A New Approach for Resolving Conflicts in Actionable Behavioral Rules

Table 1

Concepts and definitions.

Concept	Definition	Explanations
Behavioral information system (BIS)	𝐼=𝑂,𝑜^∗,𝐴,𝐷,𝜌	𝑂: a finite nonempty set of observations; 𝑜^∗∈𝑂: the projected next observation; 𝐴: a finite nonempty set of attributes; 𝐴=𝐴en∪𝐴be; 𝐷=𝑎∈𝐴𝐷𝑎 (𝐷𝑎: value domain of attribute 𝑎); 𝜌: 𝑂×𝐴→𝐷: a function associating each observation with a set of attribute values
Action	𝑡=𝑎,𝑣𝑓,𝑣𝑡	𝑎∈𝐴en; 𝑣𝑓=𝜌𝑜^∗,𝑎; 𝑣𝑡∈𝐷𝑎; If 𝑣𝑓≠𝑣𝑡, 𝑡 = standard action; else, 𝑡 = nonstandard action.; If the value of 𝑎 is changed from 𝑣𝑓 to 𝑣𝑡, 𝑡=𝑎,𝑣𝑓,𝑣𝑡 holds.
Effect	𝑒=𝑎,𝑣𝑓,𝑣𝑡	𝑎∈𝐴be; 𝑣𝑓=𝜌𝑜^∗,𝑎; 𝑣𝑡∈𝐷𝑎
Actionable behavioral rule (ABR)	𝑟=(𝑆,𝐶)	𝑆: an action set; 𝐶: a finite nonempty set of effect-probabilities; |𝐶|=𝑎∈𝐴be𝐷𝑎; 𝑟=(𝑆,𝐶): if 𝑆 holds, each effect-probability in 𝐶 will take place
Expected utility (EU)	util(𝑟)=𝑡∈𝑆util(𝑡)+𝑒𝑝∈𝐶util𝑒𝑝⋅𝑒⋅𝑒𝑝⋅𝑝	util(𝑡): utilities of action 𝑡; util(𝑒𝑝⋅𝑒): utilities of effect 𝑒𝑝⋅𝑒