Actually, PoS is much less secure than PoW.
Bitcoin’s extremely clunky and inefficient security mechanism is robust against most threats. PoS is not.
When people discuss PoW or PoS and talk about incentives, throwing around terms like “Nash equilibrium”, they are usually referring to mechanisms internal to that specific blockchain. “Bitcoin is secured by Proof of Work” is a true statement up to the point when an outside attacker has enough incentive to destroy the system.
Cardano has claimed that Proof of Work is equally as secure as Proof of Stake. I believe this only if you consider internal attack vectors. With external attack vectors, things get unpredictable.
Bitcoin is relatively secure because it would be extremely difficult, not only financially, but physically, to acquire enough hashrate to overthrow the network. Bitcoin is mined on ASICs that are difficult to put your hands on, thus controlling the network is an extremely daunting challenge, requiring the kind of determination that would lead a nation into a war with Iraq. It’s possible, but not something you for the lulz.
Proof of Stake on the other hand suffers the fact that the stake required is quickly transferrable around the globe in seconds, and the incentives governing it are fine-tuned and efficient.
For example, staking gives you a reward, which is why you would do it. You get an annual expected reward, let’s just say 3%, to stake your coins. This means you aren’t using your coins for DeFi or paying kids’ allowance or whatever. Now I could come along, with a smart contract, offering you 3.5% returns on some DeFi app over a specific time period. I have a smart contract performing all this, so it’s verifiable and secure - it’s not like you’re loaning money to your brother-in-law to do what he wants with. You will get the coins back at the end of four months with some return. Now the game theory, that is, the “rational choice assumption” says, you choose 3.5% returns instead of 3% returns. I don’t need a ton of coin to do this, I just need a little bit of coin in order to guarantee your returns will be there, so you have absolutely no risk of loaning me the coin. With a little bit of investment, I can do this on quite a large scale.
Because only a fraction of the coins (especially in a well-developed ecosystem) will be used for staking at any given time, I can lock up a controlling percentage over a period of time, while operating at epsilon loss. But this epsilon loss is in terms of the internal unit of account. If I have more to externally, this could be a profitable play for me. Try to think like Carl Icahn for a moment.
This is a slightly cartoonish example, but illustrates the basic problem. If you efficiently commoditize everything in the system, including control of the mechanism itself, you’re setting the stage for chaos. Smart people have spent considerable effort trying to thwart various attacks on the system, but ultimately this is a difficult problem to get around. It will always require complicated mechanisms and parameter tuning.
“Yes, but” people will surely say, “why would I loan my coins to someone who might attack the system.” This is because you don’t believe that your coins will be the tipping point. Either the system will withstand the attack, in which case you can make extra rewards or not, or the system will falter, in which case the rewards don’t really matter. What we have is a Prisoner’s Dilemma.