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

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 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**. 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.*blocks in the chain*are produced by him (production share)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.
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