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:
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:
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.