It makes a big difference how many chips you have, what the blinds are, when in the tourney it is. A lot of players won't fold A rag, so betting to try to get them to fold an A is futile.
As a rule I bet 3X the BB, when I have AA, KK I will go 4or5X the BB, depending on position and if someone bets before me. You want to bet kind of big to chase the 22's out, I ran into someone calling 5X the BB with 88, I had KK an 8 hit the flop, good thing for me an A hit the flop also or I would have lost a lot of chips. If the A didn't hit, I was going to go all in. I had almost half my stack in, since an A hit, I folded to build my stack up again.
I dont think a bet that wont scare an 88 out will scare a 22 out, if you flop a set of 222 you'll be just as assured that you can beat AA or KK as if you have a set of
888. Set over set is just too rare to worry about, more likely you have a cold deck if that happens.
Playing KK and QQ is hard, they're premium starting
hands but too many flops can destroy almost all their value in the hand. If we always knew when our opponent has an ace in their hand, and theres an ace on the board, there would be no question, you would fold KK in a second, you know your beat. What keeps you playing with that ace on the board is that you dont know if you're beat, and grasping at straws and hoping you arent.
Playing against someone who will call you down with A-rag is hard, but playing with someone who called you down with AQ/33 is easy on a board like A73. On the flop simply cbet, if your opponent comes out with a raise with that ace on the board congrats, you've just been outpaired/setted on the flop. If they fold then you would probably not have got any action on the turn or river unless they hit big, which you don't want them to do anyways.
If they call your bet on the flop, you have a problem. Either they're slowplaying a monster, or unsure if they have the best hand (they have A-rag) and want to see a showdown by just calling. Here you have to know your opponents. If you know your opponent is tight, give it up, they probably have a monster and are trying to extract maximum value, if they are loose, its
possible they called your cbet with nothing or something like K7 (middle pair on the example flop) or they have A-rag. In my opinion if you check the turn, K7 will check behind, and A-rag will do the same or might bet (If they bet, I would most likely fold because at the levels i play at, players dont know how to outplay someone and steal.) If checked to the river you could try and check again, but K7 would probably even bet out if you do, so you could try to just minimize your loss again by setting up how much they have to call/raise. If they raise, you know you're beat, if they call you showdown and if you lose, you paid a price but it could have been as small as 1/5 of your stack in a tournament, instead of your whole stack pushing on the flop and hoping no one had an ace. If they call and you win, the final bet was a value bet. If they fold, well, you had the best hand all along.
In the end, you dont gain as much as if you pushed on the flop, got called by weak ace/strong ace/trips and you suckout on the river and hit your third king or queen. But if you play it right, you wont lose as much if you're behind.
This is just the way i go about it when I'm not sure about how well off my big pocket pairs like KK and QQ are, it may not be the best format but its served me pretty well, and I've gotten out of a few losing situations with it.