This study examines the processing of the implicature of “might” (NOT must). The literature on implicatures contains both studies that suggest rapid computation of scalar implicatures, and studies that provide evidence for extra processing costs in generating them. The present study extends existing work by comparing “might” to “must”, and by adapting a paradigm that integrates experimental sentences into a natural discourse within a game. The experiment employed the visual world paradigm, using a guessing game with a confederate: in critical trials participants’ eye movements were recorded while they listened to utterances (guesses) made by a confederate. Our results show a delay in incorporating the ‘not must’ implicature of “might”, which is comparable in size to previous studies finding delays in implicature computation. Hence our results provide further support for the notion that implicatures incur processing cost, based on different implicature triggers and using an experimental paradigm based on natural dialogue.