This paper considers a problem for scheduling jobs on two identical parallel machines, the aim was to minimize two criteria in particular, makespan and total flow time.In order to solve this problem, two approaches were considered.A mechanism was proposed as an approach to solve this type of problem with a setting of a 2-player non-cooperative game, under the framework of a 2x2 non-sum zero matrix; each player looking after one of the criteria suggested in the scheduling problem.On the other hand, a Genetic Algorithm, known as Strength Pareto Evolutionary Algorithm (SPEA), was applied to the problem.The comparison between both approaches suggests a complementarity among rational agents approach models and machine enforced solution approaches.The resulting Pareto Front set of points were plotted and curves were compared, showing promising results for game theoretic applications to scheduling under multiple objectives.