I'd like to chime in with an opinion that is very different from everyone here.
I had a 4 year old laptop and like you got tired of the choppiness and low settings. Like you, I wanted to spend around $1,000.
I got a Lenovo 13 Yoga with an i5 with integrated graphics. I am running Sim City 2013 with 1600x900 resolution and high with consistent FPS above 30.
I have 8GB of RAM, but more than happy with the performance. As a bonus you get a great touchscreen ultrabook. Of course, if you want to play games more intense than Sim City, you may struggle.