>With cryptocurrencies, PoW works because the "good guys" (miners) and the "bad guys" (double spenders) have equal access to computing power: If the difficulty increases, both can simply add more mining hardware and stay in the game. If the "bad guys" threaten to get an advantage, the system can always increase the difficulty without risking to lock out the "good guys".
No Bitcoin PoW works because of economics and game theory; it is rational if group of people invested a lot of resources into mining and building consensus that they will stick to that consensus in order to preserve their wealth.
Read what Satoshi said in the Incentive section of the Bitcoin Whitepaper: "If a greedy attacker is able to assemble more CPU power than all the honest nodes, he would have to choose between using it to defraud people by stealing back his payments, or using it to generate new coins. He ought to find it more profitable to play by the rules, such rules that favour him with more new coins than everyone else combined, than to undermine the system and the validity of his own wealth."
That's a pretty dogma but nothing more. Unfortunately, humans aren't rational in the homo oeconomicus sense. There are all kinds of motivations a person or an organisation could have to manipulate the network that have nothing to do with making money.
I mean, even in the regular, non-crypto economy there are lots of products that are sold at a loss for strategic reasons.
But you're right, that explanation wasn't quite correct. My point was that cryptocurrencies rely on the assumption that all legitimate users taken together have more computing power than a typical malicious user. That assumption is supported by game theory and made use of with dynamic difficulty adjustment.
However for a PoW captcha, the assumption does not hold: The captcha has to be solvable by WASM on a smartphone, otherwise it would lock out legitimate users. And that is a pretty low bar for an attacker to meet, computation-wise.
Unless the attacker's whole purpose is to destroy the value of Bitcoin because of some other non-economic reason, e.g. ideological reasons, national security, state self-preservation, ...
No Bitcoin PoW works because of economics and game theory; it is rational if group of people invested a lot of resources into mining and building consensus that they will stick to that consensus in order to preserve their wealth.
Read what Satoshi said in the Incentive section of the Bitcoin Whitepaper: "If a greedy attacker is able to assemble more CPU power than all the honest nodes, he would have to choose between using it to defraud people by stealing back his payments, or using it to generate new coins. He ought to find it more profitable to play by the rules, such rules that favour him with more new coins than everyone else combined, than to undermine the system and the validity of his own wealth."