Polya Urn Process Research Articles

International Trade and Technological Competition in Markets with Dynamic Increasing Returns

We build a simple dynamic model to study the effects of technological learning, market selection and international competition in the determination of export flows and market shares. The model features two countries populated by firms with heterogeneous productivity levels and sales. Market selection in each country is driven by a finite pairwise Polya urn process. We show that market selection leads either to a national or to an international monopoly in presence of a static distribution of firm productivity levels. We then incorporate firm learning and entry-exit in the model and we show that the market structure does not converge to a monopoly. In addition, we show that the extended model is able to jointly reproduce a wide ensemble of stylized facts concerning intra-industry trade, industry and firm dynamics.

Bayesian Allocation Model: Marginal Likelihood-Based Model Selection for Count Tensors

In this article, we introduce a dynamic generative model, the Bayesian allocation model (BAM), for modeling count data. BAM covers various probabilistic nonnegative tensor factorization (NTF) and topic models under one general framework. In BAM, allocations are made using a Bayesian network, whose conditional probability tables can be integrated out analytically. We show that, when allocations are viewed as sequential, the resulting marginal process is a special type of Polya urn process, which we name as Polya-Bayes process, an integer valued self-reinforcing process. Exploiting the Polya urn construction, we develop a novel sequential Monte Carlo (SMC) algorithm for marginal likelihood estimation in BAM, leading to a unified scoring method for discrete variable Bayesian networks with hidden nodes, including various NTF and topic models. The SMC estimator for marginal likelihood has the remarkable property of being unbiased in contrast to variational algorithms which are generally biased. We also demonstrate how our novel SMC-based likelihood estimation can be integrated within a Markov chain Monte Carlo algorithm for a principled and correct (in terms of respecting the true posterior distribution) Bayesian model selection and hyperparameter estimation for BAM. We provide several numerical examples, both on artificial and real datasets, that demonstrate the performance of the algorithms for various data regimes.

Herding, Learning and Incentives for Online Reviews

We investigate the role of consumer herding and learning on the design of incentives for online customer reviews. Herding occurs when consumers are drawn to a product that appears to be popular because it has garnered a large number of reviews. Learning occurs when consumers infer product quality from reviews. We evaluate and compare three incentive policies. The first announces an incentive to all customers before purchase, the second offers an incentive after purchase, and the third rewards buyers only if they write positive, possibly fake, reviews. We use a generalized Polya urn process to model the evolution of reviews. The expected value of the resulting aggregate demand has the form of the Gompertz function. We obtain conditions under which each type of incentive is profitable, and preferred by a seller to the other incentives for reviews. The results imply that sellers should use different incentives policies depending on the quality and profit margin of a product. A pre-purchase incentive is the most profitable when product quality and profit margin are both high; an incentive offered to buyers after obtaining voluntary reviews is the most profitable when product quality is high and profit margin is low; and an incentive for only positive reviews is the most profitable when product quality and profit margin are both low. E-commerce platforms that limit their sellers to using post-purchase incentives might be more effective in curbing fake reviews if they also allow sellers to announce pre-purchase incentives to all customers.

Online popularity of destinations in Australia: An application of Polya Urn process to search engine data

The online search engine is a key source of tourism information for trip planning. In the internet, experience and feedback shared by previous tourists can be obtained with increasing ease, resulting in a feedback mechanism of interpersonal influence featured in the online peer-to-peer tourism information exchange. Such interpersonal influence may lead to a unique popularity distribution among destinations. The information exchange process is also affected by the destination choice set, which is determined by the geography of the destination countries. This study mathematically characterised the e-word-of-mouth effect in travel and tourism on the pattern formation in online destination. To achieve this, the Polya Urn process is invoked to model the clustering of destination preference and the distribution of destination popularity in an online environment. Google search volume on various destinations in Australia and France from five tourism inbound countries are used. Results show that the theoretical distribution of the Polya Urn process fits the distribution of search volume on Australian destinations, highlighting the geographical uniqueness of Australia as a closed system that possess a unique online popularity growth process and distribution.

