Strategy-proof social choice with exogenous indifference classes

There are many situations where the indifference classes of the agents’ preferences are exogenously given. We present a framework for dealing with such situations and find that the cardinality of the set SS, which consists of alternatives that every agent ranks strictly, is critical to the construction of a strategy-proof social choice function; if |S|≥3|S|≥3, then strategy-proofness implies dictatorship, if |S|=2|S|=2, then strategy-proofness and efficiency imply dictatorship, and if |S|<2|S|<2, then strategy-proofness and efficiency do not necessarily imply dictatorship.