Playing back 1080p and 720p videos (same bitrate) on a 720p display Suppose a high quality, uncompressed video is re-encoded twice, with almost identical encoding settings at a relatively low bitrate of 4mbps, but at different resolutions, first at 1080p, and then at 720p. 

So we now have two videos: One is 1080p and the other one is 720p. Both have identical bitrates. 

I know that 4mbps is considered a low bitrate for 1080p, and the video will look bad on a full-HD display due to the compression artifacts. 

However, during playback on a 720p display, the 1080p video will obviously be downscaled to 720p. 

My question is: On the 720p display, will the 1080p video (which is being downscaled to 720p in real time during playback) look as good as the 720p video (which has the same bitrate)? Or will the 720p video (which was downscaled from 1080p to 720p by the encoder) look better? Or will both of them look the same?

I would run some tests to see this for myself, but my system isn't powerful enough for 1080p encoding. I'm looking for some expert views on this.
When you ask how something will 'look' you're in the realm of the subjective. Things 'look' different to experienced professionals than they might to the average viewer.

But still, the 1080p video will not be as good no matter what you view it on. When you encode at a low bit rate you aren't removing pixels, you're mostly removing high frequencies. Doing either will reduce the amount of information, but in different ways and with different visible effects and artifacts.

The information you lose with very lossy encoding doesn't lie in the spaces that inevitably are discarded when you downscale. You'd lose in both domains. But how that 'looks' depends to some degree on who's looking.