What you have asked is the golden question! Why did Frank Stockton write the short story "The Lady or the Tiger?"
Many may find, after reading the story, that Stockton wanted to simply anger his readers. While some may find that this is the answer, given the story ends without closure for the reader, one can only imagine what he purpose was.
Given that Stockton never tried to sway the reader in any given direction, one could assume that the purpose of the story was to create great debate and conversation.
Stockton actually received so much negative criticism regarding the story with no ending that he wrote "The Discourager of Hesitancy" to tie up any concerns that readers had regarding the outcome of the young lovers fate.
In the end, an actual reason behind author intent may never be found. Instead, what one gets when pondering the question is the same answer one gets with reading Stockton's story- no true closure.