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VOL. 52 | 2006 Multi-armed bandit problem with precedence relations
Hock Peng Chan, Cheng-Der Fuh, Inchi Hu

Editor(s) Hwai-Chung Ho, Ching-Kang Ing, Tze Leung Lai


Consider a multi-phase project management problem where the decision maker needs to deal with two issues: (a) how to allocate resources to projects within each phase, and (b) when to enter the next phase, so that the total expected reward is as large as possible. We formulate the problem as a multi-armed bandit problem with precedence relations. In Chan, Fuh and Hu (2005), a class of asymptotically optimal arm-pulling strategies is constructed to minimize the shortfall from perfect information payoff. Here we further explore optimality properties of the proposed strategies. First, we show that the efficiency benchmark, which is given by the regret lower bound, reduces to those in Lai and Robbins (1985), Hu and Wei (1989), and Fuh and Hu (2000). This implies that the proposed strategy is also optimal under the settings of aforementioned papers. Secondly, we establish the super-efficiency of proposed strategies when the bad set is empty. Thirdly, we show that they are still optimal with constant switching cost between arms. In addition, we prove that the Wald’s equation holds for Markov chains under Harris recurrent condition, which is an important tool in studying the efficiency of the proposed strategies.


Published: 1 January 2006
First available in Project Euclid: 28 November 2007

zbMATH: 1268.62092
MathSciNet: MR2427850

Digital Object Identifier: 10.1214/074921706000001067

Primary: 62L05
Secondary: 62N99

Rights: Copyright © 2006, Institute of Mathematical Statistics


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