  The way that people make choices or exhibit preferences can be strongly
affected by the set of available alternatives, often called the choice set.
Furthermore, there are usually heterogeneous preferences, either at an
individual level within small groups or within sub-populations of large groups.
Given the availability of choice data, there are now many models that capture
this behavior in order to make effective predictions--however, there is little
work in understanding how directly changing the choice set can be used to
influence the preferences of a collection of decision-makers. Here, we use
discrete choice modeling to develop an optimization framework of such
interventions for several problems of group influence, namely maximizing
agreement or disagreement and promoting a particular choice. We show that these
problems are NP-hard in general, but imposing restrictions reveals a
fundamental boundary: promoting a choice can be easier than encouraging
consensus or sowing discord. We design approximation algorithms for the hard
problems and show that they work well on real-world choice data.
