticality the normalization becomes $X\downarrow Y\mid Q$9. The model is therefore competitive in the precise sense that history-dependent and history-free reinforcement rules are randomly alternated within a single urn (González-Navarrete et al., 2017).

5. Interacting urn systems, network feedback, and synchronization

A further line of work studies interacting urn schemes in which a draw from one urn governs reinforcement in another. In the deterministic cyclic model with $\mu_k(l)=\Pr(Y=l\mid X=k)$0 urns and $\mu_k(l)=\Pr(Y=l\mid X=k)$1 colours, urn $\mu_k(l)=\Pr(Y=l\mid X=k)$2 feeds urn $\mu_k(l)=\Pr(Y=l\mid X=k)$3, urn $\mu_k(l)=\Pr(Y=l\mid X=k)$4 feeds urn $\mu_k(l)=\Pr(Y=l\mid X=k)$5, and so on, with urn $\mu_k(l)=\Pr(Y=l\mid X=k)$6 feeding urn $\mu_k(l)=\Pr(Y=l\mid X=k)$7. Each urn has an $\mu_k(l)=\Pr(Y=l\mid X=k)$8 balanced replacement matrix $\mu_k(l)=\Pr(Y=l\mid X=k)$9, and the coupled system is encoded by a block-cyclic matrix

$1,\dots,n$00

If $1,\dots,n$01 is irreducible and each $1,\dots,n$02 is balanced and nonnegative, then the normalized composition of urn $1,\dots,n$03 converges almost surely to the left Perron eigenvector of the corresponding cyclic product matrix, while the aggregate normalized vector converges to the dominant left eigenvector of $1,\dots,n$04. The paper also defines a non-deterministic interaction model driven by an $1,\dots,n$05 stochastic matrix $1,\dots,n$06, with asymptotics controlled by the block matrix $1,\dots,n$07 obtained by weighting the $1,\dots,n$08 blocks by the entries of $1,\dots,n$09 (Maulik et al., 2022).

Graph-based interaction with multiple drawings yields a different asymptotic picture. On a finite connected undirected graph, each urn draws a fixed sample size $1,\dots,n$10 either from itself with probability $1,\dots,n$11 or from a uniformly chosen neighbour with probability $1,\dots,n$12. Reinforcement can be Pólya-type, where a sampled colour reinforces the same colour, or Friedman-type, where it reinforces the opposite colour; each of these comes in self, neighbour, and self-plus-neighbour variants. For the fraction vector $1,\dots,n$13 of white balls, the dynamics is written as a stochastic approximation scheme. The main convergence theorem states that in most Friedman-type cases

$1,\dots,n$14

whereas on connected regular graphs the Pólya-type models typically satisfy

$1,\dots,n$15

for a random $1,\dots,n$16. Bipartite graphs are exceptional: in certain boundary cases the two partitions synchronize internally but converge to complementary or distinct partition-wise limits (Dahiya et al., 2023).

A two-urn multi-colour variant with alternating active urns is analyzed through an embedding into a discrete-time multitype branching process. Balls are drawn without replacement from the active urn, and each drawn ball of colour $1,\dots,n$17 places a random offspring vector $1,\dots,n$18 into the other urn. If $1,\dots,n$19 has Perron eigenvalue $1,\dots,n$20 and normalized positive eigenvector $1,\dots,n$21, then

$1,\dots,n$22

for the total number of added balls of colour $1,\dots,n$23. The second-order behaviour is governed by

$1,\dots,n$24

with a trichotomy at $1,\dots,n$25: non-Gaussian oscillatory behaviour for $1,\dots,n$26, Gaussian limits with a $1,\dots,n$27 normalization for $1,\dots,n$28, and Gaussian limits with $1,\dots,n$29 normalization for $1,\dots,n$30. A distinctive feature is the presence of continuous $1,\dots,n$31-periodic scaling functions, reflecting the discrete-time branching embedding (Kolesko et al., 2023).

