AN OPTIMAL ACCEPTANCE POLICY FOR AN URN SCHEME

An urn contains m balls of value -1 and p balls of value +1. At each t urn a ball is drawn randomly, without replacement, and the player deci des before the draw whether or not to accept the ball, i.e., the bet w here the payoff is the value of the ball. The process continues until all m+p balls are drawn. Let (V) over bar (m, p) denote the value of t his acceptance (m, p) urn problem under an optimal acceptance policy. In this paper, we first derive an exact closed form for (V) over bar ( m, p) and then study its properties and asymptotic behavior. We also c ompare 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 th is optimal stopping problem. Some numerical illustrations are also pro vided.