Sometime next week, Bitcoin nodes running the BTC1 client are expected to mine a block that would be invalid under the rules of Bitcoin Core, thus creating a fork of the Bitcoin blockchain. The names for the fork and non-fork currencies are at least as controversial as the fork itself, so I’ll just call them “B2X” and “B1X”. Over 80% of Bitcoin miners have signaled their intent to mine the B2X chain, but futures markets have indicated that investors expect B1X to retain over 80% of Bitcoin’s value, with B2X retaining less than 20%.

Would such a divergence between hashrate and price be sustainable? I’m sure that some miners, traders, and exchanges have analyzed this exhaustively, but such analysis doesn’t seem to have filtered out to the broader community beyond the conclusion that “hashrate will ultimately follow price.” This post tries to justify that conclusion, arguing that supporting a less-valuable fork would likely be unsustainably expensive for miners, even in the short run. My estimate is that if the price predictions hold, subsidizing the B2X chain rather than calling off the fork could cost miners as much as $100 million in the first 10 days.

We’ll just be looking at the 10 days following the fork — or, more precisely, the period before either chain adjusts its difficulty — with a focus on what would happen if the futures prices and miner signaling both turn out to be approximately accurate, and B2X starts out with 80% of BTC’s mining power but only 20% of its value. My primary goal will be to estimate the short-term opportunity cost miners would be sacrificing if they mine the B2X chain in that scenario.

I’m not considering the longer-term incentives that miners might have for choosing to sacrifice this much to support the chain. I also do not consider any more sophisticated strategies that a mining cartel might undertake collectively, such as launching a 51% attack—or selfish mining attack—against one or both chains, or dedicating mining power to other coins such as Bitcoin Cash. Such a development is certainly possible, but will have to be covered in a future post. Finally, I’m not making any estimates of miner expenses, which are hard to determine, and are essentially irrelevant to the opportunity cost of mining one chain rather than another.

Estimating revenue

First, let’s estimate how much an individual miner would earn by mining the B2X chain versus the B1X chain until the first difficulty adjustment.

We can estimate a given miner’s revenue over a period of time using the following formula:

Rate of block generation ⋅ Time ⋅ (Block reward+ Fees) ⋅ Price

Let’s examine each of these components in turn.

Rate of block generation

The first component in the formula is the rate at which a given miner generates blocks on the chain on which they are mining.

This depends on two factors, the difficulty—a measure of how hard it is to generate a new block—and the miner’s own hashrate. Difficulty is ultimately based on total mining power of the network—as the total hashrate dedicated to mining Bitcoin grows or shrinks, the protocol adjusts the difficulty (or more precisely, the target that miners must hit) to keep the rate of new block creation close to once every 10 minutes. But difficulty is only adjusted every 2016 blocks (every two weeks, when blocks are being created at their normal rate).

It turns out that before the difficulty adjusts, the rate at which a particular miner generates blocks is independent of the actions of other miners. This is because the rate at which you generate blocks is a function of only two things: the block difficulty, and your own hashes per second. If other miners abandon the chain you are mining, then the rate at which the whole chain generates new blocks will slow down, but your rate of generating new blocks will not change until the difficulty adjusts.

This means that a miner who is earning 1 BTC per hour pre-fork in block rewards would, immediately after the fork, have the choice between earning 1 B1X per hour and 1 B2X per hour, no matter what other miners are doing.

(One caveat is that a slower blockchain may have a slightly lower orphan block rate, but orphan blocks are sufficiently rare under ordinary circumstances that this will probably not be material.)

So, an individual miner’s rate of block generation does not depend on which fork they mine or how many other miners are mining that chain (until the difficulty adjustment). However, each chain will generate blocks slower or faster, based on the total hashrate committed to that block. With 80% hashrate, the B2X chain would, on average, mine one block every 12.5 minutes. With 20% hashrate, the B1X chain would, on average, mine one block every 50 minutes.

Time

How long will it take until the difficulty adjustment?

The fork is scheduled to occur at block 494,784. For both the B1X and B2X chains, difficulty is scheduled to adjust 1152 blocks later, at block 495,936. With mining power split between two chains, it will likely take significantly longer than normal, particularly for the chain with minority hashrate.

The amount of time before the first difficulty adjustment will probably be somewhere between 8 days (if all miners join one chain, which therefore generates blocks every 10 minutes) and 16 days (if mining power is evenly split between the two chains, which therefore generate blocks every 20 minutes). It could conceivably be longer, if some BTC miners stop mining altogether after the fork, or temporarily defect to another coin such as Bitcoin Cash.

If about 80% of pre-fork BTC miners start mining B2X after the fork, the B2X fork will be about 4 times faster, and will therefore adjust its difficulty much sooner. At a rate of one block every 50 minutes, the difficulty adjustment for B1X would happen only 40 days after the fork. At a rate of one block every 12.5 minutes, the difficulty adjustment for B2X would happen approximately 10 days after the fork, so let’s consider that to be our cutoff.

Block reward

One component of coins earned per block is the block reward, which amounts to 12.5 coins per block. This will be the same for both chains — the B1X chain will give miners 12.5 B1X per block, and the B2X chain will give miners 12.5 B2X per block.

Fees

The other component of coins earned per block is transaction fees. Total transaction fees per block have mostly ranged between 1 and 4 BTC over the past six months, averaging around 2 BTC per block. Transaction fees are difficult to estimate, but since they are the less significant component of miner income, it’s not essential that we get this exactly right.

Transaction fee revenue depends, of course, on the demand for transacting on each chain. There may be more desire to transact on the faster, more secure chain (although, since B1X and B2X will be different assets, its not clear that transactions on the two chains will be treated as substitutes), and some businesses will likely follow through on their stated intention to favor the B2X chain.

