Almost perfect strategies for projection games are approximately tracial

Projection games constitute an important class of nonlocal games where, for any answer from the first player, there is a unique correct answer for the second player. This class of games captures nonlocal games arising from constraint satisfaction problems, oracularisations, and unique games. However, due to the asymmetry between the players, projection games are in general not synchronous, and therefore the powerful results constraining the structure of almost perfect strategies for synchronous games do not apply. In this work, we adapt results of Marrakchi and de la Salle for synchronous games to show that, in both the quantum and commuting-operator models, any strategy that wins with probability $1-\varepsilon$ in a projection game gives rise to a tracial strategy that wins with probability $1-O((L\varepsilon)^{1/4})$, where $L$ is the inverse of the minimal conditional probability of a question for the second player being sampled given a question to the first. For constraint system games, this strengthens the rounding result of Paddock by eliminating the dependence on number of constraints and improving the dependence on constraint size, while also generalising to the commuting-operator setting.