Potential flaw in Peter Rizun's economic model

[alkan]

In Peter Rizun's paper "A Transaction Fee Market Exists Without a Block Size Limit" the miner's expected revenue is based on the following equation: 〈V〉 = (R + M) * h/H * (1 − ℙorphan).

As we see, the expected block reward and fee revenue per block are multiplied by the miner's relative hash rate and by the probability that the block gets orphaned.

Unfortunately, this model turns out as flawed. In the equation, rewards are only counted for the blocks that make it into the blockchain. So, if a miner for example produces 10 blocks per day with an orphan rate of 0.2, he will get the rewarded for a total of 8 blocks.

Although seemingly convincing, this perspective is incorrect as block production rate will ultimately depend on the block difficulty, which is automatically adjusted according to orphan rate and hash rate of all the miners.

What actually matters for a miner's block production (and revenue) is thus not how many of his blocks make it into the chain, but how many of the blocks in the chain are produced by him (production share). This is not the same thing! The mining rewards (fees+block rewards) should be multiplied by the miner's production share since the total block production (per time unit) remains constant due to automatic difficulty adjustement.

Let's make a simple example with only 3 miners A, B and C who all have the same relative hash rate 1/3 and the same orphan rate 0.01. The miners don't engage in any form of selfish mining strategies. It's easy to see that under these conditions every miner will build 1/3 of the blocks in the chain, as everybody has the exact same chances.

Now, let's assume that A starts building bigger blocks so that his orphan rate increases to 0.2, while B und C retain their orphan rates of 0.01. To determine the fraction of the blocks (in the chain) built by the respective miners, we can calculate:

A: (0.8*1/3) / (0.8*1/3 + 0.99*1/3 + 0.99*1/3) = 0.288
B and C: (0.99*1/3) / (0.8*1/3 + 0.99*1/3 + 0.99*1/3) = 0.356

And we see that B and C can now build more blocks of the chain than their relative hash rates.
According to Peter R's equation, their success rate would be 0.99*1/3= 0.33, which is incorrect.

Mining is an entirely relativistic game!

What are the consequences of this?

Let me come back to my example of the three miners A, B and C, all with a hashrate of 1/3 and an orphan rate of 0.01.

Now, assume that A and B stick to a block size of 1mb, while C tries to find the block size that maximizes his profits. C can do so by gradually increasing the block size as long as the higher orphan rate (resulting in a lower production share) is outweighed by the higher fees. As the orphan rate follows a poisson distribution and the marginal fee income tends to decrease, there will be an equilibrium where marginal revenue = marginal costs. Let's assume that C's profits are maximized with an orphan rate 0.2, so that his current blockchain production rate will be 0.288, while that of B and C 0.356 each.

The fundamental problem arises once A and B also start using a variable block size to maximize their profits. By doing so (i.e. by increasing their own block sizes) they will not only decrease their own blockchain production shares due to their higher orphan rates, but at the same time C's blockchain production share will grow and thus destroy his individual market equlibrium. To reach equilibrium again, C will now have to increase his block size once again to collect the same fees as before. So, his optimal orphan rate will be more than 0.2. This, in turn, would place A and B in a disequilibrium, who might then increase their block sizes even more, etc. It will all end up in a doom loop.

It seems that the increasing total block space supply combined with the (probably) finite demand for transactions could make the loop converge at some upper limit. However, this equilibrium would be unstable. When a miner suddenly decreases his block size, all the others would follow suit to reach their individual market equilibrium again. The miners might even end up at an unstable lower equilibrium point

As far as I can see, no stable market equilibrium can be reached by all miners at the same time. For mining market has the peculiarity that whenever a miner increases its own supply, the supply of all the other will decrease. In contrast to regular markets where the players only compete to meet the demand, Bitcoin miners also compete to increase their own supply at the cost of their competitors since total block production remains capped even with unlimited block size.

 

[_mr_e]

Personally I think miners are incentivized to keep blocks as small as possible in order to find the sweet spot between block size and the fees that users are willing to pay. Done right, all miners will make more money. If some large miners somehow go too far off base by raising it too much, nodes will stop them. If blocks are too small and fees increase too much, nodes will up their acceptance and other miners will take that chance to grab more fees from the others.

