This paper studies learning in binary state models. For twice smooth policies, we precisely characterize the behavior of the implied value function in a general class of experimentation problems near the extreme beliefs 0 and 1. In particular, we show that under general assumptions v''(x) exists near x=0 and x=1, and satisfies v''(x)~ c₀ x^-a near x=0 and v''(x)~ c₁(1-x)^-b near x=1. Notably, we provide an exact formula for the exponents 0<a,b<1, and bound the coefficients c₀,c₁>0. An example illustrates how this result can be useful in applications.
 

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