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# The K-coefficient in blockchain networks

by Iurii Shyshatskyi, Chief Scientist at Jax.Network

The K-coefficient is the key parameter of Jax.Network economics as it regulates the issuance of JAX. It can be described as a conversion coefficient of miners’ work to JAX coins. Although this concept is native to Jax.Network, it could be applied to many other blockchain networks with completely different architectures. This post is an overview of the K-coefficient and related matters.

## Definition of the K-coefficient

We define the K-coefficient as a mathematical expectation of JAX coin block reward which the miner gets by computing a single hash. In other words, in Jax.Network, whenever a miner computes a hash in order to find a valid one, the mathematical expectation of the JAX coin block reward that he gets is equal to K. Here we need to make two important remarks.

1. In this definition aforementioned block reward doesn’t include transaction fees. We consider only the basic block subsidy issued to the miner, which often consists of freshly minted coins. The size of this subsidy is often determined by the protocol rules.
2. We don’t distinguish orphan blocks from blocks included into the main chain. We consider blocks with completed PoW puzzles regardless of whether their producers were rewarded or not.

## Single chain network case

In the case of a single-chain Proof-of-Work network such as Bitcoin, the K-coefficient could be calculated as the ratio between block reward and block difficulty. The formula is below:

K=R/D

In this formula, R is a block reward and D is a block difficulty measured in hashes.

For instance, block #740,040 has a block subsidy of 6.25 BTC and a block difficulty of 30.2833 T. Here ‘T’ stands for 10¹² Bitcoin unit difficulties of 2³² hashes:

T=10¹² * 2³² hash

However people should be careful, since on some websites T represents 2⁷² hashes.

So one can apply the formula to calculate the K-coefficient for this block:

Similarly, one can calculate the K-coefficient for Bitcoin Cash block #743753 which has been mined on the same day. Its block subsidy is 6.25 BCH and block difficulty is 175.1 G. Here ‘G’ stands for 10⁹ times 2³² hashes. So the K-coefficient for this block is

## Main property

It’s easy to notice the following fact related to the K-coefficient. If multiple blockchain networks share a similar protocol, the same coin issuance schedule, the same block interval and an identical hashing algorithm, then, most of the time, the revenue from mining a block in a particular moment of time is the same for every such network.

For example, according to the data from coinmarketcap.com, the price of BTC at 5 pm on June 9th, 2022 was about \$30,157. The price of BCH was about \$177. It’s a time when blocks #740,040 and #743753 have been mined. If we multiply prices of BTC and BCH by respective K-coefficients, we will have:

Although these values are not equal, they are close to each other. This property represents an equilibrium in the hashrate market as both mining BTC and BCH have the same profitability most of the time. It’s possible because BTC and BCH use the same hashing function SHA256D and the same PoW puzzle for miners.

Miners can use this property to build their business strategies. Although BCH block subsidies are more lucrative for mining, BTC miners collect higher transaction fees per block. Furthermore, over time BCH price depreciates compared to BTC. So miners, who prefer to hold mined coins for a while, are more interested in mining and holding BTC.

## The K-coefficient in Jax.Network

In Jax.Network, the K-coefficient naturally fits into the protocol design. Jax.Network consists of the beacon chain and multiple shard chains which work in parallel. Miners can choose a subset of shard chains for merge-mining. Every hash which they generate can complete the proof-of-work for one of multiple block candidates. These block candidates belong to different chains, they have different block difficulties and potential block rewards.

In Jax.Network, miner’s expected aggregate block rewards do not depend on the number of shards that he merge-mines. It’s possible because of the unique design of the block reward function. Whenever a miner generates a hash to mine JAX coins, the mathematical expectation of his reward equals K.

This approach to block rewards differs from the one used in Bitcoin. In Bitcoin, current block difficulty and block subsidy determine the value of the K-coefficient. In contrast, in Jax.Network, the K-coefficient determines shard block rewards. Thus we can say that Bitcoin follows a fixed block reward paradigm, while Jax.Network has a work-based or hashrate-based reward scheme.

Market conditions and economic factors change over time. That is the reason why key parameters of blockchain economics might require an adjustment. Since the K-coefficient is such a parameter, Jax.Network has a decentralized mechanism for its adjustment.

In Bitcoin and many other single-chain blockchain networks with fixed block rewards, the K-coefficient is adjusted every two weeks along with the difficulty. When the Bitcoin block difficulty is reduced, the K-coefficient of Bitcoin is increased, when the difficulty is increased, the K-coefficient decreases.

In contrast to Bitcoin, the K-coefficient adjustment is separated from the difficulty adjustment in Jax.Network. The initial value of the K-coefficient in Jax.Network is set to 2^-60 JAX/hash. Then the K-coefficient is adjusted every 4096 beacon chain blocks. This is a timespan of around 4 weeks.

K-coefficient adjustment is implemented in such a way that it doesn’t break independence of shard chains in Jax.Network. Technical aspects of this issue have been discussed in our previous post.

K-coefficient adjustment is useful for keeping a stable purchasing power of JAX and holding stable incentives for miners in Jax.Network. The economics of JAX, incentives, and other game-theoretic aspects are discussed in the dedicated paper.

## Usage of the term

Although the concept of the K-coefficient could be applied to any PoW cryptocurrency, it’s not widely used. The simple economics of Bitcoin doesn’t require the K-coefficient for its explanation. This observation is true for cryptocurrencies that follow the fixed block reward rule. However, when people design more sophisticated economics for their blockchain network proposals, they often introduce various parameters, variables, coefficients, and functions similar to the K-coefficient.

For example, Proof-of-Stake cryptocurrency ADA on the blockchain project Cardano has the so-called “K-parameter”. It represents the annual interest that Cardano stakers receive for staking their coins. The K-parameter in the PoS network has approximately the same nature as the K-coefficient in the PoW network.

There are many more examples of less-known projects or blockchain design proposals. Some of them could be found in academic papers. A good example is the proposal HaPPY-Mine by L.Kiffer et al. In this paper, the author introduces and studies the block reward design inspired by the generalized proportional allocation rule. It is very similar to the one introduced earlier in the JaxNet protocol. However, instead of the K-coefficient, the author uses other notations with upper and lowercase letter Q and their products.

## Summary

The K-coefficient is an important parameter in the economics of the blockchain network. If its value decreases, minting new coins becomes more resource-consuming. So miners get paid less for their work. Some of them might turn their rigs off or switch to mining another cryptocurrency. However, this event may make cryptocurrency more scarce and increase its price. On the other hand, the increase of the K-coefficient stimulates mining. More intense mining contributes to the security of the blockchain network.

In Jax.Network, the K-coefficient could be adjusted by miners in a decentralized way because of necessity or in response to market trends, the policy of central banks, etc. We can compare its role in Jax.Network with the role of Federal Reserve interest rates in global economics nowadays.

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