Revelation, Decision and Fascination

We model the sequential decisions made by receivers in response to multiple signals. In the first part, we model this in a special setting where the signals in the system are generated by either truth-telling or lying. We classify the signals as binary high or low, and study the information herding behavior. Then we study the decision-making in response to signals that cannot be simply classified as binary high or low. Last, we apply the Polya urn process to model the dynamics of public announcements made by the receivers in a more general setting, and discuss how senders can design their strategies to take advantage of public announcements and further increase their benefits. This research quantitatively examines people’s reactions to signals, and can be generalized and adapted to cases like the design of game shows, advertising, biding, and customer review management.

Curing Epidemics on Networks Using a Polya Contagion Model

We study the curing of epidemics of a network contagion, which is modelled using a variation of the classical Polya urn process that takes into account spatial infection among neighbouring nodes. We introduce several quantities for measuring the overall infection in the network and use them to formulate an optimal control problem for minimizing the average infection rate using limited curing resources. We prove the feasibility of this problem under high curing budgets by deriving conservative lower bounds on the amount of curing per node that turn our measures of network infection into supermartingales. We also provide a provably convergent gradient descent algorithm to find the allocation of curing under limited budgets. Motivated by the fact that this strategy is computationally expensive, we design a suit of heuristic methods that are locally implementable and nearly as effective. Extensive simulations run on large-scale networks demonstrate the effectiveness of our proposed strategies.

Spatial Distribution of Tourism Activities: A Polya Urn Process Model of Rank-Size Distribution

The power law is considered one of the most enduring regularities in human geography. This article aims to develop an understanding of the circumstances that may result in the power law distribution in the geography of tourism activities. The finite Polya urn process is adopted as a device to model the preferential attachment process in the flow of tourists. The model generates a rank-size distribution of tourism regions along with intuitively appealing parameters. Empirically examined using two independent sets of Australian inbound and outbound tourism data, results show that the rank-size distribution emerging from the finite Polya urn process is a superior fit to the conventional power law curve. This rank-size distribution (termed the Polya urn process model of visitor distribution) is compatible with tourist behaviors such as habit persistence and word-of-mouth effects, and can be adopted by tourism modelers to predict and efficiently summarize the spatiality of tourism.

Dynamic increasing returns and innovation diffusion: bringing Polya Urn processes to the empirical data

The patterns of innovation diffusion are well approximated by the logistic curves. This is the robust empirical fact confirmed by many studies in innovations dynamics. Here, we show that the logistic pattern of innovation diffusion can be replicated by the time-dependent stochastic process with positive feedbacks along the diffusion trajectory. The dynamic increasing returns process is modelled by Polya Urns. So far, Urn models have been mostly used to study the [path-dependent] limit properties. On the contrary, this work focuses on the transient [finite time] properties studying the conditions under which urn models capture the logistic trajectories which often track empirical diffusion process. As examples, we calibrate the process to match several cases of diffusion of motor ships in European countries.

Moment convergence of balanced Pólya processes

It is known that in an irreducible small Polya urn process, the composition of the urn after suitable normalization converges in distribution to a normal distribution. We show that if the urn also ...

Three faces of entropy for complex systems: Information, thermodynamics, and the maximum entropy principle

There are at least three distinct ways to conceptualize entropy: entropy as an extensive thermodynamic quantity of physical systems (Clausius, Boltzmann, Gibbs), entropy as a measure for information production of ergodic sources (Shannon), and entropy as a means for statistical inference on multinomial processes (Jaynes maximum entropy principle). Even though these notions represent fundamentally different concepts, the functional form of the entropy for thermodynamic systems in equilibrium, for ergodic sources in information theory, and for independent sampling processes in statistical systems, is degenerate, H(p)=-∑_{i}p_{i}logp_{i}. For many complex systems, which are typically history-dependent, nonergodic, and nonmultinomial, this is no longer the case. Here we show that for such processes, the three entropy concepts lead to different functional forms of entropy, which we will refer to as S_{EXT} for extensive entropy, S_{IT} for the source information rate in information theory, and S_{MEP} for the entropy functional that appears in the so-called maximum entropy principle, which characterizes the most likely observable distribution functions of a system. We explicitly compute these three entropy functionals for three concrete examples: for Pólya urn processes, which are simple self-reinforcing processes, for sample-space-reducing (SSR) processes, which are simple history dependent processes that are associated with power-law statistics, and finally for multinomial mixture processes.

