Paper is submitted to AAMAS 2027 for review. abstract A scalar rating, such as an Elo score or a leaderboard rank, presumes that performance is one-dimensional. In certain markets that clear at a shared price this presumption fails, and it fails by construction. When both competitors sell into the same downward-sloping demand, each payoff carries a large common-value component that is shared with the opponent. Only a small competitive residual decides who wins. We establish t
