In theory both sides hold exactly the sme amount of power for obvious reasons, but since men generally have a higher demand for dates than visa versa, and woman control the supply of dates for men, they have more power in the end. But only because men let them.
Is this a joke? Of course women have all the power in dating.
She literally has "final say" in everything, from whether there is contact info exchanged, to whether there is a date, whether there is a mutual kiss, if there is sex (outside of rape), if there is a monogamous commitment, etc.
If women said "no" to everything, there would be no dating! :-D
Women used to and still do to some degree, but with the rise of the promiscuous girl and the break down of traditional relationships, that power has been dwindled significantly. A man no longer has to commit to get sex or any of the other dating "benefits". Girls offer that up for free.