Because a reputation for following up brinksmanship threats means that people won't enter into deals with you at all; extortion works because, to some extent, people have to "deal" with you even if they don't want to.
This is why I saw a Walmart-monopsony (monopolistic buyer) as closer to extortion, since not trading with them is not an option.
I'm think of it this way: investigating a supplier to check they are reasonable costs $1 to Walmart. The minimum price any supplier will offer is $10. After investigating, one supplier offers $10.5. Walmart refuses, knowing the supplier will not got lower, and publicises the exchange.
The reason this is extortion, at least in the sense of this post, is that Walmart takes a cost (it will cost them at least $11 to investigate and hire another supplier) in order to build a reputation.
The connection to AI alignment is combining the different utilities of different entities without extortion ruining the combination, and dealing with threats and acausal trade.
I think the distinction is, from the point of view of the extortioner, "would it be in my interests to try and extort , *even if I know for a fact that cannot be extorted and would force me to act on my threat, to the detriment of myself in that situation?"
If the answer is yes, then it's extortion (in the meaning of this post). Trying to extort the un-extortable, then acting on the threat, makes sense as a warning to other.
That's a misspelling that's entirely my fault, and has now been corrected.
(1) You say that releasing nude photos is in the blackmail category. But who's the audience?
The other people of whom you have nude photos, who are now incentivised to pay up rather than kick up a fuss.
(2) For n=1, m large: Is an example of brinkmanship here a monopolistic buyer who will only choose suppliers giving cutrate prices?
Interesting example that I hadn't really considered. I'd say its fits more under extortion than brinksmanship, though. A small supplier has to sell, or they won't stay in business. If there's a single buyer, "I won't buy from you" is the same as "I will ruin you". Abstracting away the property rights (Walmart is definitely legally allowed to do this), this seems very much an extorsion.
Enjoyed writing it, too.