An Optimal Acceptance Policy for an Urn Scheme

An urn contains m balls of value -1 and p balls of value +1. At each turn a ball is drawn randomly, without replacement, and the player decides before the draw whether or not to accept the ball, i.e., the bet where the payoff is the value of the ball. The process continues until all m+p balls are drawn. Let V(m,p) denote the value of this acceptance (m,p) urn problem under an optimal acceptance policy. In this paper, we first derive an exact closed form for V(m,p) and then study its properties and asymptotic behavior. We also compare this acceptance (m,p) urn problem with the original (m,p) urn problem which was introduced by Shepp [Ann. Math. Statist., 40 (1969), pp. 993--1010]. Finally, we briefly discuss some applications of this acceptance (m,p) urn problem and introduce a Bayesian approach to this optimal stopping problem. Some numerical illustrations are also provided.