Honesty and Rationality

The idiom "honest majority" is extremely common when discussing PoW. The sufficiency of an honest majority (opposed to a privileged majority of two thirds) is considered one of the greatest accomplishments of Nakamoto consensus. Another tenet of the Bitcoin ethos is trustlessness: the property that all sides of all interactions do not have to trust the other side to follow suit. That the protocol protects them from malfeasance.

Ostensibly, there is quite a tension between the two: how can a system be trustless if its security hinges on an honest majority we trust to exist?

The answer is that we don't. The term honest might be a bit misleading. In the world of protocol design and algorithmic game theory, labeling a player honest is not supposed to be a testimony to the pureness of their intention, it just means that they behave the way we expect. We assume honesty because it makes analysis easier. But it does require us to further explain why we should assume miners (or other types of players) would be compelled towards the behavior we branded as "honest".

Miner behavior cannot be enforced by the protocol. If we could assure that miners are "honest" we would not need to even make assumptions. The next best thing is to encourage miners to behave as we need them to through incentive alignments. That is, we make it worthwhile for a miner to be "honest".

This, in a nutshell, is the underlying theme of the subset of algorithmic game theory called mechanism design, a beautiful field of research concerned with constructing algorithms in a way that encourages participants to behave as the protocol requires.

But what is an incentive? How do we quantify and reason about it? A more in depth discussion will be deferred to far down the line, when we analyze a blockDAG's fee market (and until that is completed, you can enjoy this post instead). But a cursory explanation is that we assume there is some utility that a so-called rational miner will want to maximize. For example, we can assume that a Bitcoin miner wants to maximize the amount of coin they gain from mining. A miner that conforms to this utility (in a sense that will be made more formal when time is due) is called rational.

Now, much like "honest", the name rational is also not a judgement call. It's not that miners that are rational are crazy, or stupid, or irrational in any other way. It just means that their utility is different. A miner that stops mining because they went into debt and are better off selling their equipment is not maximizing the utility and is therefore "not rational", despite selling her gear to cover her debts is arguably the more rational approach, and that's hardly the only example of a miner who has something to gain by deviating from the protocol.

However, if the incentive is strong enough, assuming that a majority of miners are rational (in the sense that they work toward obtaining as much coin as possible) is a very reasonable assumption. So it remains to explain why the rational behavior coincides with our idea of honesty.

Incentive Alignment in Block Chains

When we defined the block chain paradigm, we only had two expectations honest of miners:

• Always mine over the selected tip

• When discovering a new block (via mining or a peer) immediately transmit it and recompute the selected tip

But now we want rational miners to follow suit. How shall we achieve that? Well, Bitcoin's solution is simple: pay miners for creating blocks, as long as the block is within the selected chain. The said payment has two components: block rewards, and fees.

Block rewards serve functions: increase supply in a gradual manner (which is why the coins created by block rewards are often called emissions), and incentivize miners to behave even when fees are negligible.

Most PoW coins are deflationary, which means that the total emission must be finite, making fees the dominant incentive for good behavior. Hence, making the fees the only security subsidy in the long run. The fee market and the incentives it poses is obviously crucial for the long run of a deflationary proof of work. For that reason, we defer the discussion to an entire part of the book dedicated to fee market dynamics. For now, however, we treat block rewards as something fixed.

The way deflation is usually implemented is by geometric emission. That is, every fixed period of time (measured in number of blocks), the reward is decreased by a fixed ratio. If the initial reward is $R$, and it is decreased by a factor of $q$ once every $N$ blocks, then we get that the first $N$ blocks provide a reward of $R$ each, the next $N$ blocks provide $R\cdot q$ each, the next $N$ blocks provide $R\cdot q^2$ each and so on. Using the formula for a geometric series, we get that the total emissions sum to

$\sum\limits _{n=0}^{\infty}N\cdot R\cdot q^{n}=\frac{N\cdot R}{1-q}$

In Bitcoin, the initial block reward was $100$ bitcoin, and it is reduced by half once every $210,000$ blocks providing a total emission of

$\frac{210,000\cdot50}{1-\frac{1}{2}}=21,000,000$

Clearly it is not truly the case that emission resume forever, becoming infinitely smaller, as they will eventually become smaller than a single satoshi. In the exercise you will see how cutting off the tail affects the total emission.

So, how do block rewards affect rational miners?

One can prove that for Bitcoins heaviest chain rule (that we did not introduce yet), not mining over the selected tip reduces the chance of creating the next block, and thus the expected gain. This is true essentially since by not mining over the currently heaviest known chain, the miner competes against it from a disadvantage.

OK, but what about not withholding blocks? Surely a rational miner would want to avoid that too, right? Well, not quite. We will soon discuss a phenomenon called selfish mining, which shows that there is in fact some divergence between the rational and honest strategies for bitcoin miners.