  Counterfactual explanations are usually obtained by identifying the smallest
change made to an input to change a prediction made by a fixed model (hereafter
called sparse methods). Recent work, however, has revitalized an old insight:
there often does not exist one superior solution to a prediction problem with
respect to commonly used measures of interest (e.g. error rate). In fact, often
multiple different classifiers give almost equal solutions. This phenomenon is
known as predictive multiplicity (Breiman, 2001; Marx et al., 2019). In this
work, we derive a general upper bound for the costs of counterfactual
explanations under predictive multiplicity. Most notably, it depends on a
discrepancy notion between two classifiers, which describes how differently
they treat negatively predicted individuals. We then compare sparse and data
support approaches empirically on real-world data. The results show that data
support methods are more robust to multiplicity of different models. At the
same time, we show that those methods have provably higher cost of generating
counterfactual explanations under one fixed model. In summary, our theoretical
and empiricaln results challenge the commonly held view that counterfactual
recommendations should be sparse in general.
