AK is a clear choice. Since you re talking about picking what you want to have, AK or 22, that doesnt mean opponent has to have the other cards that you didnt pick. So you re much better with AK. Now lets address what you re really saying but framed it badly, you having an AK againts 22 or 22 against AK. Its almost complete luck. Now there can be some things that can change that. If you re both shorter stacks and everybody else folds that might mean that nobody had an ace or a king in their hand which improves
odds of getting them post flop and same with somebody probably having a 2 or lower cards that remove possibility of getting a straight if opponent gets an ace or a king. So i would personally pick AK just because there is a higher chance in instances like getting a different 2 pairs on the board, you re better, getting a flush, you re better, getting a straight and there is still that one ace or a king that makes you almost an instant winner cause the possibility of him getting one out of two 2 in the deck is slim and maybe even one was in somebody elses hand. So either way im picking AK even though its a slight dog against a pocket pair mathematically. I understand that every instances that i mentioned is included in the mathematicall odds of getting a win but would still pick AK.