Synchronous non-local games are a special class of two-player games where the players must respond with identical answers, given identical questions. These games have deep ties to Connes’ embedding problem in operator algebras and (equivalently) the weak Tsirelson problem in quantum information theory. In this talk, we will look at some new results on certain classes of games that are equivalent, in the sense that their game algebras are $*$-isomorphic. Notably, we show that every synchronous game is equivalent to a synchronous game with $3$ outputs, and that every synchronous game is equivalent to a bisynchronous game with equal question and answer sets. This allows us to exhibit a synchronous game with $3$ outputs that has a winning strategy in the commuting operator framework, but no winning strategy in the approximate finite-dimensional framework. We also exhibit a bisynchronous game on $20$ inputs and $20$ outputs that has a non-zero game algebra, but no winning strategies in the commuting operator framework, answering (negatively) a problem posed by Vern Paulsen and Miza Rahaman.

This work was supported by an NSERC postdoctoral fellowship.