This paper is concerned with the problem of finding an experimental design for discrimination between two rival models and for model robustness that minimizing the maximum bias simultaneously in binary response experiments. The criterion for model discrimination is based on the $T$-optimality criterion proposed in Atkinson and Fedorov (1975), which maximizes the sum of squares of deviations between the two rival models while the criterion for model robustness is based on minimizing the maximum probability bias of the two rival models. In this paper we obtain the optimum designs satisfy the above two criteria for some commonly used rival models in binary response experiments such as the probit and logit models etc.