This paper gives tight logarithmic lower bounds on the solo step complexity of leader election in an asynchronous shared-memory model with single-writer multi-reader (SWMR) registers, for randomized obstruction-free algorithms. The approach extends to lower bounds for randomized obstruction-free algorithms using multi-writer registers under bounded write concurrency, showing a trade-off between the solo step complexity of a leader election algorithm, and the worst-case contention incurred by a processor in an execution.