That's not a valid example of things that can happen in the market. You're making up ridiculously unrealistic numbers and clearly don't understand the basics of how the process works.
Share buybacks are always executed at the current market price. The company doesn't offer a higher price. A large buyback order might move the share price up a tiny bit but triggering an increase from $1 to $2 is impossible for any company traded on a major US exchange.
Yes, it was a made-up example. I feel that was obvious.
If your point [about share price jumping suddenly] was irrelevant, then maybe you shouldn't have mentioned it. How is this my problem?
I see that you edited your previous comment before replying. Very clever. Now (12:03 Pacific) you have a company worth $1000 trading on a major stock exchange. Ok.
Maybe you can make a spreadsheet similar to what I described in words, but using more believable numbers. If so, you can see the kind of effects I'm talking about. Buybacks are good for some shareholders and bad for others. Buybacks can be used to reward management, though others will be affected (+ or -) at the same time.
Or maybe you won't/can't make that spreadsheet. Again not my problem.
Pardon my bluntness, but you apparently don't understand how the process works.
I'm not claiming the price jumped from $1.10 to $2 without hitting any intermediate prices. That's your idea.