Both players know each other’s exact 2 cards.  One has AA, the other a 87cc on a 9c6c2x board.

The optimal strategy for 87 is to either bet zero, or get all in.  Obviously we will never know this in real life, but I do have a scenario where I like to use this strategy:

We have raised preflop, and get called by in position.  We hold JTs and the flop is something like Q63, rainbow one of our suit.  We bet the flop, leaving something like 2-2.5 SPR behind.  The turn is one of our best cards, one that gives us a flush draw, and either a gutter or OESD.  This could be the A/K/9/8 of our suit.  If we barrel the turn, and get jammed we will be in a spot where our call will need about ~30% to break even.  This is about what we will have, so the value of our hand is close to zero.

What if we check the turn? This goes counter to everything most of us do; flop some back doors/cbet/turn equity/fire again.  I am not saying this bluff never works, but it will leave us in an awkward spot when we are jammed.  

How does checking fair?  If Villain checks behind, that is fine for us.  We will realize all of our equity, with a free card, and still be in a good bluffing spot when we miss.  If Villain does something other than go all in, we can then jam, have fold equity, and still have the right price when he calls.