Magnitude of inefficiency

Abstract

Various types of inefficiencies of a system state like the Nash equilibrium (NE) exist, such as social inefficiency, Pareto inefficiency, etc. (A system state is inefficient if it is inferior to another realizable state.) Firstly, this article presents a general procedure to obtain each inefficiency measure. The procedure brings as each inefficiency measure, the maximum degree of corresponding inferiority of the state to some other. We examine the procedure in the game-theory context. Vastly-many people use the social-inefficiency measure (represented by the price of anarchy [PoA]). However, it cannot always serve as a Pareto-inefficiency measure. Contrarily, the Pareto-inefficiency measures are yet to establish. Secondly, we follow the procedure (to which PoA also conforms) and obtain Pareto-inefficiency measures. We confirm that they distinguish Pareto inefficiencies that PoA cannot always distinguish. We show a fixed relation between the values of the measures. Further, if a state is proportional to a Pareto-optimal state, proposed measures of strict Pareto inefficiency and Pareto inefficiency behave in identical and straightforward ways. Then, their value is the proportionality constant. Using examples, we examine the measures.