### Abstract

We propose Bell inequalities being generalizations of XOR games. In XOR games the pay-off function depends only on XOR of the outputs, i.e. on whether the outputs are correlated or not. We consider games with n outputs, and the pay-off function depends on n particularly chosen ways Alice and Bob outputs can be correlated. (this can be generalized to many parties too). Our games belong to the class of so-called "unique games". Various properties of those games are analysed. In particular, we show that they are algebraically violated by some no-signaling boxes. We also obtain some maximal classical values for particular examples of the games.