Who wins the battle between Annie or John in "Tickets, Please" by D.H. Lawrence?

There are two aspects to the conflict between Annie and John. One is the personal conflict between just the two of them. The other is the broader conflict between the women as a group and John as a symbol of all men, not just his own individual person. The reader can evaluate if one of them wins in either or both areas.
As each individual woman was dealing with John separately, they were competing with each other. In doing so, they were endowing him with the power to choose among them. Annie initially feels competitive with the other women, and this feeling is exaggerated when John moves on to his next choice. She then realizes, however, that none of the women is really benefitting in this arrangement; in fact, quite a few women have gotten pregnant and had to leave their jobs. Once she decides to unite the women to oppose him, her position changes. Taking on a leadership role and getting all the women to join in the mission, she wins. John is vanquished; the women behave “as if he had never existed,” D. H. Lawrence says. Because their “solution” is violence, however, Annie could be considered to lose, as the women have second thoughts once they beat up John. They experience a “terrifying, cold triumph.”
Viewed another way, however, even if Annie herself does not exactly win, John definitely loses. What he had valued was his power to manipulate the women into competing over him and to have a series of meaningless, sexual relationships. He will never be able to do that again—at least, not with the tram women.

Neither John nor Annie won the battle. Annie got involved with John for the thrill and excitement even though she was aware of his tendencies with women. However, as their intimate encounters continued, she grew possessive and wanted him for herself. John never had any intention of settling with one lady and used his good looks and charm to conquer as many women as he could. When John rejected Annie and moved on to the next woman, she was left wounded and plotted revenge.
The author tells us that although Annie succeeded in her revenge by ganging up with the rest of John’s victims to physically harm him, none of them felt triumphant. John staggered away defeated and in tatters, no longer appealing to those who once adored him.