Urn Problems Research Articles

Ellsberg urn problems with multiple degrees of freedom

Ellsberg urn problems with multiple degrees of freedom

Uncertain Urn Problems and Ellsberg Experiment

The basic urn problem is to determine the probability of drawing one colored ball from an urn with known composition of differently colored balls. If the composition is unknown, then it is called uncertain. This paper designs some uncertain urn problems in order to compare probability theory and uncertainty theory. It is concluded that uncertainty theory is better than probability theory to deal with uncertain urn problems. As a by-product, this paper also solves the choice problem in Ellsberg experiment by chance theory.

Strategic ambiguity and decision-making: an experimental study

We conducted a set of experiments to compare the effect of ambiguity in single-person decisions and games. Our results suggest that ambiguity has a bigger impact in games than in ball and urn problems. We find that ambiguity has the opposite effect in games of strategic substitutes and complements. This confirms a theoretical prediction made by Eichberger and Kelsey (J Econ Theory 106:436–466, 2002). In addition, we note that subjects’ ambiguity attitudes appear to be context dependent: ambiguity loving in single-person decisions and ambiguity averse in games. This is consistent with the findings of Kelsey and le Roux (Theory Decis 79:667–688, 2015).

The coupon collector urn model with unequal probabilities in ecology and evolution.

The sequential sampling of populations with unequal probabilities and with replacement in a closed population is a recurrent problem in ecology and evolution. Examples range from biodiversity sampling, epidemiology to the estimation of signal repertoire in animal communication. Many of these questions can be reformulated as urn problems, often as special cases of the coupon collector problem, most simply expressed as the number of coupons that must be collected to have a complete set. We aimed to apply the coupon collector model in a comprehensive manner to one example-hosts (balls) being searched (draws) and parasitized (ball colour change) by parasitic wasps-to evaluate the influence of differences in sampling probabilities between items on collection speed. Based on the model of a complete multinomial process over time, we define the distribution, distribution function, expectation and variance of the number of hosts parasitized after a given time, as well as the inverse problem, estimating the sampling effort. We develop the relationship between the risk distribution on the set of hosts and the speed of parasitization and propose a more elegant proof of the weak stochastic dominance among speeds of parasitization, using the concept of Schur convexity and the 'Robin Hood transfer' numerical operation. Numerical examples are provided and a conjecture about strong dominance-an ordering characteristic of random variables-is proposed. The speed at which new items are discovered is a function of the entire shape of the sampling probability distribution. The sole comparison of values of variances is not sufficient to compare speeds associated with different distributions, as generally assumed in ecological studies.

Strategic Ambiguity and Decision-Making: An Experimental Study

We conducted a set of experiments to compare the effect of ambiguity in single person decisions and games. Our results suggest that ambiguity has a bigger impact in games than in ball and urn problems. We find that ambiguity has the opposite effect in games of strategic substitutes and complements. This confirms a theoretical prediction made by Eichberger and Kelsey (2002).The experiments also test whether subjects' perception of ambiguity differs when faced by a local opponent as opposed to a foreign one. Our results show that there is little evidence of more influence of ambiguity on behaviour when faced by foreign subjects.

Ambiguity Aversion with Three or More Outcomes

Ambiguous choice problems which involve three or more outcome values can reveal aspects of ambiguity and ambiguity aversion which cannot be displayed in the classic two-outcome Ellsberg urn problems, and hence are not always captured by models designed to accommodate them. These aspects include Allais-type preferences over purely subjective acts, attitudes toward different sources involving different amounts of ambiguity, and attitudes toward ambiguity at different outcome levels. This paper presents a few such examples, and examines the standard models' predictions and performance in such cases. (JEL D81)

Population balancing with species switching

Biological organisms are composed of cells differentiated from stem cells. Emerging evidence suggests phenotypic switches of differentiated cells, called transdifferentiation, during development and under pathophysiological conditions. To examine the role of transdifferentiation for population balances, we construct a stochastic model mapped into urn problems. When a colored ball is drawn from an urn, one additional ball with the same color is put into the urn in Polya’s urn scheme while the drawn ball is replaced by another ball with different color in Ehrenfest’s urn scheme. Our population balance model is a mixture of the two classic urn problems corresponding to cell replication and transdifferentiation. Because a dominant population is more likely to be drawn, the preferential replication and transdifferentiation of the dominant population contribute to increasing and decreasing the gap between the two populations, respectively. Therefore, their competition determines population balancing. We analyze the dependence of population dynamics on the replication and the transdifferentiation rates of each population. Finally, experimentally probing the event of transdifferentiation is a challenging problem because it is indistinguishable from death of one cell type and concurrent replication of another cell type. Our analysis suggests that transdifferentiation generates fewer fluctuations in population dynamics than the combined events of cell death and replication do.

