Ok, here is a decent build.
CPU: Intel Core i5-4570 3.2GHz Quad-Core Processor ($194.99 @ NCIX US)
Motherboard: MSI Z87-G43 ATX LGA1150 Motherboard ($119.49 @ Newegg)
Memory: Kingston Black 8GB (2 x 4GB) DDR3-1600 Memory ($69.29 @ Amazon)
Storage: Seagate Barracuda 1TB 3.5" 7200RPM Internal Hard Drive ($59.99 @ NCIX US)
Video Card: MSI Radeon R9 270 2GB Video Card ($185.66 @ Newegg)
Case: NZXT Guardian 921 RB ATX Mid Tower Case ($69.99 @ Mwave)
Power Supply: Cooler Master i500 500W 80+ Bronze Certified ATX Power Supply ($54.99 @ Newegg)
Optical Drive: Samsung SH-224DB/BEBE DVD/CD Writer ($16.98 @ OutletPC)
Operating System: Microsoft Windows 8.1 - OEM (64-bit) ($94.98 @ OutletPC)
Total: $866.36 ($846 after rebate)
(Prices include shipping, taxes, and discounts when available.)
URL: http://pcpartpicker.com/p/2tlYv
If you really want to stay under the budget, then you will need to get a worse GPU. You could get a cheaper computer case, but to save like $40 it would have to be the cheapest of the cheapest cases.
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Here are the graphics cards:
MSI Radeon HD 7790 1GB
Model: MSI R7790-1GD5/OC
GPU Score: 3,007
Pros: Best value graphics card. And supports AMD's "Mantle".
$115
EVGA Nvidia GeForce GTX 650 Ti Boost 2GB
Model: EVGA 01G-P4-3655-KR
GPU Score: 3.572
Pros: Nvidia's good quality and driver support. Does not support AMD's "Mantle".
$135
Gigabyte R7 260X 2GB
Model: GV-R726XOC-2GD
GPU Score: 2,828 (The 260X is the same card as the 7790, but with more ram and a audio chip)
Pros: Comes with built in audio ("True Audio"), which takes stress off of the CPU. And supports AMD's "Mantle".
$140
Sapphire Radeon HD 7850 1GB
Model: Sapphire 11200-16-20G
GPU Score: 3,710
Pros: Higher memory bus. More graphical cores. Supports AMD's "Mantle". And step up from the 7790. And slightly better than a GTX 660.
$160
MSI R9 270 (None X) 2GB
Model: R9 270 GAMING 2G (V305)
GPU Score: 4,313
Pros: Better than all cards that are listed. Supports AMD's "Mantle".
$185
MSI Nvidia GeForce GTX 660
Model: MSI N660-2GD5/OC
GPU Score: 4,109
Pros: Nvidia's good quality and driver support. Higher performance than the 650 ti boost.
$190
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Sources:
GPU prices
http://pcpartpicker.com/parts/video-card/#c=132,114,131,81,149,154&sort=a7&m=14,18,27,41
GPU score
http://videocardbenchmark.net/high_end_gpus.html
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Ok, here they are.
Let me explain this "Mantle". It is a new API used to develop games. This API lets the developer to directly make the game for the hardware. Thus will increase performance by a lot.
And all of the cards that I have listed above, should play BF4 on high, with more than 30 FPS.
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And don't go with a AMD processor. AMD does great with their graphics cards, but they just fail on optimizing their cores to work with each other.
Just look at these benchmarks. See the "i3-3220", its a dual core. That means that it should have 4x less power than the AMD 8 cores (FX-8xxx).
http://static.techspot.com/articles-info/712/bench/CPU_03.png
http://static.techspot.com/articles-info/670/bench/CPU_01.png
http://static.techspot.com/articles-info/608/bench/CPU_01.png
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Hope this helps
-Snake