Do you think that either Da-duh or the child won their "battle of wills" in "To Da-duh, in Memoriam"?

There is a sense in which both the narrator and her
grandmother are losers in their "battle of wills," although overtly the child is the
winner in this competition. When she tells her grandmother that there are taller
buildings than the tallest tree on her island, the narrator feels she has won, but that
this victory has come at rather a great price:

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Finally, with a vague gesture that even in the
midst of her defeat still tied to dismiss me and my world, she turned and started back
through the gully, walking slowly, her steps groping and uncertain, as if she were no
longer sure of the way, while I followed triumphant yet strangely saddened
behind.

The way in which the
granddaughter is strangely saddened indicates the cost of this victory, and the way that
we could debate whether it was actually a victory at all. Note how the story ends and
the final paragraph that points towards the way in which the granddaughter actually
loses in a very significant way as well:

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She died and I lived, but always, to this day
even, within the shadow of her
death.

The narrator feels the
need to go through a period of penance when she becomes an adult, which reinforces the
impression that although she did "win" the battle of wills, it was a victory that she
came to intensely regret, and a victory that she realises was paradoxically a
defeat.