How much more energy efficient is PoS (proof of stake) than PoW (proof of work)? Are there any figures on this? I've been trying to look for some evidence showing that PoS is more energy-efficient than PoW. In theory, it's true, but if there are any figures or stats you can share on this, I'd be very grateful.

Thanks!
When providing an equivalent amount of security, I don't believe PoS can be more efficient than PoW. I do understand how it may appear this way on the surface, but I think deeper consideration shows that this is a fallacy.



The security of Bitcoin (PoW) is dependant on the economic incentives built into mining: it is the cost of attacking the system vs. the expected return that matters, not the total amount of energy spent on mining. Of course, energy has a cost, but it is important to note that it is the cost, not the energy, that is important when considering the security of the network.

Therefore, we can define a system's security relative to the cost of attacking it successfully. Less cost to attack = less secure, and vice versa.

Mining is a permission-less activity, anyone is free to join, and so the result is expected to reach a free market equilibrium: the cost of mining $1 worth of Bitcoin is expected to trend towards $1. If there is profit to be made, we would expect more miners to join the market. If costs outweigh returns, we would expect miners to leave the market. 

In PoS systems, we should expect a similar result: participants will be willing to incur costs of up to ~$1, for every $1 worth of rewards they receive in return. Anything that deviates from this will result in lowered security- you wouldn't want to pay someone $1 to provide only $0.50 worth of security! 

In PoW, a very large portion of a miner's cost is the energy used to perform computational cycles. There are additional costs to consider (hardware, overhead, employees, maintenance, etc), but the most efficient miners will be expected to minimize these costs, and spend as large a portion of their budget on energy as possible (thus maximizing their expected return). 

PoS seems to be more efficient, due to an apparently minimal number of computational cycles (compared to PoW), but when considering the above, we can arrive at a more nuanced understanding:

If a PoW network and a PoS network had the same amount of security (denominated in your favourite currency), we should expect each network's participants to spend the same amount of money to create and maintain this security. 



With this in mind I think it is interesting to reflect back upon the question at hand: does energy efficiency really matter? Shouldn't we really just be striving to maximize the security provided per dollar spent? In terms of creating a more efficient network, that is what will ultimately matter.

In that regard, I believe PoW is more efficient, as a PoW miner's costs come from an external free market mechanism, instead of an internal staking system. There is a cost to security, and so having as much of that cost tied to something universally accessible (energy) as possible seems like the most logical decision. If it is more difficult to participate in the 'validation market' (ie, act as a miner/staker), then we will expect more opportunity for validators to make profit, and thus the security of the network will be lower. This sort of 'red tape friction' creates a less efficiently secured network, dollar for dollar. 



As humanity has progressed, there has been an undeniable trend of an increased capture and utilization of energy, to power our advancements. And yet still, we utilize such an incredibly tiny fraction of the sun's output, that we are nowhere near being about to 'run out of energy'. The variable that is obviously limited is our capital, and so we should strive to use that as efficiently as possible, in order to maximize our capabilities as a technological species.