Abstract: Competitive games of complete information famously always have Nash equilibria, but it is well-known that correlation ("advice") can yield new equilibria, sometimes with preferable collective properties (social welfare, fairness, ...). While it is known that quantum correlations in the form of entanglement do not imply further correlated equilibria, this situation changes when going to so-called Bayesian games of incomplete information, where each player has to react to a privately known type, while being ignorant about the type of the other players.  In fact, when all players in the game share the same payoff function, this reproduces the "nonlocal games" studied in quantum mechanics since Bell, in which case the CHSH inequality separates classical from quantum correlation and quantum from no-signalling. In the talk, I will review, how classical correlations, shared quantum states and no-signalling correlations can create a hierarchy of sets of classically correlated, quantum and belief-invariant equilibria of conflict-of-interest games, all containing the set of Nash equilibria. I will show a simple and easy-to-analyze construction of such games based on quantum pseudo-telepathy games, which show the quantum advantage of larger social welfare and of fairer distribution of payoff.  Based on work with V. Auletta, D. Ferraioli, A. Rai and G. Scarpa (arXiv:1605.07896), and with M. Cerda (BSc thesis, UAB 2021).