Admissible interval-valued monotone comparative statics methods applied in games with strategic complements

In many theories and applications with uncertainty, using intervals to characterize uncertainty is simple and operable. It is crucial to choose a proper order for an interval method. In general, a total order is superior to a partial order for those applications in which we have to make a final decision. Motivated by this idea, we generalized interval-valued monotone comparative statics (MCS) with a partial order to interval-valued MCS with a total order, to be more precise, with an admissible order. The generalization was not trival. We obtained a necessary and sufficient condition for MCS by a series of new concepts such as an interval-valued quasi-super-modular function and an interval-valued single crossing property with an admissible order. The same condition was only sufficient in the existing literature. Furthermore, we illustrated the efficiency of the interval-valued MCS with the lexicographical orders and XY-order, which are well-known admissible orders. Finally, we applied our results in interval games with strategic complements to get the monotonity of the best response correspondence for player $i$.