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Non-local games are an important part of quantum information processing.Recently there has been an increased interest in generalizing non-local gamesbeyond the basic setup by considering games with multiple parties and/or withlarge alphabet inputs and outputs. In this paper we consider anotherinteresting generalization -- games with non-uniform inputs. Here we derive atight upper bound for the classical winning probability for a specific familyof non-local games with non-uniform input distribution, known as$\mathrm{CHSH}_q(p)$ which was introduced recently in the context ofrelativistic bit-commitment protocols by [Chakraborty et. al., PRL 115, 250501,2015].

Tight bound on the classical value of generalized Clauser-Horne-Shimony-Holt games