Optimal stopping of Markov chains and three abstract optimization problems
Isaac Mikhailovich Sonin · Stochastics · 2011
There is a well-known connection between the three problems related to the optimal stopping of Markov chains and the equality of three corresponding indices: the classical Gittins index (GI) in the ratio maximization problem, the Kathehakis–Veinott index in a restart problem and Whittle index in a family of retirement problems. In Sonin [Statist. Probab. Lett. 78 (12,1) (2008), pp. 1526–1533], these three problems and these three indices were generalized in such a way that it became possible to use the state elimination algorithm [Sonin, Math. Meth of Oper. Res. (1999), pp. 111–123] to calculate this common generalized GI α. The main goal of this note is to demonstrate that the equality of these (generalized) indices is a special case of a more general relation between three simple abstract optimization problems.