### Abstract

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 V̄(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 V̄(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.

Original language | English (US) |
---|---|

Pages (from-to) | 183-195 |

Number of pages | 13 |

Journal | SIAM Journal on Discrete Mathematics |

Volume | 11 |

Issue number | 2 |

DOIs | |

State | Published - Jan 1 1998 |

### Fingerprint

### Keywords

- Acceptance policy
- Bayesian approach
- Optimal stopping
- Urn models

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*SIAM Journal on Discrete Mathematics*,

*11*(2), 183-195. https://doi.org/10.1137/S0895480195282148

**An optimal acceptance policy for an urn scheme.** / Chen, Robert W.; Zame, Alan; Odlyzko, Andrew M.; Shepp, Larry A.

Research output: Contribution to journal › Article

*SIAM Journal on Discrete Mathematics*, vol. 11, no. 2, pp. 183-195. https://doi.org/10.1137/S0895480195282148

}

TY - JOUR

T1 - An optimal acceptance policy for an urn scheme

AU - Chen, Robert W.

AU - Zame, Alan

AU - Odlyzko, Andrew M.

AU - Shepp, Larry A.

PY - 1998/1/1

Y1 - 1998/1/1

N2 - 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 V̄(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 V̄(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.

AB - 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 V̄(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 V̄(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.

KW - Acceptance policy

KW - Bayesian approach

KW - Optimal stopping

KW - Urn models

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U2 - 10.1137/S0895480195282148

DO - 10.1137/S0895480195282148

M3 - Article

VL - 11

SP - 183

EP - 195

JO - SIAM Journal on Discrete Mathematics

JF - SIAM Journal on Discrete Mathematics

SN - 0895-4801

IS - 2

ER -