Game directional distance function in meta-frontier data envelopment analysis

•Game theory is introduced to balance the relationship between group frontier and meta-frontier.•Endogenous technique is used to determine direction in meta-frontier DDF framework.•The game process of game meta-frontier DDF is proved to be convergent.•The directions relative to different frontiers are proved to be uniform.•The results of new method are proved to achieve pareto optimality. Meta-frontier directional distance function (DDF) is an important method to evaluate the efficiency of decision-making units (DMUs) with technical heterogeneity. However, this method can only use exogenous technology to determine the direction, because the application of endogenous technique has the theoretical dilemma that DMUs have two frontiers with different data characteristics. According to this dilemma, game theory is introduced to balance the relationship between group-frontier and meta-frontier, and then their data characteristics can be unified. Sequentially, Stackelberg game and non-cooperative game are gradually applied to construct the meta-frontier DDF with endogenous technique, and their convergence, uniformity and optimality are proved; while their advantages and relationship are discussed, respectively. Compared with traditional methods, the new meta-frontier DDF methods uses an endogenous technique to determine the unified improvement direction of DMUs based on different frontiers, and then the results can be more objective and reasonable. Finally, an empirical example is used to illustrate the effectiveness of these new methods.