Controlling tail risk in randomized algorithms has received increasing attention. A recently introduced approach is to impose tail constraints that limit the probability of poor outcomes. We study the randomized online bidding problem under such tail constraints. When the tail constraints require zero probability of exceeding the prescribed thresholds, we determine the optimal expected competitive ratio. For general tail constraints that allow positive exceedance probabilities, we derive a parameter-dependent upper bound on the optimal expected competitive ratio. This work builds on the Master's thesis of Royce Kraakman, which initiated our study of tail constraints for online bidding. The optimality result established in the present paper, in particular the matching lower bound for pure tail constraints, was obtained subsequently and is not contained in the thesis. After becoming aware of independent related work by Basiak et al. on pure tail constraints, we decided to make this preliminary version publicly available while the manuscript is still under development. Some material from the thesis has not yet been incorporated into the present version.

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