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Banned
You're right in that Microsoft had a very specific goal in buying their way into the industry. Although, iirc, there was also some poor decision making with regard to their deal with NVIDIA that contributed to their losses, and that's part of why they changed vendors.
With regard to Sony's losses. I don't really know what they were thinking. Consider this: Had the PS3 not sold as abysmally as it did, those losses would have been even higher due to those large negative margins on the hardware. Were they banking on a high tie ratio? Then where was the software? It was just failure at all levels in terms of planning and execution. Although, I've seen it stated on here Kutaragi hid the cost of the PS3, although haven't been able to verify it, which may explain some of the dumbness if true.
KK hid the cost of Cell development within the electronics division, that was around $800m iirc.