They're not obligated to; boards can pretty much do whatever they want.
However, Twitter's board doesn't use Twitter and doesn't actually seem to care about it at all, so they're not exactly going to claim to shareholders they can run it better than an offer that much above its current value.
Hmm this might be hard for you to believe but I've never negotiated in my life. I always accept asked price. Didn't realize this is not the norm. This is not in business context though.
At work, I was involved in a deal with a large company. The numbers are back and forth like 10 times with adds-on and discounts. It's just standard stuff.
