In a recent paper, the authors have shown how to compute analytically the Bayesian relativ- ities for a Bonus-Malus scale superimposed on a segmented tariff. However, the percentages associated with the levels of such a scale may exhibit an abrupt rise or drop from one level to another. For commercial reasons, it is sometimes preferable that each level inflicts the same relative penalty. In this paper we will show how to obtain such linear relativities for a given Bonus-Malus System. The model allows for a priori ratemaking. It is applied to a real-life portfolio.