Strictly Pareto Inefficient Nash Equilibria

Abstract

We find `strict' Pareto inefficiency of Nash equilibria (NE) in prisoners' dilemma and Braess paradox, wherein the utilities of `all' players degrade in NE. The strict Pareto inefficiency is a narrower concept than (usually mentioned) Pareto inefficiency. In this paper, we present a measure that shows the magnitude of strict Pareto inefficiency, abbreviated as MoS. MoS distinguishes strict Pareto inefficiency of a state (an allocation/a strategy profile), say, a Nash equilibrium (NE). We present examples wherein the widely-used measure of ineffectiveness based on the social optimality, like price of anarchy (PoA), does not always distinguish strict Pareto inefficiency whereas MoS does. Furthermore, we show that, if there exists a Pareto optimum that is proportional to a state, MoS of the state is obtained as the constant of proportionality between the Pareto optimum and the state. Then, if the Pareto optimum is socially optimal, PoA of the state is identical to MoS of the state. We show that the magnitude of strict Pareto inefficiency, MoS, of NE can increase without bounds in a networking game, even though the network has a finite amount of resources and a small number of noncooperative players.