A $(2/3)n^3$ fast-pivoting algorithm for the Gittins index and optimal stopping of a Markov chain

Abstract

This paper presents a new \emph{fast-pivoting} algorithm that computes the $n$ Gittins index values of an $n$-state bandit -- in the discounted and undiscounted cases -- by performing $(2/3) n^3 + O(n^2)$ arithmetic operations, thus attaining better complexity than previous algorithms and matching that of solving a corresponding linear-equation system by Gaussian elimination. The algorithm further applies to the problem of optimal stopping of a Markov chain, for which a novel Gittins-index solution approach is introduced. The algorithm draws on Gittins and Jones' (1974) index definition via calibration, on Kallenberg's (1986) proposal of using parametric linear programming, on Dantzig's simplex method, on Varaiya et al.'s (1985) algorithm, and on the author's earlier work. The paper elucidates the structure of parametric simplex tableaux. Special structure is exploited to reduce the computational effort of pivot steps, decreasing the operation count by a factor of three relative to using conventional pivoting, and by a factor of $3/2$ relative to recent state-elimination algorithms. A computational study demonstrates significant time savings against alternative algorithms.