My friend Chaim M. wrote about the takanah of a guest being mishtateif b'perutah, contributing a "penny" to his host and thereby fulfilling the mitzvah of ner chanukah through the lighting of the ba'al habayis. http://nefeshchaim.blogspot.com/2005/12/kinyan-kesef-shel-achsenai.html He raised the issue whether giving a perutah is actually a purchase of the oil, or is it just a token gesture for the achsinai to be part of the lighting.
The gemara in pesachim debates whether the proper bracha of bedikas chameitz is "al biyur chameiyz" or "leva'eir chameitz". This leads the rishonim to an extended discussion of where the formula "al..." is used for a bracha, and where the formula "l..." is used. The Riva"m rules that where the mitzva can be accomplished by a shliach, the formula "al..." should be used, but otherwise the formula "l..." should be used. Rishonim ask: why is the bracha "l'hadlik ner chanukah" when lighting the menorah can be done through shlichus - isn't that what the achsinai (guest) accomplishing by contributing a perutah and asking the ba'al habayis to light for him?
Two answers are given: 1) the Rosh answers that the formula for brachos follows the most common case, which is a person lighting for himself, not achsinai. 2) the Ran answers that the perutah effectively converts the achsinai to owning a portion of the mitzva, not a pure shlichus case.
Perhaps these two opinions relate to how the takkanah of achsinai works. According to the Ran, there is something substantive taking place when the achsinai contributes a perutah that makes it his mitzvah and not shlichus; perhaps the Rosh deliberately avoided this formulation and considers shutfus b'perutah just a formulaic act to effect this unique form of shlichus. It is worth noting that according to the Agudah, the ba'al habayis must actually add extra oil on behalf of the achsinai for the perutah given, while other Rishonim do not adopt this position.