On the other hand, slower blocks will mean more transactions per block, and thus higher transaction fees per block. If the B1X network only mines a block every 50 minutes but the B2X network mines a block every 12.5 minutes, then, all other things equal—and in the absence of a block size limit—transaction fees per block would be about 4 times higher for the B1X chain than for the B2X chain.

Of course, the block size is not unlimited, and it’s likely that if B1X blocks are mined only every 50 minutes, most blocks would be full, which complicates the analysis. Complicating it further is the fact that the situation is not symmetrical between B1X and B2X, because B2X increases the block size by a factor of 2. But the effect of block size on total transaction fees per block is unclear, since smaller blocks tend to have higher fees per transaction. After the effective block size increase imposed by SegWit activation in late August, transaction fees per block actually went down (although there were likely some confounding factors).

Given all these counteracting factors, predicting transaction fee revenue per chain precisely is a difficult task. If B1X is significantly slower, I would predict that the latter effect would dominate and transaction fees per block on B1X would be higher, but let’s be generous and assume that transaction fees are approximately equal, and that the existing transaction fees per block for BTC are evenly split between B1X and B2X. (Additionally, if we avoid making too much use of assumptions about the total mining power on each chain—which, recall, did not affect the rate at which each miner generate blocks—our analysis will be more generally applicable to other scenarios.)

So let’s imagine that transaction fees are the same for each chain: around 1 coin per block.

Price

By the assumption we made in the introduction, B1X maintains 80% of BTC’s current price, and B2X maintains 20%. (This is actually more conservative than the futures markets, which have tended to put B1X’s share at closer to 85%.) So based on today’s price of $7200, the price of B1X would be about $5760, and the price of B2X would be about $1440.

Some might question whether it’s plausible for B1X to maintain its price for 10 days even if it has only 20% of mining power. The futures markets certainly seem to indicate an expectation that it will. Rather than go into an exhaustive analysis of this question, which is somewhat outside the scope of this post, I’ll just note that there doesn’t seem to be any direct mechanism by which a temporary drop in hashrate would drive an immediate drop in price, whereas a temporary drop in price has a clear effect on miners’ immediate incentives.

Estimating opportunity cost

Let’s now try to estimate the opportunity cost of mining the B2X chain rather than the B1X chain, assuming our assumptions about mining power and price hold.

Our main conclusion above is that for the first ten days, a miner will earn approximately the same number of coins whether they choose to mine B1X or B2X, regardless of how many other miners choose to mine each coin. This is very convenient for our analysis.

First, let’s examine the opportunity cost for an individual, “marginal” miner, who needs to decide which chain to mine, but is too small to have an impact on the speed of the chain. The calculation is simple—since the B1X coin is worth four times as much, the miner’s revenue will be four times higher if they mine B1X as if they mine B2X.

To put it in concrete terms, let’s imagine a miner who currently mines $1,000 of BTC a day. In the 10 days following the fork, thry will have to choose between mining a total of $8,000 worth of B1X, or $2,000 worth of B2X.

It seems clear that if the above assumptions hold, then until the difficulty adjustment, it will probably be individually rational to mine the chain of the most valuable coin, regardless of what other miners do. Note that this is true even if a miner disagrees with the market on the future value of the currencies— if a miner would rather hold B2X, they could earn it four times faster by mining B1X and trading it for B2X.

Does it change our analysis if we consider the short-term interests of the entire 80% of miners currently signaling support for mining B2X, as a group? After all, there is evidence that miners sometimes act more like a cartel than like a collection of individual independent actors. What is the opportunity cost for those miners, compared to abandoning B2X and continuing to mine the B1X chain?

It turns out that considering this entire group as a single actor only strengthens the conclusion that abandoning the less-valuable chain would be the rational move.

If these 80% of miners choose to mine B2X, then in the ten days following the fork, they will mine a total of 1152 blocks. If, as we estimated above, each of these blocks pays 12.5 B2X in block rewards and 1 B2X in transaction fees, for a total of 13.5 B2X, then their total revenue over those 10 days would be 15,552 B2X. If the price of B2X is $1440, that’s a total of $22 million.

In contrast, if they choose to mine B1X, those miners would still mine a total of 1152 blocks in 10 days, but would be paid in B1X. Even under the above assumptions about transaction fees and prices for B1X, they would expect to earn 15,552 B1X for a total of $89 million—making $67 million more by mining B1X rather than B2X.

But in fact, since this would effectively be equivalent to abandoning the B2X fork, the most likely outcome might be that B1X would maintain the price and transaction quantity of BTC (although, since the recent run-up in price may have anticipated some fork, it’s possible that the price would pull back a little). If those 80% of miners call off the fork, but the BTC price remains stable, and total transaction fees per block remain around 2 BTC per block, then their expected revenue would be 14.5 * $7200 * 1152 = $120 million, making the opportunity cost around $98 million for those 10 days.

Conclusion

Supporting an unpopular fork by mining it instead of a more valuable chain turns out to be extraordinarily expensive at Bitcoin’s current price. This situation is likely to be unsustainable—individual miners would have a strong incentive to defect, and cartels would need to sacrifice considerable returns to maintain a higher hashrate.

Of course, a cartel might be willing to sacrifice substantial short-term gains now if it guaranteed higher revenues in the future. In a follow-up post, I’ll examine what would happen after the B2X difficulty adjustment 10 days after the fork—which would likely make mining B2X slightly less unprofitable — and the B1X difficulty adjustment 30 days after that—which would likely make mining B1X dramatically more profitable. I may also take a closer look at other strategies that powerful miners might consider, such as 51% attacks or selfish mining attacks. [UPDATE, November 8: Turns out the follow-up post would be moot, because the 2x fork has been abandoned by its primary supporters.]