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

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