 

[solex]

@alkan
Welcome to the forum!
The likely scenario (for the short/medium term) in a Emergent Consensus block limiting model with blocks >1MB is that the miners will largely co-ordinate on excessive block (EB) levels, such that the majority always has the same value. They can publish intentions to change their limit in coinbase txns, or simply liaise via social media channels.
One constantly overlooked factor is that all the miners are incentivized to keep the Bitcoin price high and that is a greater incentive than any small profit theoretically obtainable from the many different tactics of gamesmanship in block creating.

 

[CubicEarth]

@alkan You make some good points about how a change in one miner's orphan rate necessarily changes the orphan rates of their competitors. I don't agree, however, with your conclusion that "It will all end up in a doom loop.".

First, regarding the lack of a static equilibrium, it's true that miners will need to continually optimize their blocks in response to ever changing real-world conditions if they are to maximize their profit. This is expected though, and as much is true of any entity involved in production of just about any product. Second, even as a theoretical exercise, if it is fact not a solvable equation, that still doesn't mean there will be a run-away effect. To my reasoning the optimal block size for any given miner may oscillate back and forth around some mean, or collectively it could look something like an n-body orbital simulation, which despite the appearance of some occasional chaos, as a system remains symmetrically anchored around a center of mass, and adheres to the conservation of energy.

In your examples, you make no mention of the mempool demand curve, which is how much each successive unconfirmed transaction is will to pay as a fee per byte. The fees are what the miners are after, and the orphan rate is a concession they make in pursuit of those fees, and is otherwise just something to be minimize. It's simple to think about: why wouldn't A, B and C all produce giant blocks? Well, potentially if there were enough transactions that paid high fees, the blocks could be large, but in the absence of that, any miner could stand to gain an advantage by making their blocks smaller, and having a lower orphan rate.

 

[alkan]

Thank you @_mr_e and @solex for your quick replies!

If all the miners coordinate, they can theoretically agree on a block size that maximizes their total revenue as a group. As every miner would produce blocks of the same size, the orphaning risk would be the same for everyone (assuming equal propagation time) and thus wouldn't constitute an economic cost for the miners. The latter could theoretically set an arbitrarily high block size and include as many transactions as they want, provided that safety and lifeness of the blockchain are still guaranteed. Because the miners as a group indeed have a clear incentive to keep Bitcoin price high, they wouldn't risk the soundness of the blockchain by choosing an exorbitant block size.

However, having agreed on (reasonably) large blocks, the miners would still lose a significant number of their blocks due to orphaning, whereas the marginal fee per transaction would probably be very low. So, every individual miner would have an egoistic incentive to reduce his block size in order to increase the probability of getting his blocks added to the chain. That's why the total market equilibrium wouldn't be a Nash equilibrium and the incentive to keep the Bitcoin price high wouldn't help in that context IMO due to the Prisoner's Dilemma of every single miner.

Maybe the market could stabilize around a different equilibrium though. It seems that there must be a block size where no individual miner has an incentive to deviate by building smaller or larger blocks. This equilibrium wouldn't lie at the intersection of the market demand and supply curves and wouldn't be optimal in an economical sense, but it could work out all the same. What do you think?
[doublepost=1490433343][/doublepost]I largely agree with you @CubicEarth.

The lack of a static equilibrium doesn't necessarily have to result in a doom loop. I really like your example of the n-body orbital simulation around a central mass.
As laid down my latest post, I even think now that the market may finally stabilize on a (sub-optimal) Nash equilibrium.

 

[alkan]

As laid down my latest post, I even think now that the market may finally stabilize on a (sub-optimal) Nash equilibrium.

I may have to revise my opinion once again.

It seems to me that an equilibrium where none of the miners has an incentive to build smaller or larger blocks could only exist if all the miners had the same supply function. This is not a realistic scenario though since economies-of-scale and geographic location (both in terms of network connectivity and electricity costs) result in different supply curves.

@Peter R: It would be great to know your opinion about all this.
[doublepost=1490634783][/doublepost]

Second, even as a theoretical exercise, if it is fact not a solvable equation, that still doesn't mean there will be a run-away effect. To my reasoning the optimal block size for any given miner may oscillate back and forth around some mean, or collectively it could look something like an n-body orbital simulation, which despite the appearance of some occasional chaos, as a system remains symmetrically anchored around a center of mass, and adheres to the conservation of energy.

Maybe it is possible to calculate a maximum radius (block size interval) from the average optimal block size that no rational miner would leave. Unfortunately, I don't have the math skills to further analyze this question.