  User engagement is a critical metric for evaluating the quality of
open-domain dialogue systems. Prior work has focused on conversation-level
engagement by using heuristically constructed features such as the number of
turns and the total time of the conversation. In this paper, we investigate the
possibility and efficacy of estimating utterance-level engagement and define a
novel metric, {\em predictive engagement}, for automatic evaluation of
open-domain dialogue systems. Our experiments demonstrate that (1) human
annotators have high agreement on assessing utterance-level engagement scores;
(2) conversation-level engagement scores can be predicted from properly
aggregated utterance-level engagement scores. Furthermore, we show that the
utterance-level engagement scores can be learned from data. These scores can
improve automatic evaluation metrics for open-domain dialogue systems, as shown
by correlation with human judgements. This suggests that predictive engagement
can be used as a real-time feedback for training better dialogue models.
