Math question about AA.

[Eccco]

Hi

Last night I played a SNG. Got lucky and had AA.
Blinds 50/100. Small stack raises to 200 (had 550). Big stack reraises to 400 (had 2,5k). Me with 2000 make it 1000 to go. Both fold. Entire Chips 8k (Party 20+2)

I'm sure I could have smooth called or mini raised - and for sure I thought the small stack would be committed - but I just took it down without a flop.

So I win 50+100+200+400= 750 .
That's a 37,5% increase in my stack - so I am not unhappy. But it made me think:

Is there a point where u are just satisfied with taking down the pot without a flop when u have AA?
How much chips would it need to bee in the pot - in relation to your stack / your opponents stack / before u are happy to just take it down?

Is there a mathemathical solution to this? Some kind of "flux point" where taking it down immediately is more profitable than playing them out for either your or your opponents entire stack.?

And I'm not talking about when u are a desperate shortstack, but say when u are already among the top 2-3 in chips and are facing another big stack for your tournament life.

And yeah - I like my AA enought to go all in against anybody, but they do loose on occasion. So there has got to be a point where there is enough chips in the pot that u would be happy to take it down without a flop. How could one determine this?

Anybody with some math knowledge?

Ty