A Measure of Pareto Superiority? Hisao Kameda ∗ Abstract We first confirm the notions of Pareto optimality, superiority, and inefficiency. Then, we discuss a definition of the measure of Pareto superiority. keyword Pareto superiority/inferiority, Nash equilibrium, Wardrop equilibrium 1 Introduction The notions of Pareto optimality, superiority, and inefficiency have already been established. We first confirm the notions and their definitions. Then, we discuss a definition of the measure of Pareto superiority. 2 Pareto Optimality, and Superiority We consider a system consisting of a number of users. For each state of the system, each user has its own utility. Denote a combination of utilities of all users in a system S by U(S ) = (U 1 (S ), U 2 (S ),..., U n (S )). We consider only the cases where U i (S ) > 0 has a positive real value, for all i. Denote by R the set of positive real numbers. Thus, U(S ) ∈R n . [Achievable set of utilities]: Naturally, the set of achievable U(S ) does not cover all the elements of R n . [Pareto optimality and efficiency]: There may exist a state of the system where we cannot improve the utility of each user without decreasing the utility of some other user. This is called a Pareto optimum or efficient state. In general, there are infinitely many Pareto optimum states for a system. The set of Pareto optimum points forms the border (Pareto border) separating the set of achievable U(S ) from the set of unachievable U(S ). [Pareto superiority and inferiority]: Consider an arbitrary pair of two (achievable) states of the system, S a and S b : If U i (S a ) ≤ U i (S b ) for all i and U i (S a ) < U i (S b ), then S a is Pareto inferior to S b and S b is Pareto superior to S b . Define k i  U i (S b )/U i (S a ). Then, S b is Pareto superior to S a if and only if k i > 1 for some i and k j ≥ 1 for all other j. S b is Pareto inferior to S a if and only if k i < 1 for some i and k j ≤ 1 for all other j. We define strong Pareto superiority and inferiority. That is, S b is strongly Pareto superior to S a iff k i > 1 for all i. S b is strongly Pareto inferior to S a iff k i < 1 for all i. A state to which some other state is (strongly) Pareto superior is (strongly) Pareto inefficient. [A measure of Pareto superiority/inferiority]: As we see in the above the definition of Pareto superior- ity/inferiority has already given and well accepted. It seems, however, that the measure of the degree of Pareto superiority/inferiority has not been generally accepted. The measure is necessary, e.g., for defining the degree of the inefficiency of Nash equilibria. ∗ H. Kameda is with the Graduate School of Systems and Information Engineering, University of Tsukuba, Tsukuba Science City, Ibaraki 305-8573, Japan. E-mail: kameda@osdp.cs.tsukuba.ac.jp 1