On the Conjecture of Berry Regarding a Bernoulli Two-Armed Bandit.

In this paper, we study an independent Bernoulli two-armed bandit with unknown parameters ρ and λ , where ρ and λ have a pair of priori distributions such that d R (ρ) = C R ρ r 0 (1 − ρ) r 0 ′ d μ (ρ) , d L (λ) = C L λ l 0 (1 − λ) l 0 ′ d μ (λ) and μ is an arbitrary positive measure on [ 0 , 1 ] . Berry proposed the conjecture that, given a pair of priori distributions (R , L) of parameters ρ and λ , the arm with R is the current optimal choice if r 0 + r 0 ′ < l 0 + l 0 ′ and the expectation of ρ is not less than that of λ. We give an easily verifiable equivalent form of Berry's conjecture and use it to prove that Berry's conjecture holds when R and L are two-point distributions as well as when R and L are beta distributions and the number of trials N ≤ r 0 r 0 ′ + 1 . [ABSTRACT FROM AUTHOR]