AMEX can now give retention offers through online chat as well. If you tell the chat rep directly that you want to close your card, they will transfer you to a Membership Consulting professional. A specialized representative will then handle the card closure request (or offer you a retention offer).
In general, the rep will ask why you want to close the card. You can simply say that the annual fee is too high, that you are not able to use the travel benefits, or something similar. Then the rep will explain how many points you can earn based on your spending.
After hearing that, you can ask whether there are any other offers available to offset the annual fee, and the rep will check based on your spending profile. For example, if your spending on the Platinum is fairly high, they may directly offer 50,000. Note that after receiving it, you cannot close the card for 12 months, or the retention bonus may be clawed back.
You can also ask about your other cards while you are at it. The rep will ask whether you are also planning to close other cards as well (that is standard procedure; if you do not indicate you want to close the card, the rep generally cannot check for you). I checked on the Gold and was offered 10,000 MR.
Then for the Marriott Brilliant, the offer was 20,000 points.
For the Hilton Aspire, the offer was 10,000 points. That is on the low end; reportedly some people have received as much as 70,000.
So what affects how many points you get? First, it is definitely tied to your spending. The more you spend, the more AMEX is likely to offer. Next is how long you have held the card. Generally there is no offer within the first 1 year; the longer you have had the card, the better the offer may be. The offer also depends on the specific card. Platinum tends to get much more, while Gold is usually more modest.
Have you received an AMEX Retention Offer recently? Feel free to share in the comments.