Strong reinforcement substantially complicates the interacting-urn picture. In Launay’s interacting urn model with $1,\dots,n$32 urns, each urn samples from the global pool with probability $1,\dots,n$33 and from its own local urn with probability $1,\dots,n$34, with reinforcement weights $1,\dots,n$35. The 2023 analysis distinguishes domination, meaning convergence of all urn proportions to $1,\dots,n$36 or $1,\dots,n$37 in the two-urn case, from the stronger event of monopoly, meaning that eventually only one colour is ever added anywhere in the system. For polynomial reinforcement $1,\dots,n$38, $1,\dots,n$39, it disproves the conjecture that every $1,\dots,n$40 forces monopoly almost surely, by proving that the critical threshold $1,\dots,n$41 satisfies $1,\dots,n$42; for sufficiently small positive $1,\dots,n$43, domination itself fails with positive probability. By contrast, for $1,\dots,n$44 it proves monopoly under conditions strictly weaker than the previously used monotonicity assumption on $1,\dots,n$45 (Qin, 2023).

6. Spectral asymptotics, phase transitions, and model boundaries

Some urn models are classified as competing primarily because of their interaction structure, even when the long-run asymptotics is oscillatory rather than eliminative. The cyclic urn has ball types $1,\dots,n$46, starts from one ball of type $1,\dots,n$47, and upon drawing type $1,\dots,n$48 returns it together with one new ball of type $1,\dots,n$49. For $1,\dots,n$50, the classical normalization gives a multivariate central limit theorem. For $1,\dots,n$51, the normalized centred composition does not converge; instead it admits an almost sure approximation by a periodic random vector. After subtraction of that oscillatory leading term, the residual fluctuations are asymptotically Gaussian. If $1,\dots,n$52, the covariance has rank $1,\dots,n$53 on the zero-sum hyperplane; if $1,\dots,n$54, the critical modes force an additional $1,\dots,n$55 factor and the covariance rank drops to $1,\dots,n$56 (Müller et al., 2015).

A different adjacent family is provided by interacting Ehrenfest urns. In the two-urn model with same-urn interactions, particles in the same urn contribute an energy

$1,\dots,n$57

and the resulting reversible dynamics can undergo either a second-order transition at $1,\dots,n$58, $1,\dots,n$59, or a first-order transition for sufficiently strong attraction $1,\dots,n$60. The paper computes relaxation times, Poincaré cycles, and duration-time asymmetries, using the scaling of the Poincaré cycles and the ratio of duration times as diagnostics for the order of transition and for metastability (Tseng et al., 2017). In the $1,\dots,n$61-urn equilibrium generalization with attractive all-to-all intra-urn interaction, the uniform state

$1,\dots,n$62

is stable for $1,\dots,n$63, while non-uniform phases appear for $1,\dots,n$64. Only the uniform phase and the first non-uniform phase are locally stable, and their first-order coexistence transition occurs at

$1,\dots,n$65

with an explicit energy barrier per particle at the transition (Cheng et al., 2021).

The principal terminological boundary is the Diaconis urn model revisited in 2022. There, $1,\dots,n$66 is a finite Abelian group, the urn contains balls labelled by group elements, and at stage $1,\dots,n$67 one draws $1,\dots,n$68 balls with replacement from the current composition, multiplies their labels in the group,

$1,\dots,n$69

and adds one ball labelled $1,\dots,n$70. Under the condition that some sampled size $1,\dots,n$71 satisfies $1,\dots,n$72, together with initial generators, the normalized composition converges almost surely to the uniform distribution on $1,\dots,n$73: $1,\dots,n$74 Moreover,

$1,\dots,n$75

The paper repeatedly contrasts this behaviour with standard Pólya and generalized Friedman urns, emphasizing that the group-multiplication mechanism mixes labels and forces asymptotic equidistribution. It is therefore best classified not as a classical competing urn model, but as a specialized single-urn multiplicative process with a different asymptotic logic (Yang et al., 2022).

Across these model classes, several recurrent asymptotic themes appear: conditional negative dependence in occupancy measures, almost sure synchronization, extinction to a single surviving colour or type, survival of only mutually non-interacting types, Gaussian and critical $1,\dots,n$76 fluctuation regimes, periodic modulation, and phase transitions between uniform and symmetry-broken states. What changes from one competing-urn framework to another is not the presence of interaction, but the precise mathematical form through which interaction is realized.