Together with a partner, I recently started my study of Will Tipton's Expert Heads Up No Limit Hold'em and we came across a theoretical question that is causing quite some confusion.

1. Background:
Tipton first lines out the concept of a maximally exploitative strategy (MES), that is, the most profitable response of Hero to some particular fixed strategy of an opponent. A MES is found by defining a game tree, defining Villain's ranges for each decision node, and solving for the point where Hero plays "every individual combination of hole cards as profitably as possible" vs. Villain's ranges. If we don't hold an opponent's strategy fixed anymore, the opponent and Hero iteratively adapt to each others' change in strategies and always find the most profitable response to the other player's counter-strategy until no further adjustment is possible. Together, "this pair of strategies constitues what is called a Nash equilibrium" where each player is maximally exploiting the other and neither player can change his strategy to improve his own expectation.

So: A MES can be defined against any particular strategy of Villain, as bad as it might be. But once both players play an MES against each other, the MES of each player represents his optimal equilibrum play.

2. Confusion:
So a MES is the most profitable way to "play each particular pair of hole cards individually". Further, MES "certainly do not include "loss leader" type plays where one hand is played less profitably than it might be for the sake of the whole strategy." (p.34)

This actually goes contrary to my understanding of the most fundamental concepts of optimal play. I thought than in many situations, we actually want to "protect" our sub-ranges with some holdings that could win even more money in a different branch of the tree. Consider the spot where UTG raises and BU cold calls, flop comes 9h7h3s. The UTG range has missed this flop so much that a lot of our strong overpairs will check. I always thought this is because we need to "protect our checking range" in order to not be exploitable when we check and to have enough strong hands to defend vs. a BU stab. Hence, a hand like AA could win more by betting, but it checks in order to strengthen the checking range. However, by the definition of a MES, we play each combo as profitably as possible also in a vacuum and don't sacrifize EV with one combo for the whole range.

These two logics are contradictory. How do they go together??

3. Attempt of an answer:
The only thing that I could imagine is that it is actually the incentive of a strong overpair (say AA) to check here. That the EV of e.g. AA doesn't come from barrelling on a board where Villain will have more nuts than we do (and all weaker overpairs that might pay us off would have 3-bet pre), but from checking and making money from Villain's bluffs and value hands vs. our checking range. However, then AA plays perfectly according to its own and individual incentive and does not check to "protect the checking range" in any way whatsover. It just does what is highest EV in a vacuum for this combo. But then this whole logic of range protection would be completely off, however you hear it over and over again even by the best players.

Help?