The Rishonim try to come up with rules to explain when nusach of a bracha is “al…” (like “al biyir chameitz”) and when it is “l…” (like “l’haniach tefillin). One suggested rule is that when a mitzvah can be done by another person on one’s behalf, the bracha is always is “al…”The Ran in the first perek of Pesachim challenges this rule that from the case of ner Chanukah. A ba’al habayis can light on behalf of everyone in his home, including guests. Even though the guest does nothing – he is yotzei through the ba’al habayos -- the bracha is still “l’hadlik” and not “al hadlakas...”
Ran answers that the case of a guest (achsinai) is different because the guest has to give a perutah in payment to the ba’al habayis if he wants to participate. The guest is being yotzei through his own oil, not through the oil of the ba’al habayis.It’s seems from the Ran that the mitzvah of ner chanukah is not in doing the ma’aseh hadlakah (how would contributing a perutah help in that regard?), but rather the mitzvah is in having a cheftza shel mitzvah that is lit. By contributing payment, you have a share in the oil that is burning.
Be that as it may, how does contributing a perutah help? The halacha is that a kinyan requires a meshicha to be valid, not just payment. What kind of kinyan is this?