On the Mabinogion urn model

TL;DR

This paper analyzes the Mabinogion urn model, providing formulas for expected absorption time, its asymptotic behavior under optimal control, and insights into strategies with discounting effects.

Contribution

It introduces formulas for expected absorption time and asymptotic analysis in the controlled Mabinogion urn model, expanding understanding of its dynamics.

Findings

Formulas for expected time to absorption

Asymptotic behavior under optimal control

Impact of discount factors on strategies

Abstract

In this paper we discuss the Mabinogion urn model introduced by D. Williams in Probability with Martingales (1991). Therein he describes an optimal control problem where the objective is to maximize the expected final number of objects of one kind in the Mabinogion urn model. Our main contribution is formulas for the expected time to absorption and its asymptotic behavior in the optimally controlled process. We also present results for the non-controlled Mabinogion urn process and briefly analyze other strategies that become superior if a certain discount factor is included.