In the best choice problem with random arrivals, an unknown number $n$ of rankable items arrive at times sampled from the uniform distribution. As is well known, a real-time player can ensure stopping at the overall best item with probability at least $1/e$, by waiting until time $1/e$ then selecting the first relatively best item to appear (if any). This paper discusses the issue of dominance in a wide class of stopping strategies of best choice, and argues that in fact the player faces a trade-off between success probabilities for various values of $n$. We argue that the $1/e$-strategy is not a unique minimax strategy and that it can be improved in various ways.