In a projection game, the first player answers and the second player's correct response is uniquely determined. The asymmetry is built into the rules: for any answer from Alice, there is exactly one valid answer for Bob. This makes projection games structurally simpler than general nonlocal games — simpler rules, but not simpler behavior.
Culf proves that any near-winning strategy for a projection game must be approximately tracial. If the strategy wins with probability 1-ε, it can be transformed into a tracial strategy that wins with probability at least 1-O((Lε)^{1/4}). The operators the players use must approximately commute with trace — not because traciality is assumed, but because near-optimality forces it.
The result holds in both the quantum and commuting-operator frameworks. The guarantee is uniform: the dependence is on the game structure (L, the label count) and the winning deficit (ε), not on the number of constraints. This removes a previous limitation where the bounds degraded as the game grew larger.
The structural insight: projection games select for traciality. Any strategy that nearly wins must have the algebraic structure of a trace-preserving operation. The game's rules — the deterministic projection from Alice's answer to Bob's — constrain the space of winning strategies so tightly that only approximately tracial strategies survive. The algebraic property is not an assumption about the players' resources. It is a consequence of winning the game. Near-optimal play in a projection game reveals the algebraic structure of the operators involved, whether or not you intended to measure it.