The white paper describes the mining rate formula as follows.

Translation

Pre-Mainnet Formula
The Pre-Mainnet mining formula demonstrates a meritocratic determination of a Pioneer’s hourly mining rate. Actively mining Pioneers received at least a minimum rate and were further rewarded for their contributions to security and growth of the network. The following formula determined the rate at which Pioneers mined Pi per hour:

M = I(B, S) + E(I), where

M is the total Pioneer mining rate,
I is the Individual Pioneer base mining rate,
B is the systemwide base mining rate,
S is the Security Circle reward, which is a component of the individual Pioneer base mining rate from valid Security Circle connections,
and E is the Referral Team reward from active Referral Team members.
The systemwide base mining rate B started as 3.1415926 Pi/h and halved every time the network of Engaged Pioneers increased in size by a factor of 10x, starting at 1000 Pioneers. As listed below, there have been five halving events thus far:

Engaged Pioneers Milestone Value of B (in Pi/hr, rounded to two decimals) Value of I, with full Security Circle (in Pi/hr, rounded to two decimals*)
< 1,000 3.14 6.28
1,000 1.57 3.14
10,000 0.78 1.57
100,000 0.39 0.78
1,000,000 0.19 0.39
10,000,000 0.10 0.19
Here,

I(B,S) = B + S(B)
S(B) = 0.2 • min(Sc,5) • B, where
Sc is the count of valid Security Circle connections.
E(I) = Ec • I(B,S) • 0.25, where
Ec is the count of active Referral Team members who mine concurrently.
The mining formula can also be written as a multiple of B:

M = I(B,S) + E(I)
M = [B + S(B)] + [Ec • I(B,S) • 0.25], or Pre-Mainnet Systemwide Base Mining Rate
Every active Pioneer received at least the system wide base mining rate (B). That is, if Sc = 0 and Ec = 0 in the mining formula above, then M = B. In any case, the total Pioneer mining rate was a multiple of the systemwide base mining rate. The value of B was pre-determined before the Mainnet, and as shown in the table above, it changed only five times. The max supply was undetermined due to the dynamic progress of the pre-Mainnet mining mechanism, e.g. how large the network is and how fast the network reaches the next halving event. It would only be determined when B dropped to 0. However, as explained in the next section, the value of B at Mainnet is calculated in real time, dynamically adjusting based on the total annual Pi supply and the total mining coefficient across all the Pioneers. The supply of Pi is finite at Mainnet.

In summary

The formula called Pre-Mainnet Formula is used to determine the hourly mining rate of users who mine, called Pioneers. Pioneers receive rewards based on their contribution to mining. The formula determines the rate at which Pioneers can mine Pi per hour. The systemwide base mining rate B starts with 1000 Pioneers and halves every time the number increases by 10 times. The value of B changed 5 times, and the mining rate is calculated based on that value. The formula consists of multiple terms, each assigning different rewards. The maximum supply is undetermined and will only be determined when the value of B becomes 0 on Mainnet. On Mainnet, the supply of Pi is finite, and the value of B is calculated based on the total mining coefficient of all Pioneers and the annual Pi supply.

Explanation of the formula

This formula describes the distribution of Pi tokens mined on the Pi Network. This formula determines the hourly mining rate of Pioneers (Pi Network users) based on meritocracy. Active Pioneers receive a minimum rate and are further rewarded for their contributions to network security and growth.

This formula is structured as follows:

M = I(B, S) + E(I)

Here, M represents the total Pioneer mining rate, I represents the individual Pioneer base mining rate, B represents the systemwide base mining rate, S represents the Security Circle reward, and E represents the Referral Team reward.

The systemwide base mining rate B starts with 1000 Pioneers and decreases by half every time it expands by 10 times. There have been 5 halving events thus far.

I(B, S) is represented as B + S(B). S(B) is a component of the individual Pioneer base mining rate based on the number of valid Security Circle connections, and is represented as S(B) = 0.2 x min(Sc, 5) x B.

E(I) is a reward based on the number of active Referral Team members who mine concurrently (Ec) and I(B,S), and is represented as E(I) = Ec x I(B,S) x 0.25.

From the above, the total mining rate M is represented by the following formula:

M = [B + S(B)] + [Ec x I(B,S) x 0.25]

There is also a formula based on B.

M = B x [(1 + Ec x 0.25) x (1 + 0.2 x min(Sc,5))]

In other words, the Pi Network mining rate is determined based on the systemwide base mining rate, and active Pioneers are rewarded for contributing to security and growth. The mining rate is derived from the individual base mining rate based on Security Circle and Referral Team factors, and the systemwide base mining rate changes dynamically according to the number of participating Pioneers.

Factors to increase mining rate

To increase the mining rate, the following factors are important:

1. Number of Security Circle connections: The more connections to the Security Circle, the higher the mining rate.
2. Number of active Referral Team members: The more active members in the Referral Team, the higher the mining rate.
3. Network growth: As the number of Pioneers increases, the systemwide mining rate decreases, but the individual Pioneer mining rate tends to increase.
4. Systemwide base mining rate: The higher the systemwide base mining rate, the higher the individual Pioneer mining rate.

To efficiently increase the mining rate, it is effective to increase connections to the Security Circle and increase the number of active Referral Team members. The systemwide base mining rate, which is automatically adjusted according to network growth, is also important.

Summary

Pi Network is undertaking efforts to distribute the cryptocurrency Pi using blockchain technology. To earn the cryptocurrency Pi, people called Pioneers perform mining. Their mining rate is determined by the systemwide mining rate (B), Security Circle reward (S), Referral Team reward (E), and individual base mining rate (I).

The systemwide mining rate (B) starts with 1,000 Pioneers and halves every time the number of Pioneers increases by 10 times. When the number of people participating in the Security Circle (S) increases, it is added to the individual base mining rate (I). If the number of Pioneers participating in the Referral Team (E) is large, the reward is multiplied by 0.25.

As an example calculation when the Security Circle has 5 people, the base mining rate (I) becomes 6.28 Pi/h. The systemwide mining rate (B) is 3.14 Pi/h, and the Security Circle reward (S) is calculated by min(5, Sc) × 0.2 × B. The Referral Team reward (E) is multiplied by 0.25 if the number of Pioneers is large.

Pi Network incorporates a meritocracy mechanism where rewards can be earned commensurate with effort. The systemwide mining rate (B) halves every time the number of Pioneers increases, so the mining rate can be increased as the number of Pioneers grows. Also, rewards can be increased by participating in the Security Circle or joining a Referral Team.