Steering, incompatibility, and Bell inequality violations in a class of probabilistic theories

We show that connections between a degree of incompatibility of pairs of observables and the strength of violations of Bell's inequality found in recent investigations can be extended to a general class of probabilistic physical models. It turns out that the property of universal uniform steering is sufficient for the saturation of a generalised Tsirelson bound, corresponding to maximal violations of Bell's inequality. It is also found that a limited form of steering is still available and sufficient for such saturation in some state spaces where universal uniform steering is not given. The techniques developed here are applied to the class of regular polygon state spaces, giving a strengthening of known results. However, we also find indications that the link between incompatibility and Bell violation may be more complex than originally envisaged.