How cheap do you mean cheap? The sweet spot for what you are describing is at least $250USD for a decent chipset. There are devices in the 175-250USD range which might come close, but it is the multi-touch and consistent 1080p playback are the sticklers. Many use the Telechips 8902 which has known MKV 1080P issues
I may be totally off base, but I think between US Thanksgiving and Christmas, you will see a number of units around $200-250 which may meet your expectation. But, the core restraints are three.
1. Good 1080p without stutter needs at least a Cortex A8 or a Snapdragon processor with a decent secondary GPU built into the Chipset.
2. It would be better to have a multi-processor (NOT as some market but two parallel CPUs not the CPU and GPU) and there are very few tablets with that capability. Think in PC terms a P4 vs a Core2 vs an I7 in the Intel arena.
3. Multitouch is dependent upon Firmware, screen capability and CPU processing. Multi-touch is just now reaching the Android 7" tablets. And these are at minimum $170USD +shipping
Plan for a unit which the vendor is solid. Those will be the ones which will offer a path to Gingerbread. There will be one out there, but if you pay too low, you will be disappointed. Pay too high and you will feel like a sucker when you see the same unit on sale in March for $70-100 less