Non-cooperative two-player games and linear bi-objective optimization problems

Two distinct disciplines deal with apparently common areas of finding solutions for conflicting parties and goals: the non-cooperative game theory and the traditional multi-objective optimization theory. A solution in a multiobjective optimization problem is to be a Pareto-optimal point and not an equilibrium point, because there is only a single decision-maker who makes sovereign decisions based on his preferences. When the objective holders are humans with independent interactions, as in most industrial engineering applications, a valid solution must then be a Nash equilibrium point first, and then a Pareto-optimal point, if possible. In this paper, (1) the relation between non-cooperative game theory and multi-objective optimization is established, (2) the notion of "induced game" is proposed, and (3) a new framework for finding a so-called Pareto-Optimal Equilibrium solution is presented. We present some illustrative examples to show that the proposed framework works well for linear bi-objective problems with known z-space.