פירוש על קידושין 4:9

וכן האשה שנתנה רשות לשלוחה – it was necessary for the Tanna [of our Mishnah] to inform us that the father that made [someone] his agent to betroth his daughter and a woman that made [someone] her agent to betroth herself, for had [the Mishnah] only mentioned [the case of] the father, I would think that the father that has established his traced genealogy, and when he found that he is of legitimate descent, he betrothed her to himself. And the case where say that he annulled his agent, but a woman who does not establish a traced genealogy, even though she betrothed herself, we don’t rely upon her betrothal, nor did she annul the agent, for she thought that perhaps, the agent would find a person of greater connection than this. And if we only mention about the woman, I would think that because the woman carefully investigates and marries, when she betroths herself, she voids the agent. But the man, who is not strict about his daughter, if she marries any husband, he has not voided the agency of the agent, and he who came first and betrothed her, though that perhaps he is not found. It is necessary. (See Tractate Kiddushin 79a)

Introduction Our mishnah deals with a scenario whereby a father sent an agent to betroth his daughter to a certain man and then the father betrothed her to a different man.

If a man gives permission to his agent to give his daughter in betrothal, and then he himself goes and gives her in betrothal to another, if his [betrothal] was first, his betrothal is valid; if the agent’s was first, his betrothal is valid. And if it is unknown, both must give her a divorce. And if they wish, one gives a divorce, and the other marries her. The ruling in this mishnah is straightforward: she is betrothed to who ever betrothed her first. Without the mishnah we might perhaps have thought that when the father sends an agent to betroth his daughter, and then he himself does that very same act, that he is in essence canceling the agent’s authority. Were that the case, then even if the agent’s betrothal came first, the daughter would be betrothed to the man the father betrothed her to. The mishnah teaches us that this is not so, and therefore whichever betrothal comes first is the betrothal that is valid. If it is unknown which came first, then they are both potentially married to her. Therefore, both of them must divorce her, or alternatively, one may divorce her and then the other can marry her.

Similarly, if a woman gives permission to her agent to give her in betrothal, and she goes and betroths herself [to another]: if her own preceded, her betrothal is valid; if her agent’s preceded, his betrothal is valid. And if they do not know, both must give her a divorce. And if they wish, one gives a divorce and the other marries her. This halakhah is the same, but from the woman’s perspective. She sends out an agent to accept betrothal on her behalf and then she changes her mind and accepts betrothal from someone else. The ruling is the same as above.