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If AMD releases a RX 590 with a sub-$300 price tag, AMD could possibly have a winner on its hands and potentially steal some of Nvidia's GTX 1060 market share.
The problem isn't stealing the GTX 1060's market share, but competing with Nvidia's next generation of hardware. The 1060 is well over two years old at this point, and is already long overdue for a successor, which will likely be coming within the next couple months or so. If the RX 590 only offers around 10% better performance than an RX 580, then chances are good that it's not going to outperform the competition's next sub-$300 graphics card. And while a process shrink of this existing architecture from 14 to 12nm may help efficiency slightly at a given performance level, Nvidia are also moving to 12nm, so no gains will be made against the competition there, unless Nvidia simply re-badges existing Pascal cards for their new mid-range lineup. Even if they simply called a 1070 a 2060 and tossed a $300 price tag on it, it would still likely offer better efficiency and significantly more performance than what is seen here though.
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Those are big ifs, though, to expect from Nvidia.
The 1060 6GB is currently (in the US) generally more expensive than the RX 580 8GB.
If the 2060 is on the new architecture, I can see it easily being $50 or $100 more than the current 1060 6GB. If it's a rehash of existing architecture, I don't see it performing any better, and it's already more expensive than the AMD cards in the same class, while performing slightly less or equal, depending on the game.
I very highly doubt that Nvidia is going to relabel a 1070 as a 2060. If they do, the chance that they'd match or undercut current 1060 prices, or even dip much below 1070 prices, seems extraordinarily remote.