Can a PC be built that compares in size, power, and price to the Alienware Alpha? [Serious]

I've been toying with build ideas for a low-powered, low price, and low sized gaming-capable PC to give to my mother-in-law for Christmas. My goal is to get something that's near the size of the Alienware Alpha with comparable performance at an, ideally, better price.

But I'm finding that more difficult than expected.

First of all, here's a current build idea as it stands:

PCPartPicker part list / Price breakdown by merchant

Type	Item	Price
CPU	AMD A10-7870K 3.9GHz Quad-Core Processor	$135.99 @ NCIX US
Motherboard	Gigabyte GA-F2A88XN-WIFI Mini ITX FM2+ Motherboard	$82.98 @ Newegg
Memory	PNY 8GB (2 x 4GB) DDR3-2133 Memory	$44.99 @ Amazon
Storage	Seagate 1TB 2.5" 5400RPM Hybrid Internal Hard Drive	$77.00 @ Amazon
Case	Inwin BQ656T.AD120TBL Mini ITX Desktop Case w/120W Power Supply	$57.99 @ SuperBiiz
Operating System	Microsoft Windows 8.1 (32/64-bit)	$89.88 @ OutletPC
Other	Xbox Wireless Dongle	$19.88
	Prices include shipping, taxes, rebates, and discounts
	Total	$508.71
	Generated by PCPartPicker 2015-09-14 21:06 EDT-0400

Not a bad system, I think. A couple notes:

• AMD APUs seem to love fast RAM, which is why i'm paying slightly more for the 2133MHz RAM
• I'd probably upgrade it to Windows 10
• She already owns 360 controllers so the dongle's all I'd need

Other tiny cases I considered are the M350 (probably too small for reliable cooling, and I'd have to pay extra for the pico-PSU) and the ISK 110 (the pico-PSU is likely not powerful enough so I'd have to consider that in the price).

I think the above system would be pretty great for a small gift. Powerful enough to handle most stuff at 1080p/30fps or 720/60fps.

However, the Alienware alpha, as it stands now, provides this:

Type	Item	Price
CPU	Intel Core i3 i3-4310T
Memory	4GB DDR3-1600 Memory
Storage	512GB HDD
GPU	GTX 860M(?) with 2GB GDDR5
Case	Custom Alienware Case
Operating System	Microsoft Windows 8.1
Other	Xbox 360 Wireless Controller
	Total	$450

Note that these is what's on the website now; it's presumed that an updated version will be coming this November when the Official "Steam Machines" debut.

I know the A10-7870k is a good CPU and ideal for gaming in tiny builds, and I know that technically I'm paying for more RAM and storage space, but as a gaming system can it really compete with a pre-built utilizing a notebook discrete GPU? Notebook GPUs aren't something that you can really build into your own custom system, so it's hard to compare possibilities.

I've love to do something custom, but I'm not sure if, in these parameters, it's financially practical. Ideally I would love to do it custom for the personal touch, but money is always an important consideration, and if I can get something prebuilt that's significantly more powerful at a lower price, it's hard to deny that option.

Submitted by j4nds4 Xbox 720 Release Date 2015