Each user is incentivized to to maximize their individual correlation score. But Numerai wants to maximize the metamodel's correlation score, where the metamodel is the stake weighted ensemble of all submissions.
mmc is designed to bridge this gap. Whereas
correlation rewards individual performance,
mmc rewards contribution to the metamodel's
correlation or group performance.
To calculate a user's (U)
mmc for a given round we
select a random 67% of all staking users (with replacement)
calculate the stake weighted predictions of these users
transform both the stake weighted predictions, and U's model to be uniformly distributed
neutralize U's model with respect to the uniform stake weighted predictions
calculate the covariance between U's model and the kazutsugi targets
divide this value by 0.0841 (this step is to bring the expected score up to the same magnitude as correlation)
the resultant value is an MMC score
repeat this whole process 20 times and keep the average MMC score
Stake weight is necessary to make the system unattackable
Sampling a random 67% each time is important to incentivize some redundancy.
A very large staker submitting similar predictions to yours would penalize you too much if we didn't do this.
Read more about the MMC calculation here.
Read more about MMC and profitability here.