We provide a short and elementary proof of the Gittins index theorem for the multi-armed bandit problem, for the case where each bandit is modeled as a finite-state semi-Markov process. We also indicate how this proof can be extended to the branching bandits and Klimov problems.
"A Short Proof of the Gittins Index Theorem." Ann. Appl. Probab. 4 (1) 194 - 199, February, 1994. https://doi.org/10.1214/aoap/1177005207