This question concerns the theory developed in Winning Ways for Your Mathematical Plays. The relevant Volume 1 can be found online here.
I'm unclear about the intuition behind the authors' "Bypassing Reversible Moves" result, the discussion of which begins around page 60. For convenience, this is the result:
The authors demonstrate that the resulting altered game is indeed equivalent to $G$. I follow their proof, but feel like my understanding of the result is faulty.
In particular, consider what happens if Right's option $D$ in fact has two Left options $A_1,A_2\in D^L$ that are good for Left, with one much better than the other - say $A_1\gg A_2\geq G$. Then in applying the result to $A_2$, it seems to me that we are in effect assuming that Left will respond to $D$ with $A_2$. But what right have we to disregard $A_1$ like this?
Furthermore, it seems to me that after Right's move to $D$ and Left's response to $D^L$, the result assumes that Right will definitely move again in the same component to one of $X,Y,Z,\dots$ - why would this be the case?
Thirdly, I'm not entirely sure why we need the $D^L\geq G$ condition. I would've guessed the result to take form similar to
\begin{equation*} \text{For any Right option $D$ of $G$, let $A_1\in D^L$ be Left's best response.} \\ \text{Then it will not affect the value of $G$ if we replace $D$ as a Right option} \\ \text{of $G$ by all the Right options $X,Y,Z,\dots$ of that $A_1$.} \end{equation*} Intuitively, I think of this like when one is evaluating a chess position - to any move of yours, assume your opponent makes the best response, and think about what moves you have in the resulting position.
Thank you for any clarifications in advance. I would find it particularly helpful if some examples were included.
I've thought about it some more.
Starting with my third concern: the condition $D^L\geq G$ is necessary in order to provide a disincentive for Right to ever move to $D$.
In more detail: by agreeing to play the altered game instead, Left is in effect locking himself into responding to $D$ with some particular $A_2\in D^L$ satisfying $A_2\geq G$. This might seem like a disadvantage to his cause at first (hence my first concern), but it really isn't - it's like saying "If you decide to ruin your position, then I'll commit to playing this particular good continuation." It is evident this isn't much of a concession.
Naturally, Right has no reason to disagree with the alteration either - after all, the only option he has lost is moving to $D$, but moving there would've allowed a response to $D^L$ anyway.
Finally, my second concern was simply due to inattention on my end. All that's happened is Left has committed to responding to $D$ in a certain way. Right has made no commitments.