Even custom-made the cheapest I could make that I would actually call a gaming computer is $509 using the integrated graphics of the AMD Vega CPU, which I have heard is quite capable for 1080p.
PCPartPicker part list:
https://pcpartpicker.com/list/tzYZLJ
Price breakdown by merchant: https://pcpartpicker.com/list/tzYZLJ/by_merchant/
CPU: AMD - Ryzen 3 2200G 3.5GHz Quad-Core Processor ($99.99 @ Newegg)
Motherboard: *ASRock - A320M Micro ATX AM4 Motherboard ($49.74 @ Newegg)
Memory: *GeIL - SUPER LUCE 8GB (2 x 4GB) DDR4-2400 Memory ($79.99 @ Newegg Business)
Storage: *SanDisk - SSD PLUS 240GB 2.5" Solid State Drive ($69.99 @ Newegg)
Case: *Rosewill - FBM-01 MicroATX Mini Tower Case ($19.99 @ Newegg)
Power Supply: *Corsair - TXM Gold 650W 80+ Gold Certified Semi-Modular ATX Power Supply ($89.99 @ Newegg)
Operating System: Microsoft - Windows 10 Home OEM 64-bit ($99.99 @ Newegg)
Total: $509.68
Prices include shipping, taxes, and discounts when available
*Lowest price parts chosen from parametric criteria
Generated by PCPartPicker 2018-03-02 22:20 EST-0500
Adding a Geforce 1050Ti adds about $200 more bringing the price to $709
I'd expect similar builds from manufacturers to be in the $800 range without the graphics card, and $1000 with the Geforce 1050Ti