Measure-valued Pólya urn processes

A Polya urn process is a Markov chain that models the evolution of an urn containing some coloured balls, the set of possible colours being $\{1,\ldots ,d\}$ for $d\in \mathbb{N} $. At each time step, a random ball is chosen uniformly in the urn. It is replaced in the urn and, if its colour is $c$, $R_{c,j}$ balls of colour $j$ are also added (for all $1\leq j\leq d$). We introduce a model of measure-valued processes that generalises this construction. This generalisation includes the case when the space of colours is a (possibly infinite) Polish space $\mathcal P$. We see the urn composition at any time step $n$ as a measure ${\cal M}_n$ – possibly non atomic – on $\mathcal P$. In this generalisation, we choose a random colour $c$ according to the probability distribution proportional to ${\cal M}_n$, and add a measure ${\cal R}_c$ in the urn, where the quantity ${\cal R}_c(B)$ of a Borel set $B$ models the added weight of “balls” with colour in $B$. We study the asymptotic behaviour of these measure-valued Polya urn processes, and give some conditions on the replacements measures $({\cal R}_c,c\in \mathcal P)$ for the sequence of measures $({\cal M}_n, n\geq 0)$ to converge in distribution, possibly after rescaling. For certain models, related to branching random walks, $({\cal M}_n, n\geq 0)$ is shown to converge almost surely under some moment hypothesis; a particular case of this last result gives the almost sure convergence of the (renormalised) profile of the random recursive tree to a standard Gaussian.

Technology and costs in international competitiveness: From countries and sectors to firms

This paper examines the microfoundations of the determinants of international competitiveness. It does so within the broader “technology gap” perspective whereby wide technological and organizational differences ultimately shape the patterns of trade within sectors across countries and their dynamics. First, we take stock of the incumbent evidence on the relation between cost-related and technological competition at country and sectoral level. The overall picture indeed suggests that the countries’ sectoral market shares are mainly shaped by technological factors while cost advantages/disadvantages do not seem to play any significant role. But within any sector, within any country, firms widely differ. Hence the question: does this property apply also at a micro level? Here, we first propose a heuristic model based on a generalized Polya urn process yielding such a property and, then, empirically attempt to identify the underlying dynamics at the firm level using a large panel of Italian firms, over nearly two decades. Results show that also at micro level in most sectors investments and patents correlate positively both with the probability of being an exporter and with the capacity to acquire and to increase exports, whereas labour costs show a negative effect only in some sectors. The result is reinforced when separating the short- and long-run effects, highlighting the predominant impact of technological proxies and basically the irrelevance of wage costs.

Multi‐agent model of group polarisation with biased assimilation of arguments

This study studies the opinion dynamics with group polarisation. The authors say a process of opinion formation is polarising if the final opinions after group discussion are more divergent than individuals’ initial opinions. To capture this phenomenon, they develop a multi-agent model of opinion formation in which each agent forms their opinion according to all arguments available to them, and during interacting with other agents, they assimilate the arguments of neighbours in a biased manner which can be represented by a modified Polya's urn process. They find that biased assimilation of arguments can result in group polarisation effectively, especially under the condition when the agents are relatively isolated in the network.

General randomized response techniques using Polya's Urn process as a randomization device.

In this paper, interesting improvements in [1] and [2] randomized response techniques have been proposed. The proposed randomized response technique applies Polya's urn process (see [3]) to obtain data from respondents. One of the suggested technique requires reporting the number of draws to observe a fixed number of cards of certain type. On the contrary, the number of cards of a certain type is to be reported in case of second proposed randomized response model. Based on the information collected through the suggested techniques, two different unbiased estimators of proportion of a sensitive attribute have been suggested. A detailed comparative simulation study has also been done. The results are also supported by means of a small scale survey.

