Sequential Project Selection (Multi-Armed Bandits) and the Gittins Index

A direct proof is given of the optimality of the Gittins index policy, and a related identity demonstrated for the loss function. Especial attention is paid to the case when new projects also arrive in a statistically homogeneous stream. A number of general results are obtained, of which those derived by J.M. Harrison, for example, are shown to be a special case.