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Competition-based approach to controlling the false discovery rate (FDR) recently rose to prominence when, generalizing it to sequential hypothesis testing, Barber and Cand\`es used it as part of their knockoff-filter. Control of the FDR implies that the, arguably more important, false discovery proportion is only controlled in an average sense. We present TDC-SB and TDC-UB that provide upper prediction bounds on the FDP in the list of discoveries generated when controlling the FDR using competition. Using simulated and real data we show that, overall, our new procedures offer significantly tighter upper bounds than ones obtained using the recently published approach of Katsevich and Ramdas, even when the latter is further improved using the interpolation concept of Goeman et al., Comment: The original version of this paper appeared as arxiv:2011.11939v1. That version was split into two: one branch continuing as v2 & v3 of that original submission, and the other branch is now added here as a new submission

Recently, Barber and Cand\`es laid the theoretical foundation for a general framework for false discovery rate (FDR) control based on the notion of "knockoffs." A closely related FDR control methodology has long been employed in the analysis of mass spectrometry data, referred to there as "target-decoy competition" (TDC). However, any approach that aims to control the FDR, which is defined as the expected value of the false discovery proportion (FDP), suffers from a problem. Specifically, even when successfully controlling the FDR at level $\alpha$, the FDP in the list of discoveries can significantly exceed $\alpha$. We offer FDP-SD, a new procedure that rigorously controls the FDP in the competition (knockoff / TDC) setup by guaranteeing that the FDP is bounded by $\alpha$ at any desired confidence level. Compared with the just-published general framework of Katsevich and Ramdas, FDP-SD generally delivers more power and often substantially so in simulated as well as real data., Comment: This revision focuses only on FDP-SD described in the original submission. A later submission will further develop the procedures for simultaneous bounds on the FDP