A fixed-point algorithm to estimate the Yule–Simon distribution parameter

The Yule–Simon distribution is a discrete probability distribution related to preferential attachment processes such as the growth in the number of species per genus in some higher taxon of biotic organisms, the distribution of the sizes of cities, the wealth distribution among individuals, the number of links to pages in the World Wide Web, among others. In this paper we present an algorithm to, given a set of observations stemmed from a Yule process, obtain the parameter of the Yule–Simon distribution with maximum likelihood. In order to test our algorithm, we use a modified Polya urn process simulation to generate some data that was used as input to our algorithm. We make a comparison of our algorithm with other methods and also we show an application to some empirical data.

Bi-Modal and Meta-modelling Paradigm for Cloud Computing of Habitual Entrepreneurship

A complete mathematical model of habitual entrepreneurship is developed including bi-modal and meta-modeling for cloud computing presented together with dynamic simulation in this paper. The mathematical model on which the simulation is based consists of four main elements: (1) competence-confidence regimes, (2) reinforcement learning, (3) a stochastic environment, (4) updating of outcomes through Polya urn processes. The results of the simulation illustrate differences in the temporal portfolios of entrepreneurs under four alternate competence-confidence regimes and three types of learning scenarios.

Trail formation in ants. A generalized Polya urn process

Faced with a choice of paths, an ant chooses a path with a higher concentration of pheromone. Subsequently, it drops pheromone on the path chosen. The reinforcement of the pheromone-following behavior favors the selection of an initially discovered path as the preferred path. This may cause a long path to emerge as the preferred path, were it discovered earlier than a shorter path. However, the shortness of the shorter path offsets some of the pheromone accumulated on the initially discovered longer path. In this paper, we model the trail formation behavior as a generalized Polya urn process. For k equal length paths, we give the distribution of pheromone at any time and highlight its sole dependence on the initial pheromone concentrations on paths. Additionally, we propose a method to incorporate the lengths of paths in the urn process and derive how the pheromone distribution alters on its inclusion. Analytically, we show that it is possible, under certain conditions, to reverse the initial bias that may be present in favor of paths that were discovered prior to the discovery of more efficient (shorter) paths. This addresses the Plasticity-Stability dilemma for ants, by laying out the conditions under which the system will remain stable or become plastic and change the path. Finally, we validate our analysis and results using simulations.

Path dependence: a foundational concept for historical social science

This introduction to the concept of path dependence, its pertinence for the development of historical social science, and its application in economic analysis and economic history, proceeds from intuitive general ideas about history and historicity in narratives. It provides precise definitions of what is meant by describing a dynamical process as being “historical.” Deterministic and stochastic formalizations of such dynamical systems are distinguished. The characterization of stochastic path dependent processes as “non-ergodic” is explained in non-mathematical language by reference to concepts in probability theory, and a variety of representations of such processes in formal models is surveyed (including the Polya urn-process, certain kinds of Markov chain models, branching processes, and reversible spin systems) to show that while all display path dependence, their properties in other respects are quite different. The diverse set of structural, micro-level conditions that can give rise to path dependence is examined, and a further distinction is drawn between the property of path dependence and the existence of so-called “QWERTY-effects”—characterized by decentralized competitive market failures and consequent “lock-in” to Pareto-inefficient equilibria. Concluding sections consider the implications of the existence of non-ergodic dynamics for the methods of economic policy analysis, and the nature of the guidance that can be obtained in regard to public policy affecting endogenous technological change and institutional evolution.

Value of information in the Polya urn process

A series of games based on the Polya urn process is considered. The objective is to develop an understanding of the concept of the value of information in connection to these games. It is rigorously shown that information may have negative, zero or positive incremental value to player for calculating a fair bet. It is also shown that additional information will, in many cases, not effect the betting strategy of the player.

Reinforced Random Walks and Random Distributions

Consider a classical Polya urn process on a complete binary tree. This process generates an exchangeable sequence of random variables ${Z_n}$, with values in $[0,1]$. It is shown that the empirical distribution $^\# \{ i \leq n:{Z_i} \leq s\} /n$ converges weakly and the distribution of this limit is the same as a standard Dubins-Freedman random distribution. As an application, the variance of the first moment of these Dubins-Freedman distributions is calculated.