This paper proposes a new multinomial choice model which explicitly takes into account variation in choice sets across observations. The proposed varying choice set logit model relaxes the independence of irrelevant alternatives assumption by allowing the individual random utility function to directly depend on choice set types, and can be applied to a variety of data in which some individuals can only choose from a subset of the theoretically possible responses. Both frequentist and Bayesian simulation-based estimation procedures are developed using the Monte Carlo expectation-maximization algorithm and Markov chain Monte Carlo, respectively. The proposed model can be used to analyze survey data in partially contested multiparty elections in which some political parties do not run their candidates in every district. For illustration, I apply the proposed method to the 1996 Japanese general election, where none of the districts was contested by all of the six major parties.