Very interesting spot.
Lets put Middle position on AKo AKs or AA.
KK vs that range has 47%
equity. meaning -
If UTG raised 3x there are 4.5bb in the pot. When MP re raises to 100bb, you are calling 100bb to win 204.5 OR you need 48.9% Equity to call. (So a fold is profitable)
Now lets say you add QQ+ and AKo, AKs to MP range. Now your kings have 57.8% Equity and a fold is a Losing play.
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Now lets take a look adding UTG open range into play.
If UTG is only opening 99+, AQo+ AQs+
MP is only re raising with All in with JJ+ AQs+ and AKo
Your kings in the button have 50% Equity.
(Profitable call)
If MP is only re raising all in with AKs, AKo, QQ+
Then you are supposed to fold.
Assuming UTG is folding as well.
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Finally,
Lets imagine a slightly different situation with UTG opening all in . 100bb
MP calling for 100 bb.
You in the button with KK.
You would be calling 100:201.5 and therefore need 33.1% equity
This is where you need to construct ranges for your opponents.
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Generally speaking, calling a 100bb 3bet shove on an UTG open raise, with KK is probably a losing deal in the long term.
If find your opponents overvaluing AJs+ 10+ then you could consider calling.
Not alot of good opponents will 3bet shove for 100bb with hands worse than KK.
If you are in a short stacked tournament on the other hand - i'd call a 3bet shove of 20bb with KK in a heartbeat.