Let X n ,…, X 1 be independent, identically distributed (i.i.d.) random variables with distribution function F . A statistician, knowing F , observes the X values sequentially and is given two chances to choose X s using stopping rules. The statistician's goal is to stop at a value of X as small as possible. Let equal the expectation of the smaller of the two values chosen by the statistician when proceeding optimally. We obtain the asymptotic behaviour of the sequence for a large class of F s belonging to the domain of attraction (for the minimum) 𝒟( G α ), where G α ( x ) = [1 - exp(- x α )]1( x ≥ 0) (with 1(·) the indicator function). The results are compared with those for the asymptotic behaviour of the classical one-choice value sequence , as well as with the ‘prophet value’ sequence