A Central Limit Theorem and its Applications to Multicolor Randomly Reinforced Urns

Let Xn be a sequence of integrable real random variables, adapted to a filtration (Gn). Define Cn = √{(1 / n)∑k=1nXk − E(Xn+1 | Gn)} and Dn = √n{E(Xn+1 | Gn) − Z}, where Z is the almost-sure limit of E(Xn+1 | Gn) (assumed to exist). Conditions for (Cn, Dn) → N(0, U) x N(0, V) stably are given, where U and V are certain random variables. In particular, under such conditions, we obtain √n{(1 / n)∑k=1nX_k - Z} = Cn + Dn → N(0, U + V) stably. This central limit theorem has natural applications to Bayesian statistics and urn problems. The latter are investigated, by paying special attention to multicolor randomly reinforced urns.

A Central Limit Theorem and its Applications to Multicolor Randomly Reinforced Urns

Let X n be a sequence of integrable real random variables, adapted to a filtration (G n ). Define C n = √{(1 / n)∑ k=1 n X k − E(X n+1 | G n )} and D n = √n{E(X n+1 | G n ) − Z}, where Z is the almost-sure limit of E(X n+1 | G n ) (assumed to exist). Conditions for (C n , D n ) → N(0, U) x N(0, V) stably are given, where U and V are certain random variables. In particular, under such conditions, we obtain √n{(1 / n)∑ k=1 n X_k - Z} = C n + D n → N(0, U + V) stably. This central limit theorem has natural applications to Bayesian statistics and urn problems. The latter are investigated, by paying special attention to multicolor randomly reinforced urns.

Ambiguity aversion, games against nature, and dynamic consistency

Several papers, adopting an axiomatic approach to study decision making under ambiguity aversion, have produced conflicting predictions about how decision makers would behave in simple dynamic urn problems. We explore the concepts of ambiguity aversion and dynamic consistency, with examples of dynamic games against nature. Basically, a malevolent nature puts balls into the urn, and a fair nature draws them out. Depending on the game, various choices that seem inconsistent with static notions of ambiguity aversion or dynamic consistency are consistent with subgame perfection. In the dynamic 3-color Ellsberg urn problem with 30 red balls and 60 blue or green balls, the decision maker could strictly prefer to bet on blue–green at time 0, and to switch to red–green after learning that the ball is not green.

The Testing Principle: Inductive Reasoning and the Ellsberg Paradox

We postulate the Testing Principle : that individuals ''act like statisticians'' when they face uncertainty in a decision problem, ranking alternatives to the extent that available evidence allows. The Testing Principle implies that completeness of preferences, rather than the sure-thing principle, is violated in the Ellsberg Paradox. In the experiment, subjects chose between risky and uncertain acts in modified Ellsberg-type urn problems, with sample information about the uncertain urn. Our results show, consistent with the Testing Principle, that the uncertain urn is chosen more often when the sample size is larger, holding constant a measure of ambiguity (proportion of balls of unknown colour in the urn). The Testing Principle rationalises the Ellsberg Paradox. Behaviour consistent with the principle leads to a reduction in Ellsberg-type violations as the statistical quality of sample information is improved, holding ambiguity constant. The Testing Principle also provides a normative rationale for the Ellsberg paradox that is consistent with procedural rationality.

Decomposable statistics in inverse urn problems

Decomposable statistics in inverse urn problems

Rates of Poisson convergence for some coverage and urn problems using coupling

Bounds on the rate of convergence measured by the variation distance are obtained for the number of large spacings and for two occupancy problems connected with multinomial and Pólya sampling. The bounds are derived by imbedding techniques together with the elementary coupling inequality.

Rates of Poisson convergence for some coverage and urn problems using coupling

Bounds on the rate of convergence measured by the variation distance are obtained for the number of large spacings and for two occupancy problems connected with multinomial and Pólya sampling. The bounds are derived by imbedding techniques together with the elementary coupling inequality.

On Birthday, Collectors', Occupancy and Other Classical Urn Problems

Summary An urn contains r different balls. Balls are drawn with replacement until any k balls have been obtained at least m times each. How many draws are necessary? How many balls have been drawn exactly v times? Special cases of such problems are often named as birthday, collectors', dixie cup or occupancy problems. This paper presents a unified approach to such problems by imbedding in Poisson processes. In this way we see that many classical urn problems are closely related to properties of order statistics and extreme values from the gamma distribution. Both exact and asymptotic results are derived. Finally, a brief discussion is given on other drawing schemes.

Limit Theorems for the Number of Empty Cells in an Equiprobable Scheme for Group Allocation of Particles

Limit Theorems for the Number of Empty Cells in an Equiprobable Scheme for Group Allocation of Particles