# Logarithmic ECM for staked Mithril Shares (MIS)

First off, let’s get some context.

Logarithms (log) are functions that appear frequently in nature. Logarithms look like the inverse of exponential functions. Below pictured is the basic log equation, y=log(x).

This shape is exceedingly useful. We use logarithms in our everyday lives to describe earthquake intensity, the loudness of sounds, and even the spacing of the frets of our guitars!

Utilizing the useful nature of logarithms, we can apply “staking days” to boardroom MIS. Thus, an Effective Capital Multiplier (ECM) is generated for our staked MIS as shown below:

*Where ***y** is our ECM and **t **is the number of days staked.

Effective Capital Multiplier when applied to MIS heavily rewards long term use of the system. Effective Capital Multiplier starts at 1 and increases the moment we start staking but is reset to 1 upon any unstake event. ECM can increase forever. ECM does not increase the amount of Mithril Cash minted but adjusts ratios to which shareholders are paid out.

**Function:**

Individual staking days are applied to each staking event of MIS into the boardroom. This means if someone has a 5 MIS stake that is 20 days old, and a 10 MIS stake that is 10 days old, the effective MIS (eMIS) for this staked account is equal to 5a + 10b where:

Therefore, this user has 32.025 eMIS staked. Should they unstake any eMIS, they will only receive the amount of MIS deposited (15 total MIS) and the ECM is reset to 1.

Consider our boardroom is filled with 50,000 MIS. Those who deposited this MIS wait 100 days before experiencing an epoch. Historically when an epoch is expected, large money from outside the community rapidly purchases and stakes MIS in seek of profit.

That’s where Logarithmic Staking benefits our long-term users. During the time that has passed, a 3.004 ECM has accrued on 50,000 MIS deposited by long-term shareholders, giving them a position of 150,216 eMIS. Should foreign money rapidly stake 50,000 freshly purchased MIS to profit off that epoch, this 50,000 is added to the already staked 150,216 eMIS in the boardroom totaling 200,216 effective MIS. Even though short-term buyers staked the same amount as the long term holders, those people that were invested long-term get a much better deal as they receive over 75% of the pool’s distribution.

Implementation of ECM benefits anyone who is intending to stake MIS in the boardroom for a longer period of time. ECM increases rapidly at first, only taking 9 staking days to reach a 2.0 multiplier. However, 3.0 is reached much more slowly over the next 90 days, and to reach 4.0 requires another 900 days of waiting, totaling 999. A rough estimation of ECM can be done by using powers of 10 as our days (i.e: 10, 100, or 1,000), and counting the digits. 10 is a two-digit number, so our ECM is approximately 2. 100 is a three-digit number, so after 100 staking days, our ECM is approximately 3. This means the ECM for 1,000 staking days would be approximately 4. ECM is constantly growing with each staking day but diminishes in growth rate rapidly so that no single entity can control the whole protocol.

Calculation of ECM for a specific number of days other than powers of 10 need the equation. Input 50 for our staking days, and we have a 2.707 ECM, or at 51 staking days, we’re at 2.716. If we wait for 52 staking day’s we’d have a 2.724 ECM. After staking for the average length of a US citizen’s lifespan (27,375 days), a 5.437 ECM can be accrued.

**Conclusion**

ECM is a powerful and effective incentive that benefits long-term investment in the protocol whilst reducing manipulation by outside interference. Should ECM be implemented, investors can expect increased long term benefit to stake MIS and reduced manipulation surrounding epoch events. A Potential long-term issue of implementation is lack of MIS liquidity, however as this would likely lead to increased MIS price it should not be a problem for the community.