On behalf of my wife and me
On behalf of my wife and I
On behalf of me and my wife
On behalf of myself and my wife
On behalf of my wife and myself
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My understanding is that after that phrase you would carry on talking in first person.
I looked at a bunch of style guides to see what they have to say on this subject. The vast majority of them dedicate at least a paragraph to the distinction (or nondistinction) between "in behalf of" and "on behalf of"—but not one addresses the question of how to handle "on behalf of" when used by a speaker to refer to another person and to him- or herself.
This silence leads me to believe that style guides generally agree that the same rules that apply to reflexive constructions by a speaker with other introductory phrases would also apply here. Let"s look separately at three issues that the poster"s somewhat syntactically complex sentence raises, to see how those rules (or preferences or propensities) play out.
Which indirect object comes first, the wife or the reflexive pronoun?
One idiomatic form that has long prevailed in lutz-heilmann.info is put the reference to the other person before the speaker"s self-reference—at least at a certain level of social gentility. Consider this Ngram chart of the phrase "of my wife and myself" (blue line) versus "of myself and my wife" (red line) for the period 1750–2005:
Neither expression registers on the chart until roughly the turn of the nineteenth century; but after about 1860 "of my wife and myself" becomes far more common in Google Books search results than "for myself and my wife," and it has remained more common (albeit by a shrinking margin) through 2005.
Now consider the corresponding Ngram chart for "of my wife and me" (blue line) versus "of me and my wife" (red line) for the same period:
The big advantage for the blue line in the first chart becomes far less evident in this chart, although between 1970 and 2005, "of my wife and me" manages to maintain at least a small advantage.
And finally, here is the Ngram chart for "of my wife and I" (blue line) versus "of I and my wife" (no line because of insufficient matches to plot):
In written lutz-heilmann.info, "of my wife and I" constitutes a false genteelism when used as a compound object. An editor would normally recast it as either "of my wife and me" (in nonreflexive situations) or "of my wife and myself" (in reflexive ones). Nevertheless it does appear in books by respectable authors, such as in this afterword by Wadie Said in Edward Said, From Oslo to Iraq and the Road Map: Essays (Random House, 2007):
It was perhaps this sentiment that held me back from writing about the experience of my wife and I being denied entry to the West Bank by Israeli border control officers in June 2003.
The net effect of the three charts presented in this part of my answer is to support the notion that, in "of" clauses in lutz-heilmann.info, "my wife" generally precedes "myself" or "I" and more often than not precedes "me. It also provides a fairly strong argument for dismissing "on behalf of I and my wife" from further consideration—as indeed the OP seems to have done at the outset.
Which reflexive pronouns are used with "on behalf of"?
Having identified a fairly strong preference for "of my wife and I/me/myself" over "of I/me/myself and my wife," let"s drop "my wife" out of the equation and focus on which reflexive pronouns are most commonly used in the expression "on behalf of I/me/myself." Here is the Ngram chart for "on behalf of myself" (blue line) versus "on behalf of me" (red line) versus "on behalf of I" (green line) for the period 1750–2005:
It appears that "on behalf of myself" is far more common than "on behalf of me" in Google Books search results—and that is before we remove instances where the person speaking "on behalf of me" is a third party. In fact, the Google Books searches I ran turned up just three examples of "on behalf of me" used reflexively. From a translation of a thirteenth-century charter, quoted in James Thompson, An Essay on lutz-heilmann.info Municipal History (1867):
Simon de Montfort, son of earl Simon de Montfort, lord of Leicester, to all who may see and hear the present page, health in the Lord! Know all of you that I, for the good of my soul, and the souls of my ancestors and successors, have granted, and by this my present charter have confirmed, on behalf of me and my heirs for ever, to my burgesses of Leicester and their heirs, that no Jew or Jewess in my time, or in the time of any of my heirs to the end of the world, shall inhabit or remain, or obtain a residence, in Leicester.
From India Parliament, House of the People, Parliamentary Debates: Official Report (2000):
On behalf of me and on behalf of Telugu Desam Party, I convey my heart felt condolences to the members of the bereaved family and pray God to give them necessary strength to withstand this serious loss.
And from Eric Walters, Home Team (2010), writing in the guise of a sixth-grade student:
Dear Ms. Allison,
I am writing on behalf of me and my school. We are all big basketball fans. We are particularly big Raptors fans and even bigger fans of Wayne Dawkins. I think he is a superstar.
As a reflexive form, "on behalf of myself" seems to be far more common than "on behalf of me."
A fourth alternative: "on my own behalf"
There is however a further complication, which vpn mentions in a comment beneath Robusto"s answer: Given the choice, many people would say "on my own behalf" (or perhaps simply "on my behalf") to express reflexive advocacy. Adding "on my own behalf" (yellow line) to the preceding Ngram chart, we can see how popular this alternative is:
Unmistakably, "on my own behalf" is a much more popular construction in the Google Books database than "on behalf of myself," "on behalf of me," or "on behalf of I." Unfortunately, with a compound referent, "on my behalf" becomes exceedingly awkward. If "my wife" reenters the picture, we face an unappealing choice between "on my wife"s and my behalf" and "on my and my wife"s behalf"—neither of which draws any matches in a Google Books search—and may perhaps also get a sneaking suspicion that "behalves" might sound better than "behalf" unless the interests of the speaker and his wife are not absolutely identical.
To my mind, the return of "my wife" to the mix torpedoes the "on my behalf" option, simply because "on my and my wife"s
In situations involving a speaker referring to his own advocacy for himself and for his wife, the most widely used of the many options considered in this answer appears to be "on behalf of my wife and myself." It isn"t an ideal wording, however—and it would probably be worth writing around, which the writer or speaker could do by introducing himself and his wife as a unitary party of interest in the immediately prior sentence and then using the simple phrase "on our behalf" in the example sentence:
My wife and I have been planning the heist for months. On our behalf, I persuaded the witless bank manager to supply us with a detailed floor plan of the vault room so that we could make efficient plans to "refinish the floor."
But failing that, I would go with "On behalf of my wife and myself, I persuaded..."
Postscript: A usage commentator"s take on "on behalf of myself"
I said earlier that none of the style guides I consulted have anything to say about using "on behalf of" in conjunction with both a third person and a reflexive reference to the speaker or writer. One guide does, however, address the question of reflexive use of "on behalf of." From Barbara Wallraff, Word Court (2000)
"On behalf of myself" can"t be corrected in the same way, for "on behalf of me" actually sounds worse. If the pilot had said "on my own behalf," he, or she, might have noticed that the whole phrase was a bit foolish and it would be better left off: people thanking others are presumed to be doing it on their own behalf, and it"s only when they"re doing the thinking on behalf of, say, their employers that any such thing needs to be specified.
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Wallraff doesn"t explain why ""on behalf of me" actually sounds worse," but I think that the underlying reason is that people who are trying to express themselves in self-conscious written lutz-heilmann.info overwhelmingly prefer the -self forms of pronouns (myself, ourselves, yourself, himself, herself, itself, themselves) to the non-self alternatives (me, us, you, him, her, it, them).