Parametrized preference structures and some geometrical interpretation

Order structures such as linear orders, semiorders and interval orders are often used to model preferences in decision-making problems. In this paper we introduce a family of preference structures where the mutual indifference threshold belongs to a specific family parametrized by extended reals α. This family includes interval orders (α=1), tangent circle orders (α=0) and a new preference structure called ‘diamond order’ (α=–∞). All these preference relations present an asymmetric part which is shown to be always quasi-transitive and to be transitive for α⩾1. Diamond orders present ‘forbidden configurations’ which can occur in the case of tangent circle orders.