Not exactly a budget build.
The real question comes down to how many threads your games need. ryzen 1600 will have 12 and the 1800X will have 16.
Truth is that most games can effectively use only 2-3 threads. If your games are multiplayer, then more threads are better.
The performance of the master thread is most important. Either the 1600 or the 1800X can be overclocked to about the same 3.9 performance level.
Of the two options you listed, I would opt for the stronger GTX1080ti and the 1600.
That said, if this is a purely gaming pc, I would pick a I5-8600K or the I7-8700K with 6 or 12 threads respectively.
Either can be overclocked to around 4.9.