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 .