# A Generalized Acceptance Urn Model

2010

Dissertation

Ph.D.

## Degree Granting Department

Mathematics and Statistics

## Major Professor

Stephen Suen, Ph.D.

## Committee Member

Gregory McColm, Ph.D.

## Committee Member

Nataša Jonoska, Ph.D.

## Committee Member

Brendan Nagle, Ph.D.

## Keywords

lattice paths, rotation method, ballot problem, Bayesian approach, ruin problem, generalized binomial series

## Abstract

An urn contains two types of balls: p "+t" balls and m "-s" balls, where t and s are positive real numbers. The balls are drawn from the urn uniformly at random without replacement until the urn is empty. Before each ball is drawn, the player decides whether to accept the ball or not. If the player opts to accept the ball, then the payoff is the weight of the ball drawn, gaining t dollars if a "+t" ball is drawn, or losing s dollars if a "-s" ball is drawn. We wish to maximize the expected gain for the player.

We find that the optimal acceptance policies are similar to that of the original acceptance urn of Chen et al. with s=t=1. We show that the expected gain function also shares similar properties to those shown in that work, and note the important properties that have geometric interpretations. We then calculate the expected gain for the urns with t/s rational, using various methods, including rotation and reflection. For the case when t/s is irrational, we use rational approximation to calculate the expected gain. We then give the asymptotic value of the expected gain under various conditions. The problem of minimal gain is then considered, which is a version of the ballot problem.

We then consider a Bayesian approach for the general urn, for which the number of balls n is known while the number of "+t" balls, p, is unknown. We find formulas for the expected gain for the random acceptance urn when the urns with n balls are distributed uniformly, and find the asymptotic value of the expected gain for any s and t.

Finally, we discuss the probability of ruin when an optimal strategy is used for the (m,p;s,t) urn, solving the problem with s=t=1. We also show that in general, when the initial capital is large, ruin is unlikely. We then examine the same problem with the random version of the urn, solving the problem with s=t=1 and an initial prior distribution of the urns containing n balls that is uniform.

COinS