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We establish several strong equivalences of synchronous non-local games, in the sense that the corresponding game algebras are *-isomorphic. We first show that the game algebra of any synchronous game on n inputs and k outputs is *-isomorphic to the game algebra of an associated bisynchronous game on nk inputs and nk outputs. As a result, we show that there are bisynchronous games with equal question and answer sets, whose optimal strategies only exist in the quantum commuting model, and not in<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2109.04859v1">arXiv:2109.04859v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/hp4444m5wbejjarbrizki3oh34">fatcat:hp4444m5wbejjarbrizki3oh34</a> </span>
more »... the quantum approximate model. Moreover, we exhibit a bisynchronous game with 20 questions and 20 answers that has a non-zero game algebra, but no winning commuting strategy, resolving a problem of V.I. Paulsen and M. Rahaman. We also exhibit a *-isomorphism between any synchronous game algebra with n questions and k>3 answers and a synchronous game algebra with n(k-2) questions and 3 answers.
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