First of all--not that I think you should say so--but if any agent says, "Oh yeah? Another agent offered you representation? Well, WHO? Who are they??"--or even a gentle, "Oh, that's interesting. Who?"--I would consider it pushy, rude, and a breach of etiquette.
There's a reason you're supposed to write "An agent" versus "[Name of agent]." We exist in a small world; in many cases we know each other--and just as all New Yorkers would go crazy if they were expected to talk to everyone on the subway, there are times we have to pretend we have more space and territory. We really, truly don't want to know that we're competing with friends for the same project. That's like finding out you're both writing a piece for the same publication--and comparing how much you got paid. No good can come of it.
So. That's the part that's for us. What about for you?
The minute an agent asks this question, he/she is placing you in the middle of what may be an ongoing debate/competition/industry question/drama--it's pulling you into a situation (perhaps a fight, if for some reason the agents don't get along) that just isn't fair. That's like two old friends bringing you into a generations-old battle--and you just met them. It'll cloud your judgment and make it all the more challenging to make this already difficult decision.
I've heard of nasty tricks over the years--agents badmouthing each other to potential clients; agents spreading rumors about other agents; agents putting authors in incredibly unfair situations. Oddly (or perhaps not so), a lot of this sounds intergenerational--I don't think I have to say who makes what argument. Some agents take an "all's fair in business, love, and war" approach.
I've even heard of an agent harassing once-potential clients who'd already signed with other agencies--the story involved (it sounded like) about five calls and a fax (of doctored Publishers Marketplace listings--naturally, this agent "accidentally" cut out a sizable chunk of the competition's deals--called the other agent inexperienced--and then said, "So I assume I should send you my Author-Agent Agreement?").
That's just slimy. And though you may want someone like that to represent you in certain legal cases (I suppose--perhaps if you know the other side has someone like that too), or perhaps to go on the attack if you're running for office and the other side is mud-slinging--but I don't think it'll help you much in terms of finding you a good relationship with an editor.
Will it get you more money? Well, editors won't (and can't) be bullied into offering more. They have formulas and a ceiling number from their company. And to get really large numbers, you need an auction; to get an auction, you need two (usually three) editors who can stand working with that agent. I suppose that, in some cases, preempts are also quite large--but this seems rarer.
Now, it's possible you'll come across an agent who is well-intentioned but just wants a better sense of his/her competition. Perhaps some agencies have a policy of allowing their agents to ask this question. I don't think you should assume that asking this question automatically makes an agent slimy, but I still don't think it's good form.
These things don't happen often, but the stories stick with us. No one wants to be a part of that. That said, this is all easily avoided by simply refusing to tell each agent the names of the other agents interested.
All of that said, here are some good responses. Let's practice.
Agent: So, who else made you an offer?
You: Oh, it's an agent at [choose one of the following]:
- A major agency
- A [well-known] boutique agency
- A small but well-known agency
- A company the agent started
- New York
- [If it's something else, leave off this part]
- A number of bestsellers [in the [your genre] genre]
- A strong [your genre] list
- An active interest in expanding their [your genre] list
Et voilà! You've successfully, truthfully answered their question without telling them--well, much of anything. Then again, I suppose we do this often in real life to account for others' rudeness--if someone asks how much your house cost, you say "Too much!" or "We got a good deal"--you don't give them a figure.
It's their faux pas, not yours. Now, Ms. Post (in my excellent 1940s edition) makes it quite clear that it's never polite to point out another's rudeness. It's best, methinks, to give an answer like the one above and to move on.