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Edited in response to OP's edit: I think it would be a mistake to tell them who, and I think it would be a mistake to go with an agent that is only prompted to read your full manuscript because an...
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#4: Attribution notice removed
Source: https://writers.stackexchange.com/a/43195 License name: CC BY-SA 3.0 License URL: https://creativecommons.org/licenses/by-sa/3.0/
#3: Attribution notice added
Source: https://writers.stackexchange.com/a/43195 License name: CC BY-SA 3.0 License URL: https://creativecommons.org/licenses/by-sa/3.0/
#2: Initial revision
Edited in response to OP's edit: I think it would be a mistake to tell them who, and I think it would be a mistake to go with an agent that is only prompted to read your full manuscript because _another_ agent decided it was good. You say many of their sites say to let them know if somebody else asks for the full manuscript: If the agent that **DID** ask for the full manuscript has such a request on her site, then I'd go ahead and tell the other agents somebody has asked for the full manuscript. I'd certainly give priority to the first agent that asked for it **un-prompted** and only because they thought your writing was good enough to take a closer look. Other than personality, I don't think you are going to get any better deal by looking at other agents, especially if they didn't think your query and/or sample wasn't good enough on its own merit. I expect the terms for the _agent_, with an unknown first-time author, are going to be the same. The terms a particularly successful agent might get for your book from a _publisher_ might be different. If they demand something, and your agent belongs to a large firm, you might be able to tell them which firm it is. I'm still not sure that's a good idea. Personally, I think if they asked me to be informed if somebody requested the full manuscript, I am informing them that somebody did; and if they choose not to believe me, so be it. I'd ask them to review the sample you sent, or you are happy to send them the full manuscript, and judge it on its merits, pass or fail, not on who else is interested in it.