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We address the weakly supervised video highlight detection problem for learning to detect segments that are more attractive in training videos given their video event label but without expensive supervision of manually annotating highlight segments. While manually averting localizing highlight segments, weakly supervised modeling is challenging, as a video in our daily life could contain highlight segments with multiple event types, e.g., skiing and surfing. In this work, we propose casting weakly supervised video highlight detection modeling for a given specific event as a multiple instance ranking network (MINI-Net) learning. We consider each video as a bag of segments, and therefore, the proposed MINI-Net learns to enforce a higher highlight score for a positive bag that contains highlight segments of a specific event than those for negative bags that are irrelevant. In particular, we form a max-max ranking loss to acquire a reliable relative comparison between the most likely positive segment instance and the hardest negative segment instance. With this max-max ranking loss, our MINI-Net effectively leverages all segment information to acquire a more distinct video feature representation for localizing the highlight segments of a specific event in a video. The extensive experimental results on three challenging public benchmarks clearly validate the efficacy of our multiple instance ranking approach for solving the problem.