High-profile modeling studies often project that large-scale win-win solutions are widely available, but practitioners are often skeptical of win-win narratives, due to real-world complexity. Here, we bridge this divide by showing mathematically why complexity makes win-wins elusive. We provide a general proof that, under uncertainty, the probability a manager should assign to win-win outcomes (here meaning Pareto improvements) existing strictly decreases in: the number of objectives, the number of stakeholders, and the number of constraints. We also show that a measure of tradeoff severity increases in the number of objectives, and approaches a limit unaffected by tradeoff surface curvature. This is important because most empirically estimated two-dimensional tradeoff surfaces are concave—77%, we show in a meta-analysis. Concave tradeoffs are less severe, but our theory suggests this difference dissipates in higher dimensions. Our results provide precise intuition and quantitative guidance for interpreting implications of simple tradeoff studies for complex realities.