Serious concerns have been raised about the role of `socialbots' in manipulating public opinion and influencing the outcome of elections by retweeting partisan content to increase its reach. Here we analyze the role and influence of socialbots on Twitter by determining how they contribute to retweet diffusions. We collect a large dataset of tweets during the 1st U.S. presidential debate in 2016 and we analyze its 1.5 million users from three perspectives: user influence, political behavior (partisanship and engagement) and botness. First, we define a measure of user influence based on the user's active contributions to information diffusions, i.e. their tweets and retweets. Given that Twitter does not expose the retweet structure -- it associates all retweets with the original tweet -- we model the latent diffusion structure using only tweet time and user features, and we implement a scalable novel approach to estimate influence over all possible unfoldings. Next, we use partisan hashtag analysis to quantify user political polarization and engagement. Finally, we use the BotOrNot API to measure user botness (the likelihood of being a bot). We build a two-dimensional "polarization map" that allows for a nuanced analysis of the interplay between botness, partisanship and influence. We find that not only are socialbots more active on Twitter -- starting more retweet cascades and retweeting more -- but they are 2.5 times more influential than humans, and more politically engaged. Moreover, pro-Republican bots are both more influential and more politically engaged than their pro-Democrat counterparts. However we caution against blanket statements that software designed to appear human dominates politics-related activity on Twitter. Firstly, it is known that accounts controlled by teams of humans (e.g. organizational accounts) are often identified as bots. Secondly, we find that many highly influential Twitter users are in fact pro-Democrat and that most pro-Republican users are mid-influential and likely to be human (low botness).