I don't believe that the PS4 can beat the One on home turf (i.e; North America), not even one bit. If the global marketshare balance between the PS and xbox brands will change in any significant way for the 8th gen it will be all down to Europe and, to a lesser extent, Japan and the emerging markets later on in the console life cycle when the price is a lot lower.
I just don't think that the Ps brand is strong enough in NA to beat the xbox in the long run, day one million sales be damned. Remember, the 360 had a million selling week last holiday, black friday after 6-7 years and still such massive sales speak volumes of the xbox brand's power in the Americas.
I'll prove you wrong with that claim, using some very simple math. But before this, just remember that PS2 sold a lot more than Xbox in US despite the so called Xbox image. So this "image" of Xbox is not THAT strong as you imagine.
Now let's crunch the numbers right from this site, VGChartz...
You'll see that X360 sold 44.9 mil consoles in Us versus PS3 selling only 27.4 mil. But we need to adjust for a few things here. First 360 was in the market longer compared to PS3, 8 vs 7 years; so the average annual sales is a more meaningful comparison, especially considering that XB1 doesn't have a one year headstart this time.
This brings the average sales to 5.61 vs 3.91. But there is also the "price gap", which America always proves to be very sensitive to (which is also the main reason why 360 won the last gen). The price gap was usually around 25%, so I will just give the PS3 average sales a 25% boost, creating a scenario where PS3 matched 360's price. But at the same time, I will assume the market size is constant, so the overall sales will not change (9.53).
The 25% boost to PS3 due to lower price increases its average sales increase from 3.91 to 4.89, stealing sales from X360 whose sales decrease from 5.61 to 4.63 annually. Finally 8 years of X360 will yield total sales of 37 million vs 34 million for PS3 in 7 years, reaching at a 52%-48% equilibrium.
All this ignores the "network effects & externalities", which worked against PS3 last gen, but would actually work in favor of PS3, had its price been lower.
* Note :
The lower average HD console price is likely to steal sales from Wii as well, but this is much more likely to work in favor of PS3 since the additional HD consumers already knew the X360's price.
In short, the only reasons Xbox dominated in US are
a) It was launched early, which created network effects and positive externatilities
b) It was cheaper all along
None of which exists today. On the contrary, it is suffering from bad press, underwhelming perfomance, and HIGH PRICE.
So much for the Xbox Brand!