Gemara

9a (Rava): If a man wrote 'your daughter is Mekudeshes to me', he may give this to her father, or to her with her father's consent, if she is not yet a Bogeres.

19a (Beraisa - R. Yosi b'Rebbi Yehudah): "Ya'adah v'Hefdah" - when her master makes Yi'ud, she must be able to be redeemed. I.e. there is enough time left (in her six years of service) for her to do a Perutah's worth of labor.

Inference: He holds that the initial money (given to buy her) was not given for Kidushin. (Rather, he is Mekadesh her by pardoning her remaining servitude.)

Rejection (Rav Nachman bar Yitzchak): Really, the money was given for Kidushin. The Torah decreed that Yi'ud works only if she could have been redeemed.

(Rava): From R. Yosi b'Rebbi Yehudah, we learn that one can tell his daughter Tze'i v'Kabli Kidushayich (go receive your Kidushin money). He holds that the initial money was not given for Kidushin (Yi'ud is through pardon of her remaining servitude). It must be that a man can say Tze'i v'Kabli (even without selling her)!

Rishonim

The Rif (5a and 6a) brings our Gemara.

Rambam (Hilchos Ishus 3:14): One can make his daughter a Shali'ach to receive her Kidushin when she is in his Reshus. One can tell his minor daughter Tze'i v'Kabli Kidushayich.'

Rosh (1:15): The case of giving a document of Kidushin to a girl with her father's consent if when he said Tze'i v'Kabli. The Ra'avad says that it is not Shelichus. A minor cannot be a Shali'ach. Rather, it works like 'give a loaf to my dog', or 'put money on my rock, and my daughter will be Mekudeshes to you' (8b).

Rebuttal (Rosh 1:15,25): If so, what is Rava's Chidush? 'Give the money to Ploni' works like Arvus (an Arev obligates himself through another's receiving). All the more so, one can be Mekadesh his daughter to one who gives money to her! Rather, Arvus does not apply here, for her father did not speak with the Mekadesh. Also, he authorized her to receive Kidushin even through a document, which gives to her no benefit!

Ran (5a DH v'Chosav): It is not due to Din Arev. Rather, what she receives benefits her father. A father receives anything his Na'arah or minor acquires.

Question (Rosh): We do not learn from R. Yosi (b'Rebbi Yehudah) that a girl can receive her Kidushin even though she cannot be Zocheh (acquire). If so, how does she acquire the money? If mid'Oraisa she can be Zocheh, why must we learn from R. Yosi? If we learn from R. Yosi that a girl can be Zocheh mid'Oraisa, Rava should have taught about (saying that the Mekadesh may give the money to) any girl, not only one's daughter! Also, we could not learn from Yi'ud, which is mere pardon. All agree that a minor cannot be Zocheh (Bava Metzi'a 12a)!

Answer (Rosh): We learn from Gitin 64b that when Da'as Acheres Makneh (an adult transfers ownership to her), she has Zechiyah mid'Oraisa. An orphan minor can acquire money this way, but she cannot be Makneh herself to anyone. We learn from R. Yosi that a man can give his power (to be Mekadesh his daughter) to her. Selling her is like saying 'if you want, accept Kidushin from your master.' She is his Shali'ach to receive her Kidushin money. It is as if her father received. Her Yad is not like his Chatzer. An orphan minor has no Yad to receive Kidushin! A minor cannot be a Shali'ach to help others, but he can to help himself. (She can even accept a Shtar Kidushin). We learn about Shelichus from divorce, Terumah and Pesach. In each case it says "Ish". Even if the Halachah follows Chachamim, who learn from elsewhere that the initial money was given for Kidushin, the proof from R. Yosi is valid. No one argues about whether a father can say Tze'i v'Kabli.

Rashba (19a DH Omar): Tze'i v'Kabli is not exactly like R. Yosi's law. One who sold his daughter cannot retract. Presumably, one can retract from Tze'i v'Kabli before she received Kidushin. The Rif rules that a minor can acquire for himself, but not for others. Even though a man acquires what his Na'arah acquires, her Zechiyah was valid. Alternatively, her Yad is like his to acquire for him when there is Da'as Acheres Makneh. However, a moving Chatzer does not acquire. Tosfos rules like Rav Yehudah, that a minor who knows to guard things and return them can acquire even for others. Tosfos can say that Tze'i v'Kabli works with such a minor.

Poskim

Shulchan Aruch (EH 37:7): A man can be Mekadesh his daughter through a Shali'ach, or through her herself. He tells her 'Tze'i v'Kabli Kidushayich.'

Chelkas Mechokek: According to the Ra'avad, this is through the law of Arev. He could designate any girl to receive the money. The Rosh holds that she can be a Shali'ach only for herself. It cannot work through another girl.

Rema: He must say this in front of witnesses. A Shali'ach Kabalah (to receive Kidushin) needs witnesses. If it is clear to all that he prepares her for Chupah and to receive her Kidushin, it is as if he told her in front of witnesses.

Chelkas Mechokek (9): Some are stringent if he told her without witnesses.

Rema (ibid.): The primary opinion holds that this works for a Na'arah or minor. Some say that a minor cannot receive her Kidushin. Her father must receive it.

Mordechai (517, cited in Beis Yosef DH Aval): The Re'em says that Rava's law is like R. Yosi, but the Halachah follows Chachamim, who disagree. If a minor received Kidushin without her father's knowledge, she needs a Get, lest he consented (44b). Rabbeinu Baruch derives that all agree that she is Mekudeshes if he consented, or if he previously discussed Kidushin with the Mekadesh, and all the more so if he said Tze'i v'Kabli. One who is stringent will not lose. R. Yitzchak says that even Chachamim agree to Rava's law. It is no worse than saying 'give money to Ploni, and my daughter (Hagahos ha'Bach) will be Mekudeshes to you.' Rava mentioned R. Yosi, for we can derive the law from his opinion without the law of Arev.

Beis Yosef (DH Aval): The Rif omitted this law. He holds that Rava's law is only according to R. Yosi. The Halachah does not follow him.

Beis Shmuel (8): Our text of the Rif brought Rava's law! And even if it was not in the Beis Yosef's text, surely the Rif agrees to the law, for he discusses (5a) a document of Kidushin that she received herself. How can the Re'em reject that Sugya? Perhaps he holds that it discusses a Na'arah. This is difficult.

Beis Yosef (DH uvi'Terumas, citing Terumas ha'Deshen 213): The Austrian version of the Mordechai says that one may not be Mekadesh his minor daughter (until she knows whom she wants), but he may tell her Tze'i v'Kabli and she accepts Kidushin. I did so with my own daughter.

Rema (ibid.): To evade Safek, he should hold his daughter's hand when she gets the Kidushin, or stand by her at the time, and it is as if he himself received.

Source (Terumas ha'Deshen ibid.): All agree that this works. The argument is only when the father is not there at the time. Even though the Mekadesh says you are (as opposed to your daughter is) Mekudeshes to me, we know that he means 'you are Mekudeshes to me through your father's power.'