Matching-adjusted indirect comparison (MAIC) has been increasingly employed in health technology assessments (HTA). By reweighting subjects from a trial with individual participant data (IPD) to match the covariate summary statistics of another trial with only aggregate data (AgD), MAIC facilitates the estimation of a treatment effect defined with respect to the AgD trial population. This manuscript introduces a new class of methods, termed arbitrated indirect treatment comparisons, designed to address the ``MAIC paradox'' -- a phenomenon highlighted by Jiang et al.~(2025). The MAIC paradox arises when different sponsors, analyzing the same data, reach conflicting conclusions regarding which treatment is more effective. The underlying issue is that each sponsor implicitly targets a different population. To resolve this inconsistency, the proposed methods focus on estimating treatment effects in a common target population, specifically chosen to be the overlap population.