Comparing two Models on Preference and Preference Change Fenrong Liu February 28, 2007 1 Some general points about preference There are different notions of preference, expressed in various ways in natural language. It occurs in many research areas as well. • Typically, we use preference to draw comparison between two things explicitly. • Depending on the real situations, things under comparison can be possible states of affairs, objects, actions, means, and so on, as listed in [Wri63]. • Logical modelling is an abstraction, we can only focus some aspects. It is dangerous to always try to match the formal preference with our intuitions of preference. [Hal57] described several aspects of preference usually ignored. In the following we are going to compare the recent two approaches of modelling preference and preference change, proposed in [BL07] (R-preference for short) and in [JL06] (C-preference for short), respectively. We start with reviewing the basics. Then we will discuss several issues, to show how these two ways of modelling can benefit from each other. 2 Review on R-Preference: the basics 2.1 Basic static system Definition 2.1 (model) An epistemic preference model is a tuple M=(S, {∼ i | i ∈ I }, { i | i ∈ I }, V), S, ∼ i and V are standard,  i is a reflexive and transitive relation over the worlds. We read s  i t as ‘t is at least as good as s’ for agent i. Intuitively, worlds stand for situations that are compared. Definition 2.2 (language) Take a set of propositional variables P and a set of agents I , with p ranging over P and i over I . The epistemic preference language is given by: ϕ ::= ⊥| p |¬ϕ | ϕ ∧ ψ |〈K〉 i ϕ |〈pref 〉 i ϕ | Eϕ. 〈pref 〉 i ϕ says that some worlds which the agent considers as least as good as the current one satisfy ϕ. E is an auxiliary existential modality. U is the dual of E. Definition 2.3 (truth conditions) We define M,s | = ϕ (formula ϕ is true in M at s) by induction on ϕ, here are two new clauses: 1. M,s | = 〈pref 〉 i ϕ iff for some t : s  i t and M,t | = ϕ 2. M,s | = Eϕ iff for some t: M,t | = ϕ. 1