The transition to a low-carbon power system requires growing the share of generation from (intermittent) renewables while ensuring security of supply. Policymakers and economists increasingly see a capacity mechanism as a way to deal with this challenge. Yet this raises new concerns about the exercise of market power by large players via the capacity auction. We present a new modelling approach that captures such strategic behaviour together with a set of ex ante empirical estimates for the new Irish electricity market design (I-SEM) in which a single firm controls 44% of generation capacity (excluding wind). We find significant costs of strategic behaviour, even with new entry: In our baseline scenarios, procurement costs in the capacity auction are around 150-400 million EUR (or 40-100%) above the competitive least-cost solution. From a policy perspective, we also examine how market power can be measured and mitigated through auction design.