  Incentive compatibility (IC) is one of the most fundamental properties of an
auction mechanism, including those used for online advertising. Recent methods
by Feng et al. and Lahaie et al. show that counterfactual runs of the auction
mechanism with different bids can be used to determine whether an auction is
IC. In this paper we show that a similar result can be obtained by looking at
the advertisers' envy, which can be computed with one single execution of the
auction. We introduce two metrics to evaluate the incentive-compatibility of an
auction: IC-Regret and IC-Envy. For position auction environments, we show that
for a large class of pricing schemes (which includes e.g. VCG and GSP), IC-Envy
$\ge$ IC-Regret (and IC-Envy = IC-Regret when bids are distinct). We consider
non-separable discounts in the Ad Types environment of Colini-Baldeschi et al.
where we show that for a generalization of GSP also IC-Envy $\ge$ IC-Regret.
Our final theoretical result is that in all these settings IC-Envy be used to
bound the loss in social welfare due advertiser misreports.
  Finally, we show that IC-Envy is useful as a feature to predict IC-Regret in
auction environments beyond the ones for which we show theoretical results. In
particular, using IC-Envy yields better results than training models using only
price and value features.
